diff --git a/-9E1T4oBgHgl3EQfogQj/content/tmp_files/2301.03320v1.pdf.txt b/-9E1T4oBgHgl3EQfogQj/content/tmp_files/2301.03320v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d18e7a9a11f4017f03baca57159a39a7e14065b --- /dev/null +++ b/-9E1T4oBgHgl3EQfogQj/content/tmp_files/2301.03320v1.pdf.txt @@ -0,0 +1,524 @@ +New results for thermal interquark bottomonium +potentials using NRQCD from the HAL QCD method +Thomas Spriggs,𝑎,∗ Chris Allton,𝑎 Timothy Burns𝑎 and Seyong Kim𝑏 +𝑎Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom +𝑏Department of Physics, Sejong University, Seoul 143-747, Korea +E-mail: {t.spriggs.996870,c.allton,t.burns}@swansea.ac.uk, +skim@sejong.ac.kr +We report progress in the calculation of the thermal interquark potential of bottomonium using the +HAL QCD method applied to bottom quarks in the non-relativistic approximation (i.e. NRQCD). +We exploit the fast Fourier transform algorithm, using a momentum space representation, to +efficiently calculate NRQCD correlation functions of non-local mesonic S-wave states, and thus +obtain the potential for temperatures in both the hadronic and plasma phases. This work was +performed on our anisotropic 2+1 flavour “Generation 2" FASTSUM ensembles. +The 39th International Symposium on Lattice Field Theory, LATTICE2022 8th–13th August, 2022 Bonn, +Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.03320v1 [hep-lat] 9 Jan 2023 + +Thermal interquark bottomonium potentials +Thomas Spriggs +1. +Introduction +The interquark potential of quarkonia was one of the first quantities studied in the quest for a +deeper understanding of the nature of the strong interaction. Pioneering studies include [1] where +the Cornell potential was used to calculate the spectrum of charmonium states using a quantum +mechanical formalism. In thermal QCD, the temperature dependence of the interquark potential +results in quarkonium states melting at different temperatures [2]. These considerations strongly +motivate a study of the thermal behaviour of the quarkonia interquark potential. +Slowly moving heavy quarks, interacting via QCD, can be studied using non-relativistic QCD +(NRQCD) which allows significant benefits. For example, NRQCD calculations of bottomonia +are typically accurate at the percent level or less and is an excellent ground for quantitative tests. +In this work we use NRQCD to determine the interquark potential in bottomonia using the HAL +QCD approach [3]: Correlation functions of bottomonia operators are studied where the quark +and antiquark are spatially separated, and this allows an access to the Nambu-Bethe-Salpeter +wavefunction in the quarkonium rest frame. Using this wavefunction in the Schrödinger equation +leads to the interquark potential. We find indications of the weakening of the potential as the +temperature increases, as expected. This work is a continuation of the work in [4] and extends +previous studies of the interquark potential by the FASTSUM Collaboration in the charmonium +system [5, 6]. Other work in this area includes [7]. +2. +NRQCD and lattice setup +NRQCD is an effective theory with a power counting in the heavy quark velocity, 𝑣. In this +theory, the heavy quark and antiquark fields decouple and so virtual heavy quark-antiquark loops +cannot form. The NRQCD quark propagator is calculated via an initial value problem, rather +than via a boundary value problem (as is the case for relativistic quarks). NRQCD is particularly +amenable for lattice simulations because NRQCD quarkonium correlation functions do not have +“backward movers” which means the full extent of the lattice in the temporal direction can be used +in the analysis. +Our NRQCD formulation incorporates both O(𝑣4) and the leading spin-dependent corrections. +The 𝑏-quark mass is tuned by setting the “kinetic” mass (i.e. from the dispersion relation) of the +spin-averaged 1𝑆 states to its experimental value. Full details of our NRQCD setup appear in [8]. +All our results were obtained using our FASTSUM 𝑁 𝑓 = 2+1 flavour “Generation 2” ensembles +which have the parameters listed in Table 1. +𝑁𝜏 +16 +20 +24 +28 +32 +36 +40 +T [MeV] +352 +281 +235 +201 +176 +156 +141 +𝑁configurations +1050 +950 +1000 +1000 +1000 +500 +500 +Table 1: An overview of the FASTSUM Generation 2 correlation functions used in this work. Lattice +volumes are (24𝑎𝑠)3 × (𝑁𝜏𝑎𝜏) with 𝑎𝑠 = 0.1227(8)fm and 𝑎𝜏 = 35.1(2)am. For these ensembles with a +pion mass of 𝑀𝜋 = 384(4)MeV, the pseudo-critical temperature Tpc = 181(1)MeV [9]. +2 + +Thermal interquark bottomonium potentials +Thomas Spriggs +3. +Method +3.1 The HAL QCD method +To calculate the potential between two quarks in a bottomonium - the interquark potential, 𝑉(𝑟) +- we use the method from the HAL QCD collaboration [3]. In brief, this method uses the point-split +correlation function and the time independent Schrödinger equation to calculate the interquark +potential. +The point-split correlation function is defined by +𝐶Γ(r, 𝜏) = +∑︁ +x +⟨𝐽Γ(x, 𝜏; r)𝐽† +Γ(0; 0)⟩, +(1) +where the non-local mesonic operators are defined +𝐽Γ(𝑥; r) = ¯𝑞(𝑥)Γ𝑈(𝑥, 𝑥 + r)𝑞(𝑥 + r). +(2) +The quark and antiquark fields, 𝑞 and ¯𝑞, are separated in space by r. The gauge field 𝑈(𝑥, 𝑥 + r) is +required to ensure gauge invariance and Γ signifies the channel being considered; in this work we +consider vector and pseudoscalar S-wave states. The correlator in (1) is depicted in Figure 1. +(0,0) +(x,𝜏) +(x+r,𝜏) +𝐽† +Γ(0; 0) +𝐽Γ(x, 𝜏; r) +Source +Sink +¯𝑏 +𝑏 +Figure 1: A representation of the point-split correlation function, as defined in (1) +As usual, the correlation function can be expressed as a sum over eigenstates of the Hamiltonian, +𝐶Γ(r, 𝜏) = +∑︁ +𝑗 +Ψ𝑗(r)𝑒−𝐸𝑗 𝜏, +(3) +where 𝐸 𝑗 is the energy of a given state 𝑗, and the unnormalised wavefunction +Ψ 𝑗(r) = +𝜓∗ +𝑗(0)𝜓 𝑗(r) +2𝐸 𝑗 +(4) +is defined in terms of the Nambu-Bethe–Salpeter wavefunction 𝜓 𝑗(r). +We introduce the time-independent Schrödinger equation, +� +−∇2 +𝑟 +2𝜇 + 𝑉Γ (𝑟) +� +Ψ 𝑗 (𝑟) = 𝐸 𝑗Ψ 𝑗 (𝑟) , +(5) +where 𝑉Γ(𝑟) is the potential for the channel Γ and 𝜇 is the reduced quark mass. We apply the +Schrödinger equation to the point-split correlation function in (3) through the following steps +−𝜕𝐶Γ(r, 𝜏) +𝜕𝜏 += +∑︁ +𝑗 +𝐸 𝑗Ψ 𝑗(r)𝑒−𝐸𝑗 𝜏 = +∑︁ +𝑗 +� +−∇2 +𝑟 +2𝜇 + 𝑉Γ (𝑟) +� +Ψ 𝑗 (𝑟) 𝑒−𝐸𝑗 𝜏 += +� +−∇2 +𝑟 +2𝜇 + 𝑉Γ (𝑟) +� +𝐶Γ(r, 𝜏). +(6) +3 + +Thermal interquark bottomonium potentials +Thomas Spriggs +This yields the form of the interquark potential for a given channel, 𝑉Γ, as +𝑉Γ(𝑟) = +1 +𝐶Γ(r, 𝜏) +�∇2 +𝑟 +2𝜇 − 𝜕 +𝜕𝜏 +� +𝐶Γ(r, 𝜏). +(7) +Note that in the continuum limit, we expect the potential to be function of 𝑟 = |r|. There is explicit +time dependency in this form for the potential, and this will be studied in Section 4.1. Section 4.2 +will discuss how the reduced quark mass, 𝜇, is set. +It is convenient to define the central potential, 𝑉C, obtained via the usual spin-average [10] +𝑉C = 1 +4𝑉Pseudo Scalar + 3 +4𝑉Vector. +(8) +3.2 Using momentum space to reformulate the calculation +This work is a continuation of [4] where more detail about the HAL QCD method can be +found. We build upon [4] by using an efficient computation of the point-split correlation function, +𝐶Γ(r, 𝜏). +For each 𝜏, a direct calculation of (1) requires a loop over all lattice sites x for each value of r +which is an expensive operation scaling as O(V2) where V is the spatial volume. What follows is +a method to reduce the cost of this computation by introducing a momentum space representation +for the propagator and correlation function, see the Appendix of [6]. +We introduce quark propagators, 𝐷−1(𝑥; 𝑦), by Wick contracting the quark fields in the point- +split correlation function, (1), +𝐶Γ(r, 𝜏) = − +∑︁ +x +⟨𝐷−1(x + r, 𝜏; 0, 0)Γ𝛾5 +� +𝐷−1(x, 𝜏; 0, 0) +�† +𝛾5Γ†⟩. +(9) +Note that we have gauge fixed our configurations to the Coulomb gauge, and have replaced +the gauge connection, 𝑈(𝑥, 𝑥 + r) in (2) by unity. We now implicitly define the corresponding +momentum space quark propagator via +𝐷−1(y, 𝜏; 0, 0) = 1 +𝑉 +∑︁ +p +˜𝐷−1(p, 𝜏)𝑒𝑖y·p, +(10) +in terms of the 3-momentum, p, which is conjugate to the position y. Introducing this momentum- +space quark propagator into (9) yields +𝐶Γ(r, 𝜏) = 1 +𝑉 +∑︁ +p +⟨ ˜𝐷−1(p, 𝜏)Γ𝛾5 ˜𝐷−1(−p, 𝜏)𝛾5Γ†⟩𝑒𝑖p·r, +(11) +which we will use to implicitly define the momentum-space correlator, ˜𝐶Γ(p, 𝜏), i.e. +𝐶Γ(r, 𝜏) = 1 +𝑉 +∑︁ +p +˜𝐶Γ(p, 𝜏)𝑒𝑖p·r. +(12) +We note that once we have calculated ˜𝐶Γ(p, 𝜏), we can determine the desired correlator 𝐶Γ(r, 𝜏) +for any r using (12). +4 + +Thermal interquark bottomonium potentials +Thomas Spriggs +At first sight, the conversion to momentum space does not produce any savings, because the +calculation of 𝐶Γ and ˜𝐷−1, defined via (10) and (12), are both O(V2) in the number of operations, +i.e. the same as the direct method. However both (10) and (12) are Fourier transforms, and so +significant speed-up for these steps can be achieved using the fast Fourier transform (FFT) algorithm +which scales as O(V log V). +4. +Results +For better comparison with [4], and as progress towards the treatment of 𝐶Γ(r, 𝜏) for all r, we +consider here only the on-axis r data. Extensions to this will be discussed in Section 5. +4.1 Time dependence +The potential is defined in (7) where there is an apparent explicit dependence on time, 𝜏, from +the correlation function. In Figure 2, the potential, 𝑉Vector, from (7) is plotted against 𝜏 for a variety +of distances r for our two extreme temperatures, 𝑇 = 141 and 352 MeV. We can see a clear 𝜏 +dependence for small 𝜏 which increases with r. However, for various ranges of 𝜏 and r there are +clear plateau. +In addition, we note that we would like to uncover temperature effects in the potential. The +most accurate way of doing this is to compare different temperatures’ potentials obtained with the +same time window to avoid contamination by systematic artefacts. +Based on these considerations, we restrict the range of 𝑟 and 𝜏 used in the determination of +the potential to those listed in Table 2. Notice that in selecting a time window, there is a trade-off +between the ranges of 𝑟 and 𝑇 for which the potential can be extracted: larger time windows give +access to a larger range of 𝑟, but over a smaller range of 𝑇. +In Figure 3 we show four determinations of the central potential, corresponding to the first four +time windows identified in Table 2. In each plot we show the potentials for several temperatures, and +since these have been obtained by averaging over the same range of 𝜏, the temperature dependence +can be ascribed to temperature effects, rather than fitting artefacts. +We find that the potential +consistently flattens as the temperature increases above 𝑇pc, as expected. There is little thermal +variation in the potential for 𝑇 ⪅ 𝑇pc. +In Figure 3, the error bars show statistical errors only. The curves are fits to the Cornell +potential, which will be discussed in Section 4.3. +4.2 Quark mass dependence +Equation (7) contains the reduced quark mass, 𝜇, which needs to be defined. In [7], the 1S +and 2S states were used to determine the bottom quark mass 𝑚𝑏, and thus the reduced quark mass. +In our simulations we do not have access to the 2S state. We instead use the simple argument: +𝜇 ≡ 1 +2𝑚𝑏 ≈ 1 +2 𝑀Υ, with 𝑀Υ from [11]. We have tested the sensitivity of the potential on the quark +mass and found that the variation (within sensible 𝜇 ranges) is minimal. +5 + +Thermal interquark bottomonium potentials +Thomas Spriggs +Figure 2: Time dependence in the potential restricting the range of 𝑟 that we can consider valid. Shown for +two temperatures using the vector channel as an example. +Time window [𝑎𝜏] +𝑟 range [𝑎𝑠] +𝑟 range [fm] +Temperatures [MeV] +13-14 +1-3 +0.12-0.37 +352-141 +17-18 +1-4 +0.12-0.49 +281-141 +19-22 +1-5 +0.12-0.61 +235-141 +21-26 +1-5 +0.12-0.61 +201-141 +24-30 +1-6 +0.12-0.74 +176-141 +24-33 +1-6 +0.12-0.74 +156-141 +Table 2: Range of displacements and temperatures allowed to best approximate time independence in𝑉(𝑟, 𝜏). +Note that 𝑇pc = 181 MeV and thus the time windows below the solid line do not span this pseudocritical +temperature. +4.3 Cornell potential fits +The Cornell potential [12] is a phenomenological description of a confining potential applicable +to heavy quarks in QCD and is given by +𝑉(𝑟) = −𝛼 +𝑟 + 𝜎𝑟 + 𝐷. +(13) +Fits using (13) to our potential data are shown as solid curves in Figure 3. As can be seen these +reproduce the data well. When the string tension, 𝜎, in the Cornell potential is zero, this implies a +deconfined potential. In all cases above 𝑇pc, we find that 𝜎 decreases with increasing temperature, +confirming the expected thermal behaviour in the bottomonium system. Below𝑇pc the string tension +does not change within statistical errors. +5. +Conclusion +The temperature dependence of the central interquark potential in the bottomonium system +using NRQCD quarks was explored. This work was an extension of [4] and use a momentum-space +approach which can improve the efficiency of the calculation. Clear thermal effects in this potential +6 + +T = 141 MeV +8 +Preliminary +X +6 +[GeV] +* +4 +* +Vector +来来 +2 +米 +10 +15 +20 +25 +30 +0 +5 +35 +40 +t/aT = 352 MeV +8 +Preliminary +6 +TI +[GeV] +* +4 +2 +10 +15 +0 +5Thermal interquark bottomonium potentials +Thomas Spriggs +Figure 3: The central potential calculated from (7) (points), overlaid with a fit of these data to the Cornell +potential (13) (curves). Each plot contains all temperatures and r ranges listed in Table 2. +were observed using a method which decoupled systematic “time window” artefacts from physical, +thermal effects. A systematic flattening of the potential with increasing temperature above 𝑇pc was +observed, with no statistically significant variation in the potential for temperatures below 𝑇pc. +This work will be extended in a number of directions. The potential will be calculated at all +possible spatial separations, r, rather than just the on-axis values used here, and channels beyond +the pseudoscalar and vector S-wave states will be included. Also, a more robust definition of the +reduced quark mass will be developed. Finally, a direct comparison will be made between these +bottomonium results and those obtained for the charmonium potential using the same ensembles in +[6]. +Acknowledgments +This work is supported by STFC grant ST/T000813/1. SK is supported by the National Research +Foundation of Korea under grant NRF-2021R1A2C1092701andgrantNRF-2021K1A3A1A16096820, +funded by the Korean government (MEST). This work used the DiRAC Extreme Scaling service at +the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre and the DiRAC +7 + +Time window: 13-14 +2.4 +- - T= 352 MeV +2.2 +T = 281 MeV +T = 235 MeV +2.0 +T = 201 MeV +1.8 +T = 176 MeV +[GeV] +T = 156 MeV +1.6 +T= 141 MeV +1.4 +C +1.2 +1.0 +0.8 +Preliminary +0.6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +r [fm]Time window: 17-18 +2.4 +T = 281 MeV +2.2 +T = 235 MeV +T = 201 MeV +2.0 +T = 176 MeV +1.8 +T = 156 MeV +[GeV] +T = 141 MeV +1.6 +1.4 +1.2 +1.0 +0.8 +Preliminary +0.6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +r [fm]Time window: 19-22 +2.4 +T = 235 MeV +2.2 +T = 201 MeV +T = 176 MeV +2.0 +T = 156 MeV +1.8 +T = 141 MeV +[GeV] +1.6 +1.4 +1.2 +1.0 +0.8 +Preliminary +0.6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +r [fm]Time window: 21-26 +2.4 +- T = 201 MeV +2.2 + T = 176 MeV + T = 156 MeV +2.0 + T = 141 MeV +1.8 +[GeV] +1.6 +1.4 +1.2 +1.0 +0.8 +Preliminary +0.6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +r [fm]Thermal interquark bottomonium potentials +Thomas Spriggs +Data Intensive service operated by the University of Leicester IT Services on behalf of the STFC +DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via +STFC capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC Opera- +tions grants ST/R001006/1 and ST/R001014/1. DiRAC is part of the UK National e-Infrastructure. +This work was performed using PRACE resources at Cineca (Italy), CEA (France) and Stuttgart +(Germany) via grants 2015133079, 2018194714, 2019214714 and 2020214714. We acknowledge +the support of the Swansea Academy for Advanced Computing, the Supercomputing Wales project, +which is part-funded by the European Regional Development Fund (ERDF) via Welsh Government, +and the University of Southern Denmark and ICHEC, Ireland for use of computing facilities. We +are grateful to the Hadron Spectrum Collaboration for the use of their zero temperature ensemble. +References +[1] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T.-M. Yan, Phys. Rev. D 17 (1978) +3090. +[2] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416–422. +[3] N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99 (2007) 022001, [nucl-th/0611096]. +[4] T. Spriggs, C. Allton, T. Burns and S. Kim, PoS LATTICE2021 (2022) 569, +[arXiv:2112.09092]. +[5] P. W. M. Evans, C. R. Allton and J. I. Skullerud, Phys. Rev. D 89 (2014) 071502, +[arXiv:1303.5331]. +[6] C. Allton, W. Evans, P. Giudice and J.-I. Skullerud, arXiv:1505.06616. +[7] R. Larsen, S. Meinel, S. Mukherjee and P. Petreczky, Phys. Rev. D 102 (2020) 114508, +[arXiv:2008.00100]. +[8] G. Aarts, C. Allton, T. Harris, S. Kim, M. P. Lombardo, S. M. Ryan et al., JHEP 07 (2014) +097, [arXiv:1402.6210]. +[9] G. Aarts et al., PoS LATTICE2019 (2019) 075, [arXiv:1912.09827]. +[10] S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189–231. +[11] Particle Data Group collaboration, R. L. Workman and Others, PTEP 2022 (2022) +083C01. +[12] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane and T.-M. Yan, Phys. Rev. D 21 (1980) +203. +8 + diff --git a/-9E1T4oBgHgl3EQfogQj/content/tmp_files/load_file.txt b/-9E1T4oBgHgl3EQfogQj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a6e7fa8cf29fbd100beae90c7c11846c0af702a --- /dev/null +++ b/-9E1T4oBgHgl3EQfogQj/content/tmp_files/load_file.txt @@ -0,0 +1,322 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf,len=321 +page_content='New results for thermal interquark bottomonium potentials using NRQCD from the HAL QCD method Thomas Spriggs,𝑎,∗ Chris Allton,𝑎 Timothy Burns𝑎 and Seyong Kim𝑏 𝑎Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom 𝑏Department of Physics, Sejong University, Seoul 143-747, Korea E-mail: {t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='spriggs.' metadata={'source': 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calculation of the thermal interquark potential of bottomonium using the HAL QCD method applied to bottom quarks in the non-relativistic approximation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' NRQCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We exploit the fast Fourier transform algorithm, using a momentum space representation, to efficiently calculate NRQCD correlation functions of non-local mesonic S-wave states, and thus obtain the potential for temperatures in both the hadronic and plasma phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This work was performed on our anisotropic 2+1 flavour “Generation 2" FASTSUM ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory, LATTICE2022 8th–13th August, 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='03320v1 [hep-lat] 9 Jan 2023 Thermal interquark bottomonium potentials Thomas Spriggs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Introduction The interquark potential of quarkonia was one of the first quantities studied in the quest for a deeper understanding of the nature of the strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Pioneering studies include [1] where the Cornell potential was used to calculate the spectrum of charmonium states using a quantum mechanical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In thermal QCD, the temperature dependence of the interquark potential results in quarkonium states melting at different temperatures [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' These considerations strongly motivate a study of the thermal behaviour of the quarkonia interquark potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Slowly moving heavy quarks, interacting via QCD, can be studied using non-relativistic QCD (NRQCD) which allows significant benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' For example, NRQCD calculations of bottomonia are typically accurate at the percent level or less and is an excellent ground for quantitative tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In this work we use NRQCD to determine the interquark potential in bottomonia using the HAL QCD approach [3]: Correlation functions of bottomonia operators are studied where the quark and antiquark are spatially separated, and this allows an access to the Nambu-Bethe-Salpeter wavefunction in the quarkonium rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Using this wavefunction in the Schrödinger equation leads to the interquark potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We find indications of the weakening of the potential as the temperature increases, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This work is a continuation of the work in [4] and extends previous studies of the interquark potential by the FASTSUM Collaboration in the charmonium system [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Other work in this area includes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' NRQCD and lattice setup NRQCD is an effective theory with a power counting in the heavy quark velocity, 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In this theory, the heavy quark and antiquark fields decouple and so virtual heavy quark-antiquark loops cannot form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The NRQCD quark propagator is calculated via an initial value problem, rather than via a boundary value problem (as is the case for relativistic quarks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' NRQCD is particularly amenable for lattice simulations because NRQCD quarkonium correlation functions do not have “backward movers” which means the full extent of the lattice in the temporal direction can be used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Our NRQCD formulation incorporates both O(𝑣4) and the leading spin-dependent corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The 𝑏-quark mass is tuned by setting the “kinetic” mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' from the dispersion relation) of the spin-averaged 1𝑆 states to its experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Full details of our NRQCD setup appear in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' All our results were obtained using our FASTSUM 𝑁 𝑓 = 2+1 flavour “Generation 2” ensembles which have the parameters listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 𝑁𝜏 16 20 24 28 32 36 40 T [MeV] 352 281 235 201 176 156 141 𝑁configurations 1050 950 1000 1000 1000 500 500 Table 1: An overview of the FASTSUM Generation 2 correlation functions used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Lattice volumes are (24𝑎𝑠)3 × (𝑁𝜏𝑎𝜏) with 𝑎𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1227(8)fm and 𝑎𝜏 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1(2)am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' For these ensembles with a pion mass of 𝑀𝜋 = 384(4)MeV, the pseudo-critical temperature Tpc = 181(1)MeV [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 2 Thermal interquark bottomonium potentials Thomas Spriggs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1 The HAL QCD method To calculate the potential between two quarks in a bottomonium - the interquark potential, 𝑉(𝑟) we use the method from the HAL QCD collaboration [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In brief, this method uses the point-split correlation function and the time independent Schrödinger equation to calculate the interquark potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The point-split correlation function is defined by 𝐶Γ(r, 𝜏) = ∑︁ x ⟨𝐽Γ(x, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' r)𝐽† Γ(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 0)⟩, (1) where the non-local mesonic operators are defined 𝐽Γ(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' r) = ¯𝑞(𝑥)Γ𝑈(𝑥, 𝑥 + r)𝑞(𝑥 + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (2) The quark and antiquark fields, 𝑞 and ¯𝑞, are separated in space by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The gauge field 𝑈(𝑥, 𝑥 + r) is required to ensure gauge invariance and Γ signifies the channel being considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' in this work we consider vector and pseudoscalar S-wave states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The correlator in (1) is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (0,0) (x,𝜏) (x+r,𝜏) 𝐽† Γ(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 0) 𝐽Γ(x, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' r) Source Sink ¯𝑏 𝑏 Figure 1: A representation of the point-split correlation function, as defined in (1) As usual, the correlation function can be expressed as a sum over eigenstates of the Hamiltonian, 𝐶Γ(r, 𝜏) = ∑︁ 𝑗 Ψ𝑗(r)𝑒−𝐸𝑗 𝜏, (3) where 𝐸 𝑗 is the energy of a given state 𝑗, and the unnormalised wavefunction Ψ 𝑗(r) = 𝜓∗ 𝑗(0)𝜓 𝑗(r) 2𝐸 𝑗 (4) is defined in terms of the Nambu-Bethe–Salpeter wavefunction 𝜓 𝑗(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We introduce the time-independent Schrödinger equation, � −∇2 𝑟 2𝜇 + 𝑉Γ (𝑟) � Ψ 𝑗 (𝑟) = 𝐸 𝑗Ψ 𝑗 (𝑟) , (5) where 𝑉Γ(𝑟) is the potential for the channel Γ and 𝜇 is the reduced quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We apply the Schrödinger equation to the point-split correlation function in (3) through the following steps −𝜕𝐶Γ(r, 𝜏) 𝜕𝜏 = ∑︁ 𝑗 𝐸 𝑗Ψ 𝑗(r)𝑒−𝐸𝑗 𝜏 = ∑︁ 𝑗 � −∇2 𝑟 2𝜇 + 𝑉Γ (𝑟) � Ψ 𝑗 (𝑟) 𝑒−𝐸𝑗 𝜏 = � −∇2 𝑟 2𝜇 + 𝑉Γ (𝑟) � 𝐶Γ(r, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (6) 3 Thermal interquark bottomonium potentials Thomas Spriggs This yields the form of the interquark potential for a given channel, 𝑉Γ, as 𝑉Γ(𝑟) = 1 𝐶Γ(r, 𝜏) �∇2 𝑟 2𝜇 − 𝜕 𝜕𝜏 � 𝐶Γ(r, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (7) Note that in the continuum limit, we expect the potential to be function of 𝑟 = |r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' There is explicit time dependency in this form for the potential, and this will be studied in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 will discuss how the reduced quark mass, 𝜇, is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' It is convenient to define the central potential, 𝑉C, obtained via the usual spin-average [10] 𝑉C = 1 4𝑉Pseudo Scalar + 3 4𝑉Vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 Using momentum space to reformulate the calculation This work is a continuation of [4] where more detail about the HAL QCD method can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We build upon [4] by using an efficient computation of the point-split correlation function, 𝐶Γ(r, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' For each 𝜏, a direct calculation of (1) requires a loop over all lattice sites x for each value of r which is an expensive operation scaling as O(V2) where V is the spatial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' What follows is a method to reduce the cost of this computation by introducing a momentum space representation for the propagator and correlation function, see the Appendix of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We introduce quark propagators, 𝐷−1(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 𝑦), by Wick contracting the quark fields in the point- split correlation function, (1), 𝐶Γ(r, 𝜏) = − ∑︁ x ⟨𝐷−1(x + r, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 0, 0)Γ𝛾5 � 𝐷−1(x, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 0, 0) �† 𝛾5Γ†⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (9) Note that we have gauge fixed our configurations to the Coulomb gauge, and have replaced the gauge connection, 𝑈(𝑥, 𝑥 + r) in (2) by unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We now implicitly define the corresponding momentum space quark propagator via 𝐷−1(y, 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 0, 0) = 1 𝑉 ∑︁ p ˜𝐷−1(p, 𝜏)𝑒𝑖y·p, (10) in terms of the 3-momentum, p, which is conjugate to the position y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Introducing this momentum- space quark propagator into (9) yields 𝐶Γ(r, 𝜏) = 1 𝑉 ∑︁ p ⟨ ˜𝐷−1(p, 𝜏)Γ𝛾5 ˜𝐷−1(−p, 𝜏)𝛾5Γ†⟩𝑒𝑖p·r, (11) which we will use to implicitly define the momentum-space correlator, ˜𝐶Γ(p, 𝜏), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 𝐶Γ(r, 𝜏) = 1 𝑉 ∑︁ p ˜𝐶Γ(p, 𝜏)𝑒𝑖p·r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (12) We note that once we have calculated ˜𝐶Γ(p, 𝜏), we can determine the desired correlator 𝐶Γ(r, 𝜏) for any r using (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 4 Thermal interquark bottomonium potentials Thomas Spriggs At first sight, the conversion to momentum space does not produce any savings, because the calculation of 𝐶Γ and ˜𝐷−1, defined via (10) and (12), are both O(V2) in the number of operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' the same as the direct method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' However both (10) and (12) are Fourier transforms, and so significant speed-up for these steps can be achieved using the fast Fourier transform (FFT) algorithm which scales as O(V log V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Results For better comparison with [4], and as progress towards the treatment of 𝐶Γ(r, 𝜏) for all r, we consider here only the on-axis r data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Extensions to this will be discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1 Time dependence The potential is defined in (7) where there is an apparent explicit dependence on time, 𝜏, from the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In Figure 2, the potential, 𝑉Vector, from (7) is plotted against 𝜏 for a variety of distances r for our two extreme temperatures, 𝑇 = 141 and 352 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We can see a clear 𝜏 dependence for small 𝜏 which increases with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' However, for various ranges of 𝜏 and r there are clear plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In addition, we note that we would like to uncover temperature effects in the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The most accurate way of doing this is to compare different temperatures’ potentials obtained with the same time window to avoid contamination by systematic artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Based on these considerations, we restrict the range of 𝑟 and 𝜏 used in the determination of the potential to those listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Notice that in selecting a time window, there is a trade-off between the ranges of 𝑟 and 𝑇 for which the potential can be extracted: larger time windows give access to a larger range of 𝑟, but over a smaller range of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In Figure 3 we show four determinations of the central potential, corresponding to the first four time windows identified in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In each plot we show the potentials for several temperatures, and since these have been obtained by averaging over the same range of 𝜏, the temperature dependence can be ascribed to temperature effects, rather than fitting artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We find that the potential consistently flattens as the temperature increases above 𝑇pc, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' There is little thermal variation in the potential for 𝑇 ⪅ 𝑇pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In Figure 3, the error bars show statistical errors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The curves are fits to the Cornell potential, which will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 Quark mass dependence Equation (7) contains the reduced quark mass, 𝜇, which needs to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In [7], the 1S and 2S states were used to determine the bottom quark mass 𝑚𝑏, and thus the reduced quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In our simulations we do not have access to the 2S state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We instead use the simple argument: 𝜇 ≡ 1 2𝑚𝑏 ≈ 1 2 𝑀Υ, with 𝑀Υ from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We have tested the sensitivity of the potential on the quark mass and found that the variation (within sensible 𝜇 ranges) is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 5 Thermal interquark bottomonium potentials Thomas Spriggs Figure 2: Time dependence in the potential restricting the range of 𝑟 that we can consider valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Shown for two temperatures using the vector channel as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Time window [𝑎𝜏] 𝑟 range [𝑎𝑠] 𝑟 range [fm] Temperatures [MeV] 13-14 1-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='37 352-141 17-18 1-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='49 281-141 19-22 1-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='61 235-141 21-26 1-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='61 201-141 24-30 1-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='74 176-141 24-33 1-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='12-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='74 156-141 Table 2: Range of displacements and temperatures allowed to best approximate time independence in𝑉(𝑟, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Note that 𝑇pc = 181 MeV and thus the time windows below the solid line do not span this pseudocritical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='3 Cornell potential fits The Cornell potential [12] is a phenomenological description of a confining potential applicable to heavy quarks in QCD and is given by 𝑉(𝑟) = −𝛼 𝑟 + 𝜎𝑟 + 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' (13) Fits using (13) to our potential data are shown as solid curves in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' As can be seen these reproduce the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' When the string tension, 𝜎, in the Cornell potential is zero, this implies a deconfined potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' In all cases above 𝑇pc, we find that 𝜎 decreases with increasing temperature, confirming the expected thermal behaviour in the bottomonium system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Below𝑇pc the string tension does not change within statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Conclusion The temperature dependence of the central interquark potential in the bottomonium system using NRQCD quarks was explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This work was an extension of [4] and use a momentum-space approach which can improve the efficiency of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Clear thermal effects in this potential 6 T = 141 MeV 8 Preliminary X 6 [GeV] 4 Vector 来来 2 米 10 15 20 25 30 0 5 35 40 t/aT = 352 MeV 8 Preliminary 6 TI [GeV] 4 2 10 15 0 5Thermal interquark bottomonium potentials Thomas Spriggs Figure 3: The central potential calculated from (7) (points), overlaid with a fit of these data to the Cornell potential (13) (curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Each plot contains all temperatures and r ranges listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' were observed using a method which decoupled systematic “time window” artefacts from physical, thermal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' A systematic flattening of the potential with increasing temperature above 𝑇pc was observed, with no statistically significant variation in the potential for temperatures below 𝑇pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This work will be extended in a number of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' The potential will be calculated at all possible spatial separations, r, rather than just the on-axis values used here, and channels beyond the pseudoscalar and vector S-wave states will be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Also, a more robust definition of the reduced quark mass will be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Finally, a direct comparison will be made between these bottomonium results and those obtained for the charmonium potential using the same ensembles in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' Acknowledgments This work is supported by STFC grant ST/T000813/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' SK is supported by the National Research Foundation of Korea under grant NRF-2021R1A2C1092701andgrantNRF-2021K1A3A1A16096820, funded by the Korean government (MEST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This work used the DiRAC Extreme Scaling service at the University of Edinburgh, operated by the Edinburgh Parallel Computing Centre and the DiRAC 7 Time window: 13-14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 - T= 352 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 T = 281 MeV T = 235 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 T = 201 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 T = 176 MeV [GeV] T = 156 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 T= 141 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 Preliminary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='7 r [fm]Time window: 17-18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 T = 281 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 T = 235 MeV T = 201 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 T = 176 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 T = 156 MeV [GeV] T = 141 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 Preliminary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='7 r [fm]Time window: 19-22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 T = 235 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 T = 201 MeV T = 176 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 T = 156 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 T = 141 MeV [GeV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 Preliminary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='7 r [fm]Time window: 21-26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 T = 201 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 T = 176 MeV T = 156 MeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 T = 141 MeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 [GeV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='8 Preliminary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='7 r [fm]Thermal interquark bottomonium potentials Thomas Spriggs Data Intensive service operated by the University of Leicester IT Services on behalf of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This equipment was funded by BEIS capital funding via STFC capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC Opera- tions grants ST/R001006/1 and ST/R001014/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' DiRAC is part of the UK National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' This work was performed using PRACE resources at Cineca (Italy), CEA (France) and Stuttgart (Germany) via grants 2015133079, 2018194714, 2019214714 and 2020214714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9E1T4oBgHgl3EQfogQj/content/2301.03320v1.pdf'} +page_content=' We acknowledge the support of the Swansea Academy for Advanced Computing, the Supercomputing Wales project, which is part-funded by the European Regional Development Fund (ERDF) via Welsh Government, and the University of Southern Denmark and ICHEC, Ireland for use of computing facilities.' metadata={'source': 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F. Faria ∗ +Centro de Ciˆencias da Natureza, +Universidade Estadual do Piau´ı, +64002-150 Teresina, PI, Brazil +Abstract +We study the dynamics of the early universe in massive conformal +gravity. In particular, we show that the theory is consistent with the +observed values of the primordial abundances of light elements if we +consider the existence of right-handed sterile neutrinos. +PACS numbers: 04.62.+v, 04.60-m, 12.60.-i +* felfrafar@hotmail.com +arXiv:2301.11954v1 [physics.gen-ph] 27 Jan 2023 + +1 +Introduction +It is well known that the standard ΛCDM cosmological model is consistent +with most observations of the universe at both early and late times [1, 2]. +However, for this consistency to occur, a very small value for the cosmological +constant (Λ) is required, which by far does not match with the huge value pre- +dicted by quantum field theory (see [3] for a nice review). This discrepancy +between the cosmological and quantum values of Λ is known as the cosmolog- +ical constant problem [4]. Another important problem of ΛCDM is that the +primordial lithium abundance from the early universe nucleosynthesis pre- +dicted by it differs by about a factor of three from the observed abundance +[5], which is known as the lithium problem. Despite several attempts over +the years, no alternative cosmological model has succeeded in solving these +two problems and being consistent with other cosmological observations at +the same time. +One of such models comes from massive conformal gravity (MCG), which +is a conformally invariant theory of gravity in which the gravitational action +is the sum of the Weyl action with the Einstein-Hilbert action conformally +coupled to a scalar field [6]. Among so many cosmological models, we chose +the MCG model because it fits well with the Type Ia supernovae (SNIa) +data without the cosmological constant problem [7]. In addition, the theory +is free of the van Dam-Veltman-Zakharov (vDVZ) discontinuity [8], can re- +produce the orbit of binaries by the emission of gravitational waves [9] and +is consistent with solar system observations [10]. Furthermore, MCG is a +power-counting renormalizable [11, 12] and unitary [13] quantum theory of +gravity. +In this paper, we want to see if the MCG cosmology is consistent with +the observed primordial abundances of light elements without the lithium +problem. In Sec. 2, we describe the MCG cosmological equations. In Sec. +3, we derive the matter energy-momentum tensor used in the theory. In Sec. +4, we study the dynamics of the early MCG universe. In Sec. 5, we compare +the early universe nucleosynthesis of MCG with cosmological observations. In +Sec. 6, we analyze the evolution of the baryon density of the MCG universe. +Finally, in Sec. 7, we present our conclusions. +1 + +2 +Massive conformal gravity +The total MCG action is given by1 [8] +S = +� +d4x √−g +� +ϕ2R + 6∂µϕ∂µϕ − +1 +2α2CαβµνCαβµν +� ++ 1 +c +� +d4xLm, +(1) +where ϕ is a scalar field called dilaton, α is a coupling constant, +CαβµνCαβµν = RαβµνRαβµν − 4RµνRµν + R2 + 2 +� +RµνRµν − 1 +3R2 +� +(2) +is the Weyl tensor squared, Rαµβν = ∂βΓα +µν + · · · is the Riemann tensor, +Rµν = Rαµαν is the Ricci tensor, R = gµνRµν is the scalar curvature, and +Lm = Lm(gµν, Ψ) is the Lagrangian density of the matter field Ψ. It is worth +noting that besides being invariant under coordinate transformations, the +action (1) is also invariant under the conformal transformations +˜Φ = Ω(x)−∆ΦΦ, +(3) +where Ω(x) is an arbitrary function of the spacetime coordinates, and ∆Φ is +the scaling dimension of the field Φ, whose values are −2 for the metric field, +0 for gauge bosons, 1 for scalar fields, and 3/2 for fermions. +The variation of (1) with respect to gµν and ϕ gives the MCG field equa- +tions +ϕ2Gµν +6∂µϕ∂νϕ−3gµν∂ρϕ∂ρϕ+gµν∇ρ∇ρϕ2 −∇µ∇νϕ2 −α−2Wµν = 1 +2cTµν, +(4) +� +∇µ∇µ − 1 +6R +� +ϕ = 0, +(5) +where +Wµν += +∇ρ∇ρRµν − 1 +3∇µ∇νR − 1 +6gµν∇ρ∇ρR + 2RρσRµρνσ − 1 +2gµνRρσRρσ +−2 +3RRµν + 1 +6gµνR2 +(6) +1This action is obtained from the action of Ref. [8] by rescaling ϕ → +�� +32πG/3 +� +ϕ +and considering m = +� +3/64πGα. +2 + +is the Bach tensor, +Gµν = Rµν − 1 +2gµνR +(7) +is the Einstein tensor, +∇ρ∇ρϕ = +1 +√−g∂ρ �√−g∂ρϕ +� +(8) +is the generally covariant d’Alembertian for a scalar field, and +Tµν = − +2 +√−g +δLm +δgµν +(9) +is the matter energy-momentum tensor. +Before we proceed, it is important to note that both the symmetries of +the theory allow us to introduce in (1) a quartic self-interacting term of the +dilaton λ +� √−gϕ4 as well as interaction terms of the dilaton with the matter +fields. In the case of the dilaton self-interaction term, we do not include it +in the MCG action because this inclusion makes the flat metric no longer a +solution of the field equations, which invalidates the S-matrix formulation. +Although such a term is reintroduced in the effective action by quantum +corrections, we can consider the renormalized value of the coupling constant +λ equal zero so that the self-interacting term is present in the renormalized +action only to cancel out the corresponding divergent term. In addition, we +neglect the couplings between the dilaton and the matter fields because they +make the field equation (5) no longer valid. This equation is fundamental to +cancel non-renormalizable divergent terms that appear in the effective action +[14]. +At scales below the Planck scale, the dilaton field acquires a spontaneously +broken constant vacuum expectation value ϕ0 [15]. In this case, the field +equations (4) and (5) become +ϕ2 +0Gµν − α−2Wµν = 1 +2cTµν, +(10) +R = 0. +(11) +In addition, for ϕ = ϕ0, the MCG line element ds2 = (ϕ/ϕ0)2 gµνdxµdxν +reduces to +ds2 = gµνdxµdxν. +(12) +The full dynamics of the MCG universe can be described by (10)-(12) without +loss of generality. +3 + +3 +Dynamical perfect fluid +In order to find the MCG matter energy-momentum tensor, we consider the +conformally invariant matter Lagrangian density [16] +Lm = −√−gc +� +S2R+6∂µS∂µS+λS4+ i +2ℏ +� +ψγµDµψ − Dµψγµψ +� +−ℏµSψψ +� +, +(13) +where S is a scalar Higgs field2, λ and µ are coupling constants, ψ = ψ†γ0 is +the adjoint fermion field, Dµ = ∂µ + [γν, ∂µγν]/8 − [γν, γλ]Γλµν/8 (Γλµν is the +Levi-Civita connection), and γµ are the general relativistic Dirac matrices, +which satisfy the anti-commutation relation {γµ, γν} = 2gµν. +By varying (13) with respect to S, ψ and ψ, we obtain the field equations +12∇µ∇µS − 2RS − 4λS3 + ℏµψψ = 0, +(14) +iγµDµψ − µSψ = 0, +(15) +iDµψγµ + µSψ = 0. +(16) +Additionally, the substitution of (13) into (9) gives +Tµν +c += +12∂µS∂νS − 6gµν∂ρS∂ρS + 2gµν∇ρ∇ρS2 − 2∇µ∇νS2 ++ 2S2Gµν − gµν +� +λS4 + i +2ℏ +� +ψγρDρψ − Dρψγρψ +� +− ℏµSψψ +� ++ i +4ℏ +� +ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ +� +. +(17) +Then, using (14)-(16) and ∇µ∇νS2 = 2(S∇µ∇νS + ∂µS∂νS) in (17), we find +the energy-momentum tensor +Tµν += +c (8∂µS∂νS − 2gµν∂ρS∂ρS − 4S∇µ∇νS + gµνS∇ρ∇ρS) ++ 2cS2 +� +Rµν − 1 +4gµνR +� ++ T f +µν, +(18) +where +T f +µν = i +4cℏ +� +ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ +� +− 1 +4gµνcℏµSψψ (19) +2Although the Higgs field is actually a doublet, and it is more likely that we must have +two more scalar fields to get the correct quantum phenomenology at low energies [17], +considering only a scalar Higgs field will not change the classical results of the theory. +4 + +is the fermion energy-momentum tensor. +Considering that, at scales below the electroweak scale, the Higgs field +acquires a spontaneously broken constant vacuum expectation value S0, and +making some algebra, we find that (15) and (18) become +� +DµDµ − +�mc +ℏ +�2� +ψ = 0, +(20) +Tµν(S0, gµν) = 2cS2 +0 +� +Rµν − 1 +4gµνR +� ++ T f +µν(S0, gµν), +(21) +where +T f +µν(S0, gµν) = i +4cℏ +� +ψγµDνψ−Dνψγµψ+ψγνDµψ−Dµψγνψ +� +− 1 +4gµνmc2ψψ, +(22) +with m = µS0ℏ/c being the fermion mass. In flat spacetime, is not difficult +to see that (20) and (22) reduce to +� +∂µ∂µ − +�mc +ℏ +�2� +ψ = 0, +(23) +T f +µν(S0, ηµν) = i +4cℏ +� +ψγµ∂νψ − ∂νψγµψ + ψγν∂µψ − ∂µψγνψ +� +− 1 +4ηµνmc2ψψ, +(24) +where now the Dirac matrices satisfy the anti-commutation relation {γµ, γν} = +2ηµν. +The normalized plane wave solution to (23) is given by +ψ = +1 +√V Ek +uk eikµxµ, +(25) +where V is the volume, Ek = +√ +k2c2 + m2c4 is the energy, uk is a spinor +which satisfies [γµkµ + mc/ℏ] uk = 0, and kµ = (Ek/cℏ,⃗k/ℏ) is the wave +vector, with ⃗k being the momentum and k = |⃗k|. By substituting (25) and +its adjoint into (24), and using ukuk = −mc2, we obtain +T f +µν(S0, ηµν) = +� c2ℏ2 +V Ek +� +kµkν + +� m2c4 +4V Ek +� +ηµν. +(26) +5 + +Incoherently adding to (26) the individual contributions of a set of six plane +waves moving in the ± x, ± y and ± z directions, all with the same Ek and +k, we can write the energy-momentum tensor (26) in the perfect fluid form +T f +µν(S0, ηµν) = +� +ρ + p +c2 +� +uµuν + ηµνp + ηµνc2ρΛ, +(27) +where +c2ρ = 6Ek +V +(28) +is the energy density of the fluid, +p = 2k2c2 +V Ek +(29) +is the pressure of the fluid, +c2ρΛ = 3m2c4 +2V Ek +(30) +is the vacuum energy (dark energy) density, and uµ is the four-velocity of the +fluid, which is normalized to uµuµ = −c2. It follows from (28)-(30) that +p = 0, +ρΛ = 1 +4ρ, +(31) +for a non-relativistic perfect fluid (k2c2 ≪ m2c4), and +p = 1 +3c2ρ, +ρΛ = 0, +(32) +for a relativistic perfect fluid (k2c2 ≫ m2c4). +In curved spacetime, the perfect fluid energy-momentum tensor (27) is +generalized to +T f +µν(S0, gµν) = +� +ρ + p +c2 +� +uµuν + gµνp + gµνc2ρΛ. +(33) +Finally, the insertion of (33) into (21) gives the energy-momentum tensor of +a dynamical perfect fluid +Tµν(S0, gµν) = 2cS2 +0 +� +Rµν − 1 +4gµνR +� ++ +� +ρ + p +c2 +� +uµuν +gµνp+gµνc2ρΛ. (34) +6 + +Taking the trace of (34), and substituting into the trace of (10), whose left +hand side is zero due to the field equation (11) and the tracelessness of the +Bach tensor (W = gµνWµν = 0), we arrive at +T = gµνTµν = 3p − c2ρ + 4c2ρΛ = 0. +(35) +We can see from (31) and (32) that both non-relativistic and relativistic +perfect fluids satisfies the tracelessness relation (35). For simplicity, we could +isolate ρΛ in (35) and replace it in (34) as done in Ref. [7]. In this case, it is +made clear that the vacuum energy density does not contribute directly to +the dynamic evolution of the MCG universe, which solves the cosmological +constant problem found in the ΛCDM model. However, here we will keep ρΛ +so we don’t miss any physical details during the calculations. +By substituting (34) into (10), and considering (11), we find +� +ϕ2 +0 − S2 +0 +� +Rµν − α−2Wµν = 1 +2c +�� +ρ + p +c2 +� +uµuν + gµνp + gµνc2ρΛ +� +, +(36) +which is the field equation that we will use in the study of the dynamics of +the early MCG universe in the next section. But before that, it is important +to compare MCG with another conformally invariant theory of gravity called +conformal gravity (CG)3, whose action is given by [18] +S = − 1 +2α2 +� +d4x √−g +� +CαβµνCαβµν +� ++ 1 +c +� +d4xLm. +(37) +By varying (37) with respect to gµν, we obtain the field equation +− α−2Wµν = 1 +2cTµν, +(38) +where Tµν is given by (18). We can easily see the difference between the two +theories by comparing (38) with (10) and (11). Just to stay within the scope +of this paper, it is worth noting that CG does not pass the early universe +nucleosynthesis test [19]. +3Although the difference between the two theories is quite obvious, as we will readily +show next, MCG is often confused with CG. Perhaps this is because CG is much older +and known than MCG. +7 + +4 +Early universe +As usual, we consider that the geometry of the universe is described by the +Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) line element +ds2 = −c2dt2 + a(t)2 +� +dr2 +1 − Kr2 + r2dθ2 + r2 sin2 θdφ2 +� +, +(39) +where a = a(t) is the scale factor and K = -1, 0 or 1 is the spatial curvature. +By substituting (39) and the fluid four-velocity uµ = (c, 0, 0, 0) into (36), we +obtain4 +¨a +a = − +c +6 (ϕ2 +0 − S2 +0) +� +c2ρ − c2ρΛ +� +, +(40) +¨a +a + 2 +� ˙a +a +�2 ++ 2Kc2 +a2 += +c +2 (ϕ2 +0 − S2 +0) +� +p + c2ρΛ +� +, +(41) +where the dot denotes d/dt. +Subtracting (40) from (41), and considering that5 +ϕ2 +0 = +3c3 +32πG ≫ S2 +0, +(42) +we obtain +� ˙a +a +�2 += 8πG +9c2 +� +c2ρ + 3p + 2c2ρΛ +� +− Kc2 +a2 . +(43) +The combination of (43) with (40) then gives the energy continuity equation +c2 ˙ρ + 3 ˙a +a +� +c2ρ + p +� +− c2 ˙ρΛ = 0, +(44) +which can also be obtained by the conservation law ∇µT f +µν = 0, with T f +µν +being the perfect fluid energy-momentum tensor (33). +Using either (31) or (32) in (44), we get +˙ρ + 4 ˙a +aρ = 0, +(45) +4It is worth noting that Wµν = 0 for the FLRW spacetime. +5This value of ϕ0 is necessary for the theory to be consistent with solar system obser- +vations [10]. +8 + +which, consequently, is valid for both non-relativistic and relativistic dynam- +ical perfect fluids. As usual, we can write the solution to (45) in the form +ρ = ρ0 +�a0 +a +�4 +, +(46) +where, from now on, the subscript 0 denotes values at the present time t0. +In the case of the early universe, which is composed by a very hot plasma +dominated by relativistic particles (radiation), we find that (43) becomes +˙a2 = 16πGa4 +0 +9a2 +ρr0 − Kc2. +(47) +where we used (32) and (46), with ρr being the mass density of the radiation. +Since a is small in the early universe, we can neglect the curvature term on +the right hand side of (47) and write it in the approximate form +˙a2 = 16πGa4 +0 +9a2 +ρr0, +(48) +whose solution is given by +a(t) = +�64πGa4 +0ρr0 +9 +�1/4 +t1/2. +(49) +Finally, inserting (49) into the Hubble constant +H = ˙a +a, +(50) +we obtain +H = 1 +2t, +(51) +which is the same relation between the Hubble constant and time that occurs +in the early ΛCDM universe. However, since the MCG scale factor (49) is +equal 0.9 times the value of the ΛCDM scale factor, the expansion of the early +MCG universe is slower than the expansion of the early ΛCDM universe, +which will give a difference in the values of the two Hubble constants, as we +will show in the next section. +9 + +5 +Nucleosynthesis +The abundances of light chemical elements in the early universe are mainly +determined by one cosmological parameter, namely, the baryon-to-photon +ratio η = nb/nγ, where nb and nγ are the number densities of baryons and +photons in the universe. As usual, to find η we must first write the Hubble +constant in function of temperature T using the Stefan-Boltzmann law +ρr = +�g∗aB +2c2 +� +T 4, +(52) +where aB is the radiation energy constant and g∗ counts the number of rela- +tivistic particle species determining the energy density in radiation. Substi- +tuting (52) and (49) into (46), we obtain +t = +� +9c2 +32πGg∗aB +�1/2 1 +T 2. +(53) +It then follows from (51) and (53) that +H = +�8πGg∗aB +9c2 +�1/2 +T 2, +(54) +which is equal 0.82 times the value of the ΛCDM Hubble constant. +In order to describe the thermal history of the early MCG universe, we +must compare the Hubble constant in the form (54) with the collision rate +of particle interactions +Γ = nσv, +(55) +where n is the number density of particles, σ is their interaction cross section +and v is the average velocity of the particles. A specific temperature that is of +particular importance for the outcome of the early universe nucleosynthesis +(EUN) is the one at which the thermal equilibrium between neutrons and +protons begins to break down, which happens when H ∼ Γν, where +Γν ≈ G2 +F +c6ℏ7(kBT)5 +(56) +is the collision rate of a neutrino with electrons or positrons, with GF being +the Fermi constant and kB the Boltzmann constant. +10 + +By equating (54) with (56), and assuming that at the onset of the electron- +positron annihilation the remaining relativistic particles are photons, elec- +trons, positrons and left-handed neutrinos, for which g∗ = 10.75, we obtain +kBTeq = 0.75 MeV. +(57) +We can see from (57) that the thermal equilibrium between neutrons and +protons is maintained at temperatures above Teq = 8.7 × 109 K in the early +MCG universe. At that time, the neutron-to-proton ratio was +�nn +np +� +eq += e−Q/kBTeq = 0.178, +(58) +where we used (57) and the neutron-proton energy difference Q = 1.239 MeV. +Using (58), we can make a rough estimate that the final freeze-out neutron +abundance is given by +X∞ +n ∼ Xeq +n = +e−Q/kBTeq +1 + e−Q/kBTeq = 0.15. +(59) +Including the neutron decay in our calculation, we find +Xn(t) = X∞ +n e−t/τn = 0.15 e−t/τn, +(60) +where τn = 879.4 s is the neutron mean lifetime [20]. +The first light element formed in the early universe was deuterium (D), +whose ratio to proton is approximately given by +nD +np +≈ 6.9η +� kBT +mnc2 +�3/2 +exp +� BD +kBT +� +, +(61) +where we used (58) and BD = 2.2 MeV is the binding energy of deuterium. +Noting that the EUN starts when nD ∼ np, it follows from (61) that +6.9ηEUN +�kBTEUN +mnc2 +�3/2 +exp +� +BD +kBTEUN +� +≈ 1, +(62) +where ηEUN and TEUN are the baryon-to-photon ratio and temperature of +the EUN. We can see from (62) that we need the value of TEUN to find +11 + +ηEUN. Fortunately, we can find such value from the primordial helium (4He) +abundance +YP ≡ 4n4He +nH += +2Xn(tEUN) +1 − Xn(tEUN), +(63) +where tEUN is the time of the EUN. +The substitution of (60) and the observed value of the helium abundance +YP = 0.245 [21] into (63) gives +tEUN ≈ 279.7 s. +(64) +Then, by inserting (64) into (53), and considering that the electrons and +protons are no longer relativistic after their annihilation, which gives g∗ = +3.36, we obtain +TEUN ≈ 8.8 × 108 K. +(65) +Finally, using (65) in (62), we arrive at +ηEUN ≈ 5.12 × 10−8, +(66) +which produces abundances of other light elements besides helium orders +of magnitude below the primordial abundances inferred from current obser- +vations [22]. +However, this result does not automatically rule out MCG. +If we consider that the theory has low energy (≲ eV) right-handed sterile +neutrinos6, then we must replace g∗ = 10.75 by g∗ = 16.125 prior to the +electron-positron annihilation and g∗ = 3.36 by g∗ = 5.04 after the electron- +positron annihilation due to the contribution of the sterile neutrinos to the +relativistic energy content of the universe. These replacements lead to the +standard value +ηEUN ≈ 6 × 10−10, +(67) +which is consistent with the observed abundances of all light elements with +the exception of lithium7. +6The existence of such neutrinos is allowed by the symmetries of the theory and may +be responsible for the small masses of the left-handed neutrinos found in nature [23]. +7It is possible that the decay of the sterile neutrinos solves the inconsistency between +the predicted and observed values of the lithium abundance [24]. +12 + +6 +Baryon density +Another important cosmological parameter that is determined by η is the +baryon mass density ρb of the universe. In order to find the relation between +these two parameters in the MCG universe, we start from the definitions of +the baryon and photon number densities +nb = ρb +mN +, +(68) +nγ = 2ζ(3)8π +c3 +�kBT +h +�3 +≈ 2 × 107T 3, +(69) +where mN is the nucleons mass. The combination of (68), (69) and (52), +with g∗ = 2, then gives the relation +η = +aB +2 × 107mNc2 +ρb +ργ +T, +(70) +which is valid for any cosmological model. Noting that both ρb and ργ obey +(46) in MCG, we can write (70) in the form +η = +aB +2 × 107mNc2 +ρb0 +ργ0 +T, +(71) +which means that the baryon-to-photon ratio evolves over time in the MCG +universe8, different to what happens in the ΛCDM universe where η is con- +stant after the EUN. +Using the current temperature of the universe T0 = 2.73 K in (52), with +g∗ = 2, we find +ργ0 = 4.65 × 10−31 kg/m3. +(72) +In addition, the use of (67) in (62), with 6.9 replaced by 6.5 due to the +different value of (58) which leads to (67), gives +TEUN ≈ 7.56 × 108 K. +(73) +8It would be important to check if (71) at the time of recombination is consistent with +the value of η measured by cosmic microwave background (CMB) anisotropies. However, +a theory for the growth of inhomogeneities in MCG has not yet been developed due to the +complexity generated by the contribution of the Bach tensor in (10). Therefore, we will +leave this analysis for future works. +13 + +Finally, substituting (67), (72) and (73) into (71), we obtain the current +baryon mass density +ρb0 = 1.46 × 10−36 kg/m3. +(74) +Since ρr and ρb evolve at the same rate in MCG, it follows from (72) and +(74) that radiation always dominates the MCG universe. +In fact, the scale factor is big at late times such that we can neglect +the density term on the right hand side of (47), which makes the late MCG +universe curvature dominated. In this case, we must impose K = −1, which +gives the approximated solution +a(t) = ct +(75) +in the late MCG universe. It is not difficult to show that for an open uni- +verse with the scale factor (75) such as the late MCG universe, we have the +luminosity distance +dL(z) = c +H0 +�(1 + z)2 − 1 +2 +� +, +(76) +which fits well to SNIa data9 [6]. We intend to check if (75) provides good +fits to other low redshift data in future works. +Just to finish, it is important to note that the evolution of the baryon- +to-photon ratio (71) causes the number of baryons Nb to decrease over time +in the MCG universe. We can see this explicitly by substituting (46) and +V ∼ a3 in +Nb = nbV = ρbV +mN +, +(77) +which gives +Nb ∼ ρb0a4 +0 +mNa. +(78) +Using (75), we find that the number of baryons evolves over time according +to +Nb ∼ +�ρb0c3t4 +0 +mN +� +t−1 +(79) +in the late MCG universe. +It follows from the energy continuity equation (44) that +˙ρb + 3Hρb = ˙ρΛ. +(80) +9It is worth noting that the density term has not been neglected in Ref. [6], which in +practice does not change the SNIa data fitting. +14 + +By comparing (80) with the standard adiabatic conservation equation, and +noting that ˙ρΛ < 0, we conclude that the decrease in the number of baryons +(79) is due to the decay of the baryons into dynamic vacuum10, which clearly +leads to a violation of the conservation of the quantum numbers. However, +we can see from (79) that the variation of the number of baryons should only +be significant on cosmological time scales, which makes the decay of baryons +into vacuum not observable in the laboratory. +On the other hand, the non-conservation of baryons can have an im- +portant impact on the evolution of inhomogeneous structures of the universe +from the end of recombination until today. Due to the decrease in the amount +of baryons in the MCG universe, it is expected that the formation of struc- +tures happen much later than is observed or not happen at all. However, +the evolution of cosmological structures does not depend only on baryons +but also on dark matter, whose existence is necessary in MCG to explain the +galaxy rotation curves and the deflection of light by galaxies [10]. Therefore, +although the theory possibly has an extra scalar field that is a good candidate +for dark matter [14], much still has to be studied to find out if the evolution +of cosmological structures predicted by MCG is consistent with observations +or not. +7 +Final remarks +Here we have shown that the abundances of light elements, including lithium, +predicted by the early MCG cosmology are consistent with the observed val- +ues provided the theory has right-handed sterile neutrinos, which is allowed +by the symmetries of the theory. Even though we still need to check the +existence of such neutrinos in experiments like the Mini Booster Neutrino +Experiment (MiniBooNE) [25], this result is quite encouraging for us to con- +tinue with the study of the theory. +In addition, it was shown in this paper that the baryon-to-photon ratio +of the MCG universe evolves over time. Although further studies are needed +to verify whether this evolution is consistent with the value of the baryon- +to-photon ratio determined by the CMB anisotropies, who knows it solves +other early universe problems found in the ΛCDM model such as the baryon +asymmetry problem. We intend to study this and other MCG cosmological +predictions in future works. +10This decaying process can be accounted by the Yukawa interaction µSψψ in (19). +15 + +References +[1] A.G. Riess et al., Astron. J. 116, 1009 (1998); S. Perlmutter et al., ApJ +517, 565 (1999). +[2] N. Aghanim et al. [Planck Collab.], Planck 2018 results. VI. 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Smirnov, JHEP 2017, +113 (2017). +[18] P. D. Mannheim, Prog. Part. Nucl. Phys. 56, 340 (2006). +[19] L. Knox and A. Kosowsky, arXiv:9311006 [astro-ph]. +16 + +[20] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 +(2018). +[21] E. Aver et al., JCAP 03, 027 (2021). +[22] P. A. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020). +[23] K. A. Meissner and H. Nicolai, Phys. Lett. B 648, 312 (2007). +[24] L. Salvati et al., JCAP 08, 022 (2016). +[25] A. A. Aguilar-Arevalo et al. (MiniBooNE Collaboration), Phys. Rev. +Lett. 121, 221801 (2018). +17 + diff --git a/4tFKT4oBgHgl3EQf9i5P/content/tmp_files/load_file.txt b/4tFKT4oBgHgl3EQf9i5P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9477ab1fc5809ff10b317af636b9ddc1db9bb6d1 --- /dev/null +++ b/4tFKT4oBgHgl3EQf9i5P/content/tmp_files/load_file.txt @@ -0,0 +1,355 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf,len=354 +page_content='Early universe nucleosynthesis in massive conformal gravity F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Faria ∗ Centro de Ciˆencias da Natureza, Universidade Estadual do Piau´ı, 64002-150 Teresina, PI, Brazil Abstract We study the dynamics of the early universe in massive conformal gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In particular, we show that the theory is consistent with the observed values of the primordial abundances of light elements if we consider the existence of right-handed sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='+v, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='60-m, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='-i felfrafar@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='11954v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='gen-ph] 27 Jan 2023 1 Introduction It is well known that the standard ΛCDM cosmological model is consistent with most observations of the universe at both early and late times [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, for this consistency to occur, a very small value for the cosmological constant (Λ) is required, which by far does not match with the huge value pre- dicted by quantum field theory (see [3] for a nice review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' This discrepancy between the cosmological and quantum values of Λ is known as the cosmolog- ical constant problem [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Another important problem of ΛCDM is that the primordial lithium abundance from the early universe nucleosynthesis pre- dicted by it differs by about a factor of three from the observed abundance [5], which is known as the lithium problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Despite several attempts over the years, no alternative cosmological model has succeeded in solving these two problems and being consistent with other cosmological observations at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' One of such models comes from massive conformal gravity (MCG), which is a conformally invariant theory of gravity in which the gravitational action is the sum of the Weyl action with the Einstein-Hilbert action conformally coupled to a scalar field [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Among so many cosmological models, we chose the MCG model because it fits well with the Type Ia supernovae (SNIa) data without the cosmological constant problem [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In addition, the theory is free of the van Dam-Veltman-Zakharov (vDVZ) discontinuity [8], can re- produce the orbit of binaries by the emission of gravitational waves [9] and is consistent with solar system observations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Furthermore, MCG is a power-counting renormalizable [11, 12] and unitary [13] quantum theory of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In this paper, we want to see if the MCG cosmology is consistent with the observed primordial abundances of light elements without the lithium problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 2, we describe the MCG cosmological equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 3, we derive the matter energy-momentum tensor used in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 4, we study the dynamics of the early MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 5, we compare the early universe nucleosynthesis of MCG with cosmological observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 6, we analyze the evolution of the baryon density of the MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 7, we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 1 2 Massive conformal gravity The total MCG action is given by1 [8] S = � d4x √−g � ϕ2R + 6∂µϕ∂µϕ − 1 2α2CαβµνCαβµν � + 1 c � d4xLm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (1) where ϕ is a scalar field called dilaton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' α is a coupling constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' CαβµνCαβµν = RαβµνRαβµν − 4RµνRµν + R2 + 2 � RµνRµν − 1 3R2 � (2) is the Weyl tensor squared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Rαµβν = ∂βΓα µν + · · · is the Riemann tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Rµν = Rαµαν is the Ricci tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' R = gµνRµν is the scalar curvature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' and Lm = Lm(gµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Ψ) is the Lagrangian density of the matter field Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' It is worth noting that besides being invariant under coordinate transformations, the action (1) is also invariant under the conformal transformations ˜Φ = Ω(x)−∆ΦΦ, (3) where Ω(x) is an arbitrary function of the spacetime coordinates, and ∆Φ is the scaling dimension of the field Φ, whose values are −2 for the metric field, 0 for gauge bosons, 1 for scalar fields, and 3/2 for fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' The variation of (1) with respect to gµν and ϕ gives the MCG field equa- tions ϕ2Gµν +6∂µϕ∂νϕ−3gµν∂ρϕ∂ρϕ+gµν∇ρ∇ρϕ2 −∇µ∇νϕ2 −α−2Wµν = 1 2cTµν, (4) � ∇µ∇µ − 1 6R � ϕ = 0, (5) where Wµν = ∇ρ∇ρRµν − 1 3∇µ∇νR − 1 6gµν∇ρ∇ρR + 2RρσRµρνσ − 1 2gµνRρσRρσ −2 3RRµν + 1 6gµνR2 (6) 1This action is obtained from the action of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' [8] by rescaling ϕ → �� 32πG/3 � ϕ and considering m = � 3/64πGα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 2 is the Bach tensor, Gµν = Rµν − 1 2gµνR (7) is the Einstein tensor, ∇ρ∇ρϕ = 1 √−g∂ρ �√−g∂ρϕ � (8) is the generally covariant d’Alembertian for a scalar field, and Tµν = − 2 √−g δLm δgµν (9) is the matter energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Before we proceed, it is important to note that both the symmetries of the theory allow us to introduce in (1) a quartic self-interacting term of the dilaton λ � √−gϕ4 as well as interaction terms of the dilaton with the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In the case of the dilaton self-interaction term, we do not include it in the MCG action because this inclusion makes the flat metric no longer a solution of the field equations, which invalidates the S-matrix formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Although such a term is reintroduced in the effective action by quantum corrections, we can consider the renormalized value of the coupling constant λ equal zero so that the self-interacting term is present in the renormalized action only to cancel out the corresponding divergent term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In addition, we neglect the couplings between the dilaton and the matter fields because they make the field equation (5) no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' This equation is fundamental to cancel non-renormalizable divergent terms that appear in the effective action [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' At scales below the Planck scale, the dilaton field acquires a spontaneously broken constant vacuum expectation value ϕ0 [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In this case, the field equations (4) and (5) become ϕ2 0Gµν − α−2Wµν = 1 2cTµν, (10) R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (11) In addition, for ϕ = ϕ0, the MCG line element ds2 = (ϕ/ϕ0)2 gµνdxµdxν reduces to ds2 = gµνdxµdxν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (12) The full dynamics of the MCG universe can be described by (10)-(12) without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 3 3 Dynamical perfect fluid In order to find the MCG matter energy-momentum tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' we consider the conformally invariant matter Lagrangian density [16] Lm = −√−gc � S2R+6∂µS∂µS+λS4+ i 2ℏ � ψγµDµψ − Dµψγµψ � −ℏµSψψ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (13) where S is a scalar Higgs field2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' λ and µ are coupling constants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' ψ = ψ†γ0 is the adjoint fermion field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Dµ = ∂µ + [γν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' ∂µγν]/8 − [γν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' γλ]Γλµν/8 (Γλµν is the Levi-Civita connection),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' and γµ are the general relativistic Dirac matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' which satisfy the anti-commutation relation {γµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' γν} = 2gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' By varying (13) with respect to S, ψ and ψ, we obtain the field equations 12∇µ∇µS − 2RS − 4λS3 + ℏµψψ = 0, (14) iγµDµψ − µSψ = 0, (15) iDµψγµ + µSψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (16) Additionally, the substitution of (13) into (9) gives Tµν c = 12∂µS∂νS − 6gµν∂ρS∂ρS + 2gµν∇ρ∇ρS2 − 2∇µ∇νS2 + 2S2Gµν − gµν � λS4 + i 2ℏ � ψγρDρψ − Dρψγρψ � − ℏµSψψ � + i 4ℏ � ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (17) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' using (14)-(16) and ∇µ∇νS2 = 2(S∇µ∇νS + ∂µS∂νS) in (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' we find the energy-momentum tensor Tµν = c (8∂µS∂νS − 2gµν∂ρS∂ρS − 4S∇µ∇νS + gµνS∇ρ∇ρS) + 2cS2 � Rµν − 1 4gµνR � + T f µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (18) where T f µν = i 4cℏ � ψγµDνψ − Dνψγµψ + ψγνDµψ − Dµψγνψ � − 1 4gµνcℏµSψψ (19) 2Although the Higgs field is actually a doublet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' and it is more likely that we must have two more scalar fields to get the correct quantum phenomenology at low energies [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' considering only a scalar Higgs field will not change the classical results of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 4 is the fermion energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Considering that, at scales below the electroweak scale, the Higgs field acquires a spontaneously broken constant vacuum expectation value S0, and making some algebra, we find that (15) and (18) become � DµDµ − �mc ℏ �2� ψ = 0, (20) Tµν(S0, gµν) = 2cS2 0 � Rµν − 1 4gµνR � + T f µν(S0, gµν), (21) where T f µν(S0, gµν) = i 4cℏ � ψγµDνψ−Dνψγµψ+ψγνDµψ−Dµψγνψ � − 1 4gµνmc2ψψ, (22) with m = µS0ℏ/c being the fermion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In flat spacetime, is not difficult to see that (20) and (22) reduce to � ∂µ∂µ − �mc ℏ �2� ψ = 0, (23) T f µν(S0, ηµν) = i 4cℏ � ψγµ∂νψ − ∂νψγµψ + ψγν∂µψ − ∂µψγνψ � − 1 4ηµνmc2ψψ, (24) where now the Dirac matrices satisfy the anti-commutation relation {γµ, γν} = 2ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' The normalized plane wave solution to (23) is given by ψ = 1 √V Ek uk eikµxµ, (25) where V is the volume, Ek = √ k2c2 + m2c4 is the energy, uk is a spinor which satisfies [γµkµ + mc/ℏ] uk = 0, and kµ = (Ek/cℏ,⃗k/ℏ) is the wave vector, with ⃗k being the momentum and k = |⃗k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' By substituting (25) and its adjoint into (24), and using ukuk = −mc2, we obtain T f µν(S0, ηµν) = � c2ℏ2 V Ek � kµkν + � m2c4 4V Ek � ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (26) 5 Incoherently adding to (26) the individual contributions of a set of six plane waves moving in the ± x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' ± y and ± z directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' all with the same Ek and k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' we can write the energy-momentum tensor (26) in the perfect fluid form T f µν(S0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' ηµν) = � ρ + p c2 � uµuν + ηµνp + ηµνc2ρΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (27) where c2ρ = 6Ek V (28) is the energy density of the fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' p = 2k2c2 V Ek (29) is the pressure of the fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' c2ρΛ = 3m2c4 2V Ek (30) is the vacuum energy (dark energy) density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' and uµ is the four-velocity of the fluid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' which is normalized to uµuµ = −c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' It follows from (28)-(30) that p = 0, ρΛ = 1 4ρ, (31) for a non-relativistic perfect fluid (k2c2 ≪ m2c4), and p = 1 3c2ρ, ρΛ = 0, (32) for a relativistic perfect fluid (k2c2 ≫ m2c4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In curved spacetime, the perfect fluid energy-momentum tensor (27) is generalized to T f µν(S0, gµν) = � ρ + p c2 � uµuν + gµνp + gµνc2ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (33) Finally, the insertion of (33) into (21) gives the energy-momentum tensor of a dynamical perfect fluid Tµν(S0, gµν) = 2cS2 0 � Rµν − 1 4gµνR � + � ρ + p c2 � uµuν +gµνp+gµνc2ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (34) 6 Taking the trace of (34), and substituting into the trace of (10), whose left hand side is zero due to the field equation (11) and the tracelessness of the Bach tensor (W = gµνWµν = 0), we arrive at T = gµνTµν = 3p − c2ρ + 4c2ρΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (35) We can see from (31) and (32) that both non-relativistic and relativistic perfect fluids satisfies the tracelessness relation (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' For simplicity, we could isolate ρΛ in (35) and replace it in (34) as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In this case, it is made clear that the vacuum energy density does not contribute directly to the dynamic evolution of the MCG universe, which solves the cosmological constant problem found in the ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, here we will keep ρΛ so we don’t miss any physical details during the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' By substituting (34) into (10), and considering (11), we find � ϕ2 0 − S2 0 � Rµν − α−2Wµν = 1 2c �� ρ + p c2 � uµuν + gµνp + gµνc2ρΛ � , (36) which is the field equation that we will use in the study of the dynamics of the early MCG universe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' But before that, it is important to compare MCG with another conformally invariant theory of gravity called conformal gravity (CG)3, whose action is given by [18] S = − 1 2α2 � d4x √−g � CαβµνCαβµν � + 1 c � d4xLm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (37) By varying (37) with respect to gµν, we obtain the field equation − α−2Wµν = 1 2cTµν, (38) where Tµν is given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' We can easily see the difference between the two theories by comparing (38) with (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Just to stay within the scope of this paper, it is worth noting that CG does not pass the early universe nucleosynthesis test [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 3Although the difference between the two theories is quite obvious, as we will readily show next, MCG is often confused with CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Perhaps this is because CG is much older and known than MCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 7 4 Early universe As usual, we consider that the geometry of the universe is described by the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) line element ds2 = −c2dt2 + a(t)2 � dr2 1 − Kr2 + r2dθ2 + r2 sin2 θdφ2 � , (39) where a = a(t) is the scale factor and K = -1, 0 or 1 is the spatial curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' By substituting (39) and the fluid four-velocity uµ = (c, 0, 0, 0) into (36), we obtain4 ¨a a = − c 6 (ϕ2 0 − S2 0) � c2ρ − c2ρΛ � , (40) ¨a a + 2 � ˙a a �2 + 2Kc2 a2 = c 2 (ϕ2 0 − S2 0) � p + c2ρΛ � , (41) where the dot denotes d/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Subtracting (40) from (41), and considering that5 ϕ2 0 = 3c3 32πG ≫ S2 0, (42) we obtain � ˙a a �2 = 8πG 9c2 � c2ρ + 3p + 2c2ρΛ � − Kc2 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (43) The combination of (43) with (40) then gives the energy continuity equation c2 ˙ρ + 3 ˙a a � c2ρ + p � − c2 ˙ρΛ = 0, (44) which can also be obtained by the conservation law ∇µT f µν = 0, with T f µν being the perfect fluid energy-momentum tensor (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Using either (31) or (32) in (44), we get ˙ρ + 4 ˙a aρ = 0, (45) 4It is worth noting that Wµν = 0 for the FLRW spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 5This value of ϕ0 is necessary for the theory to be consistent with solar system obser- vations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 8 which, consequently, is valid for both non-relativistic and relativistic dynam- ical perfect fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' As usual, we can write the solution to (45) in the form ρ = ρ0 �a0 a �4 , (46) where, from now on, the subscript 0 denotes values at the present time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In the case of the early universe, which is composed by a very hot plasma dominated by relativistic particles (radiation), we find that (43) becomes ˙a2 = 16πGa4 0 9a2 ρr0 − Kc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (47) where we used (32) and (46), with ρr being the mass density of the radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Since a is small in the early universe, we can neglect the curvature term on the right hand side of (47) and write it in the approximate form ˙a2 = 16πGa4 0 9a2 ρr0, (48) whose solution is given by a(t) = �64πGa4 0ρr0 9 �1/4 t1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (49) Finally, inserting (49) into the Hubble constant H = ˙a a, (50) we obtain H = 1 2t, (51) which is the same relation between the Hubble constant and time that occurs in the early ΛCDM universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, since the MCG scale factor (49) is equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='9 times the value of the ΛCDM scale factor, the expansion of the early MCG universe is slower than the expansion of the early ΛCDM universe, which will give a difference in the values of the two Hubble constants, as we will show in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 9 5 Nucleosynthesis The abundances of light chemical elements in the early universe are mainly determined by one cosmological parameter, namely, the baryon-to-photon ratio η = nb/nγ, where nb and nγ are the number densities of baryons and photons in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' As usual, to find η we must first write the Hubble constant in function of temperature T using the Stefan-Boltzmann law ρr = �g∗aB 2c2 � T 4, (52) where aB is the radiation energy constant and g∗ counts the number of rela- tivistic particle species determining the energy density in radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Substi- tuting (52) and (49) into (46), we obtain t = � 9c2 32πGg∗aB �1/2 1 T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (53) It then follows from (51) and (53) that H = �8πGg∗aB 9c2 �1/2 T 2, (54) which is equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='82 times the value of the ΛCDM Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In order to describe the thermal history of the early MCG universe, we must compare the Hubble constant in the form (54) with the collision rate of particle interactions Γ = nσv, (55) where n is the number density of particles, σ is their interaction cross section and v is the average velocity of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' A specific temperature that is of particular importance for the outcome of the early universe nucleosynthesis (EUN) is the one at which the thermal equilibrium between neutrons and protons begins to break down, which happens when H ∼ Γν, where Γν ≈ G2 F c6ℏ7(kBT)5 (56) is the collision rate of a neutrino with electrons or positrons, with GF being the Fermi constant and kB the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 10 By equating (54) with (56), and assuming that at the onset of the electron- positron annihilation the remaining relativistic particles are photons, elec- trons, positrons and left-handed neutrinos, for which g∗ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='75, we obtain kBTeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='75 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (57) We can see from (57) that the thermal equilibrium between neutrons and protons is maintained at temperatures above Teq = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='7 × 109 K in the early MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' At that time, the neutron-to-proton ratio was �nn np � eq = e−Q/kBTeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='178, (58) where we used (57) and the neutron-proton energy difference Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='239 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Using (58), we can make a rough estimate that the final freeze-out neutron abundance is given by X∞ n ∼ Xeq n = e−Q/kBTeq 1 + e−Q/kBTeq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (59) Including the neutron decay in our calculation, we find Xn(t) = X∞ n e−t/τn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='15 e−t/τn, (60) where τn = 879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='4 s is the neutron mean lifetime [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' The first light element formed in the early universe was deuterium (D), whose ratio to proton is approximately given by nD np ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='9η � kBT mnc2 �3/2 exp � BD kBT � , (61) where we used (58) and BD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='2 MeV is the binding energy of deuterium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Noting that the EUN starts when nD ∼ np, it follows from (61) that 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='9ηEUN �kBTEUN mnc2 �3/2 exp � BD kBTEUN � ≈ 1, (62) where ηEUN and TEUN are the baryon-to-photon ratio and temperature of the EUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' We can see from (62) that we need the value of TEUN to find 11 ηEUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Fortunately, we can find such value from the primordial helium (4He) abundance YP ≡ 4n4He nH = 2Xn(tEUN) 1 − Xn(tEUN), (63) where tEUN is the time of the EUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' The substitution of (60) and the observed value of the helium abundance YP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='245 [21] into (63) gives tEUN ≈ 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='7 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (64) Then, by inserting (64) into (53), and considering that the electrons and protons are no longer relativistic after their annihilation, which gives g∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='36, we obtain TEUN ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='8 × 108 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (65) Finally, using (65) in (62), we arrive at ηEUN ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='12 × 10−8, (66) which produces abundances of other light elements besides helium orders of magnitude below the primordial abundances inferred from current obser- vations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, this result does not automatically rule out MCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' If we consider that the theory has low energy (≲ eV) right-handed sterile neutrinos6, then we must replace g∗ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='75 by g∗ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='125 prior to the electron-positron annihilation and g∗ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='36 by g∗ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='04 after the electron- positron annihilation due to the contribution of the sterile neutrinos to the relativistic energy content of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' These replacements lead to the standard value ηEUN ≈ 6 × 10−10, (67) which is consistent with the observed abundances of all light elements with the exception of lithium7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 6The existence of such neutrinos is allowed by the symmetries of the theory and may be responsible for the small masses of the left-handed neutrinos found in nature [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 7It is possible that the decay of the sterile neutrinos solves the inconsistency between the predicted and observed values of the lithium abundance [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 12 6 Baryon density Another important cosmological parameter that is determined by η is the baryon mass density ρb of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In order to find the relation between these two parameters in the MCG universe, we start from the definitions of the baryon and photon number densities nb = ρb mN , (68) nγ = 2ζ(3)8π c3 �kBT h �3 ≈ 2 × 107T 3, (69) where mN is the nucleons mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' The combination of (68), (69) and (52), with g∗ = 2, then gives the relation η = aB 2 × 107mNc2 ρb ργ T, (70) which is valid for any cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Noting that both ρb and ργ obey (46) in MCG, we can write (70) in the form η = aB 2 × 107mNc2 ρb0 ργ0 T, (71) which means that the baryon-to-photon ratio evolves over time in the MCG universe8, different to what happens in the ΛCDM universe where η is con- stant after the EUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Using the current temperature of the universe T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='73 K in (52), with g∗ = 2, we find ργ0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='65 × 10−31 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (72) In addition, the use of (67) in (62), with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='9 replaced by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='5 due to the different value of (58) which leads to (67), gives TEUN ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='56 × 108 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (73) 8It would be important to check if (71) at the time of recombination is consistent with the value of η measured by cosmic microwave background (CMB) anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, a theory for the growth of inhomogeneities in MCG has not yet been developed due to the complexity generated by the contribution of the Bach tensor in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Therefore, we will leave this analysis for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 13 Finally, substituting (67), (72) and (73) into (71), we obtain the current baryon mass density ρb0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='46 × 10−36 kg/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (74) Since ρr and ρb evolve at the same rate in MCG, it follows from (72) and (74) that radiation always dominates the MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In fact, the scale factor is big at late times such that we can neglect the density term on the right hand side of (47), which makes the late MCG universe curvature dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In this case, we must impose K = −1, which gives the approximated solution a(t) = ct (75) in the late MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' It is not difficult to show that for an open uni- verse with the scale factor (75) such as the late MCG universe, we have the luminosity distance dL(z) = c H0 �(1 + z)2 − 1 2 � , (76) which fits well to SNIa data9 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' We intend to check if (75) provides good fits to other low redshift data in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Just to finish, it is important to note that the evolution of the baryon- to-photon ratio (71) causes the number of baryons Nb to decrease over time in the MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' We can see this explicitly by substituting (46) and V ∼ a3 in Nb = nbV = ρbV mN , (77) which gives Nb ∼ ρb0a4 0 mNa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (78) Using (75), we find that the number of baryons evolves over time according to Nb ∼ �ρb0c3t4 0 mN � t−1 (79) in the late MCG universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' It follows from the energy continuity equation (44) that ˙ρb + 3Hρb = ˙ρΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' (80) 9It is worth noting that the density term has not been neglected in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' [6], which in practice does not change the SNIa data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 14 By comparing (80) with the standard adiabatic conservation equation, and noting that ˙ρΛ < 0, we conclude that the decrease in the number of baryons (79) is due to the decay of the baryons into dynamic vacuum10, which clearly leads to a violation of the conservation of the quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, we can see from (79) that the variation of the number of baryons should only be significant on cosmological time scales, which makes the decay of baryons into vacuum not observable in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' On the other hand, the non-conservation of baryons can have an im- portant impact on the evolution of inhomogeneous structures of the universe from the end of recombination until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Due to the decrease in the amount of baryons in the MCG universe, it is expected that the formation of struc- tures happen much later than is observed or not happen at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' However, the evolution of cosmological structures does not depend only on baryons but also on dark matter, whose existence is necessary in MCG to explain the galaxy rotation curves and the deflection of light by galaxies [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Therefore, although the theory possibly has an extra scalar field that is a good candidate for dark matter [14], much still has to be studied to find out if the evolution of cosmological structures predicted by MCG is consistent with observations or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 7 Final remarks Here we have shown that the abundances of light elements, including lithium, predicted by the early MCG cosmology are consistent with the observed val- ues provided the theory has right-handed sterile neutrinos, which is allowed by the symmetries of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Even though we still need to check the existence of such neutrinos in experiments like the Mini Booster Neutrino Experiment (MiniBooNE) [25], this result is quite encouraging for us to con- tinue with the study of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' In addition, it was shown in this paper that the baryon-to-photon ratio of the MCG universe evolves over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Although further studies are needed to verify whether this evolution is consistent with the value of the baryon- to-photon ratio determined by the CMB anisotropies, who knows it solves other early universe problems found in the ΛCDM model such as the baryon asymmetry problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' We intend to study this and other MCG cosmological predictions in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 10This decaying process can be accounted by the Yukawa interaction µSψψ in (19).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Yeh, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 88, 015004 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Faria, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' 2014, 520259 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} +page_content=' Faria, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFKT4oBgHgl3EQf9i5P/content/2301.11954v1.pdf'} 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+1,1640 @@ +’0x / N.N. and N.N. +(Guest Editors) +Volume 0 (200x), Number 0 +Interactive Control over Temporal Consistency +while Stylizing Video Streams +Sumit Shekhar1∗ +, Max Reimann1∗ +, Moritz Hilscher1, Amir Semmo1,2 +, +Jürgen Döllner1, and Matthias Trapp1 +1Hasso Plattner Institute for Digital Engineering, University of Potsdam, Germany +2Digital Masterpieces GmbH, Germany +(“*” denotes equal contribution) +Abstract +With the advent of Neural Style Transfer (NST), stylizing an image has become quite popular. A convenient way for extending +stylization techniques to videos is by applying them on a per-frame basis. However, such per-frame application usually lacks +temporal consistency expressed by undesirable flickering artifacts. Most of the existing approaches for enforcing temporal +consistency suffers from one or more of the following drawbacks: They (1) are only suitable for a limited range of techniques, (2) +typically do not support live processing as they require the complete video as input, (3) cannot provide consistency for the task +of stylization, or (4) do not provide interactive consistency-control. Note that existing consistent video-filtering approaches aim +to completely remove flickering artifacts and thus do not respect any specific consistency-control aspect. For stylization tasks, +however, consistency-control is an essential requirement where a certain amount of flickering can add to the artistic look and +feel. Moreover, making this control interactive is paramount from a usability perspective. To achieve the above requirements, we +propose an approach that can stylize video streams while providing interactive consistency-control. For achieving interactive +performance, we develop a lite optical-flow network that operates at 80 Frames per second (FPS) on desktop systems with +sufficient accuracy. Further, we employ an adaptive combination of local and global consistent features and enable interactive +selection between the two. By objective and subjective evaluation, we show that our method is superior to state-of-the-art +approaches. +CCS Concepts +• Computing methodologies ,..., Image-based rendering; Non-photorealistic rendering; Image processing; +1. Introduction +For thousands of years, paintings have served as a tool for vi- +sual communication and expression. However, it was not until +the late 20th century that computers were used to simulate paint- +ings [Hae90]. In the course of following decades, the field of artistic +stylization [KCWI13] has significantly developed and extended by +NSTs [SID17,JYF∗20]. Even though a large number of image styl- +ization techniques exist, extending these to video remains challeng- +ing. A major obstacle in this regard is the enforcement of temporal +coherence between stylized video frames. With the proliferation of +video streaming applications, stylizing video streams has become +popular, however, the requirements of low-latency processing add +additional challenges. Most of the existing methods, to address the +above, can be classified into one of the following four categories: +Style Specific. A common approach is to develop a specific +method for a particular artistic style and exploit its characteris- +tics for temporal coherency [BNTS07]. Such methods work ef- +fectively for the specific target style, however, do not generalize +well. Many of these specialized approaches have been discussed +by Bénard et al. [BTC13]. +Coherent Noise. Another class of techniques adopt and transform +a generic, temporally-coherent noise function to yield a visually +plausible stylized output [BLV∗10, KP11]. Compared to target- +based coherence enforcement [BNTS07], these are applicable to +a wider range of techniques but are limited for scenarios with +rapid temporal changes. +Stylization by Example. More recently, authors have adopted a +stylization-by-example approach to support a wide range of styl- +ization techniques [BCK∗13, JST∗19, TFK∗20, FKL∗21]. How- +ever, this approach requires the paring of the complete video and +keyframe marking. Thus, by design it is not applicable to video +streams. +Consistent Video Filtering. One can also enable stylization of +video streams using consistent video filtering techniques. Exist- +ing approaches are either not well-suited for Image-based Artis- +tic Rendering (IB-AR) [BTS∗15,YCC17] (Fig. 1) or do not pro- +vide interactive consistency control [LHW∗18,TDKP21], which +submitted to 200x. +arXiv:2301.00750v1 [cs.GR] 2 Jan 2023 + +2 +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +Table 1: Comparing existing consistent video filtering methods with ours with regards to consistency-control. Here, the color green denotes +the aspect which is favourable to interactive consistency-control while the color red denotes otherwise (“NA” stands for Not-Applicable). +Bonneel et al. [BTS∗15] +Yao et al. [YCC17] +Lai et al. [LHW∗18] +Shekhar et al. [SST∗19] +Thiomonier et al. [TDKP21] +Ours +Requires pre-processing? +No +Yes +No +Yes +No +No +Provides consistency-control at inference time? +Yes +No +No +Yes +No +Yes +Is the consistency-control interactive? +No +NA +NA +Yes +NA +Yes +(a) Input +(b) Processed +(c) Ours +(d) Lai et al. [LHW∗18] +(e) Bonneel et al. [BTS∗15] +Figure 1: For the top-row: first two columns depicts (a) input and (b) processed result for frame-24, column three to five depict the correspond- +ing consistent output using (c) Ours (d) Lai’s, and (e) Bonneel’s method. For the mid-row: depict the corresponding results for frame-80. +For the bottom-row: we show the Temporal Slice Image (TSI) for the entire video sequence depicting long-term temporal similarity with the +per-frame processed output. Note, that our method is able to preserve the look and feel of the per-frame processed result in comparison to +the method of Lai et al. which suffers from color bleeding artifacts while the stylized textures are lost for the output of Bonneel et al. . Please +see the supplementary material for video results. +is an essential requirement for artistic rendering [FLJ∗14]. Cur- +rently, the only method that provides interactive consistency- +control is limited to offline processing and requires pre- +processing [SST∗19]. +We aim to develop a temporal-consistency enforcement ap- +proach for artistic stylization techniques that provides (1) interac- +tive consistency-control and (2) online processing to facilitate the +application to video streams. +A determining factor towards the slow performance of ex- +isting online and interactive consistent video filtering tech- +nique [BTS∗15] is the costly step of optic-flow computation. Pre- +vious works using learning-based methods are able to achieve a +considerable accuracy for optic-flow estimation [TD20, JCL∗21]. +However, we argue that such a high accuracy is not particularly +necessary to enforce temporal consistency for artistic stylization +tasks. To validate our conjecture, we conduct a user study, wherein +the participants prefer the final consistent video output generated +using our flow network as compared to that being obtained us- +ing State-of-the-art (SOTA) approaches. We define artistic styl- +ization as the adaptation of colors, textures, and strokes. While +our approach is effective for most image-based stylization tech- +niques (e.g., NSTs, algorithmic filtering), it is not able to han- +dle significant shape or content inconsistencies between frames in- +troduced by semantically-driven image synthesis (e.g., image-to- +image diffusion-based models [RBL∗22]), as flow-based warping +is insufficient to enforce consistency in these cases. +In contrast to accuracy, little attention has been paid to improve +the run-time performance of optic-flow estimation, but which is +essential for online-interactive editing. To this end, we develop a +lite optic-flow neural network that runs at a high-speed (approx. +80 FPS on mid-tier desktop GPUs) while maintaining sufficient +accuracy. The compact network is also deployable on mobile de- +vices (iPhones and iPads) where it runs at interactive frame rates +(24 FPS on iPad Pro 2020). We use the optic-flow output from the +above network to enforce warping-based consistency at interactive +frame rates. Moreover, we construct an adaptive consistency prior +which allows for global and local temporal-consistency control. To +summarize we present the following contributions: +1. A novel approach for making per per-frame stylized videos tem- +porally consistent via adaptive combination of local and global +consistency features which allows for interactive consistency- +control. +2. A lite optic-flow network, to achieve interactive performance, +that runs at 80 FPS on a mid-tier desktop PC and at 24 FPS on a +mobile device while achieving reasonable accuracy. +2. Background & Related Work +Consistent Video Filtering. Lang et al. +[LWA∗12] propose +a solution to enforce temporal consistency for a large-class of +optimization-based problems via iterative filtering along the mo- +tion path. Dong et al. [DBZY15] address the problem of temporal +inconsistency for enhancement algorithms by dividing individual +video frames into multiple regions and performing a region-based +spatio-temporal optimization. Bonneel et al. +[BTS∗15] was the +submitted to 200x. + +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +3 +𝐼𝑡−1 +𝐼𝑡−1 +𝐼𝑡𝐼𝑡 +𝐼𝑡+1 +𝐼𝑡+1 +𝑃𝑡 +𝑃𝑡 +𝑃𝑡−1 +𝑃𝑡−1 +𝑃𝑡+1 +𝑃𝑡+1 +𝑂𝑡−1 +𝑂𝑡−1 +𝑤𝑝 +𝑤𝑛 +ϴ𝑙 +ϴ𝑙 +Linear +combination +ϴ𝑔 +ϴ𝑔 +Linear +combination +𝐴𝑡 +𝐴𝑡 +Optimization +Solving +𝑂𝑡 +𝑂𝑡 +𝑤𝑝 +Use 𝑤𝑝 and 𝑤𝑛 +for combining +1 +2 +3 +4 +5 +Estimate 𝑤𝑐 +for optimization +Figure 2: Schematic overview of our approach: (1) We start by calculating the warping weights wp and wn using Eqn. 3. (2) The computed +weights are used to linearly combine Pt, Pt−1, and Pt+1 to obtain the locally consistent image Lt, see Eqn. 2. (3) To obtain the globally +consistent version Gt we warp the output at previous time instance Ot−1 as depicted in Eqn. 4. (4) The local and global consistent images, Lt +and Gt, are linearly combined to obtain a temporally smooth version At, see Eqn. 5. (5) To include high-frequency details from the per-frame +processed result, At and Pt is adaptively combined via the optimization in Eqn. 1 using the weights wc (Eqn. 7) to obtain the final result Ot. +first to present a generalized approach for consistent video filter- +ing which is agnostic to the type of filtering applied on individ- +ual video-frames. The method combines gradient-based character- +istics of the per-frame processed result with the warped version of +the previous-frame output using a gradient-domain based optimiza- +tion scheme. Yao et al. [YCC17] propose a similar approach how- +ever considers multiple key-frames for warping-based consistency +to avoid problems due to occlusion. Both of the approaches assume +that the gradient of the processed video is similar to that of the +input video and thus cannot handle artistic rendering tasks where +new gradients resembling brush strokes are generated as part of +the stylization process. Moreover, due to slow optic-flow computa- +tion they are non-interactive in nature. Shekhar et al. [SST∗19] +employs a similar formulation as Bonneel et al. , with the dif- +ference of using a temporally denoised version of the current- +frame for consistency guidance. However, the temporal denois- +ing requires the complete video as input making the method of- +fline in nature. Lai et al. [LHW∗18] propose the first learning- +based technique in this context. The authors employ perceptual +loss to enforce similarity with the processed frames and for con- +sistency make use of short-term and long-term temporal losses. +Thimonier et al. [TDKP21] employ a ping-pong loss and a cor- +responding training procedure for temporal consistency. Both the +learning based technique are faster than their optimization-based +counterpart since they do not perform optic-flow computation at +test time. However, these learning based techniques do not allow +to control the degree of consistency in the final output which is +vital for the task of stylization. Thus, the above discussed meth- +ods are either non-interactive/offline or do not provide any con- +sistency control at inference time. Our approach addresses these +limitations (Tab. 1). +Optic Flow for Consistent Filtering. Both Booneel et al. and +Yao et al. use the PatchMatch algorithm [BSFG09] for flow-based +warping, however, the slow performance of PatchMatch makes +them non-interactive. Lai et al. use FlowNet 2.0 [IMS∗17] for flow- +based warping to design their short-term and long-term temporal +consistency losses. FlowNet 2.0 is on par with the quality of state- +of-the-art classical methods, however, due to large number of pa- +rameters and operations, achieves only interactive frame rates even +on high-end desktop Graphical Processing Units (GPUs). An im- +proved compact optic-flow Convolutional Neural Network (CNN) +is proposed by Sun et al. [SYLK18] – PWC-Net. It combines +coarse-to-fine estimation with pyramidal image features, correla- +tion, warping, and CNN-based estimation. Furthermore, a refine- +ment CNN is stacked at the end to improve the final flow estimate. +PWC-Net is orders of magnitude smaller than FlowNet 2.0, runs +at real-time frame rates using desktop GPUs. Liu et al. [LZH∗20] +employ their approach to train a similar architecture in an un- +supervised setting and achieve reasonable accuracy – ARFlow. +LiteFlowNet and its successor LiteFlowNet2, both proposed by +Hui et al. [HTL18, HTL20], have similar compact architectures. +Further improvement in accuracy is achieved by models using iter- +ative refinement, such as RAFT [TD20] and transformer modules +such as GMA [JCL∗21], however they heavily trade runtime for ac- +curacy. Based on a runtime-accuracy comparison (see Sec. 3.2), we +select PWC-Net as a base network to develop a "Lite" flow network +with improved performance for interactive consistent filtering. +Temporal +Consistency +for +Video +Stylization. Litwinow- +icz [Lit97] describes a technique to apply an impressionist effect +on images and videos. For enforcing temporal coherence, optical +flow was used to transform the brush strokes from one frame to +the next. Winnemöller et al. [WOG06] develop a real-time video +and image abstraction framework. The authors employ soft quan- +tization that spreads over a larger area, thus significantly reducing +temporal incoherence. Bousseau et al. [BNTS07] advects texture +in forward and backward direction using optical flow for coherent +water-colorization of videos. Numerous such specialized video- +based approaches have been discussed by Bénard et al. [BTC13]. +The above classical IB-AR techniques approximate rendering +primitives by modifying traditional image filters. Most often, +they use low-level image features for modeling and fail to model +structures resembling a particular style. Recently, deep CNNs +were successfully used to transfer high-level style attributes from +a painting onto a given image [GEB16]. Various methods have +been proposed to extend the above for videos [HWL∗17,CLY∗17, +GJAF17,RDB18,LLKY19,PP19,DTD∗21]. Ruder et al. [RDB18] +submitted to 200x. + +4 +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +propose novel initialization technique and loss functions for +consistent stylized output even in cases with large motion and +strong occlusion. The methods of Gupta et al. +[GJAF17], +Chen et al. [CLY∗17], and Huang et al. [HWL∗17] enforce con- +sistency via certain formulation of temporal loss and use optical +flow based warping only during the training phase thus achieving +fast performance. Puy and Pérez [PP19] develop a flexible deep +CNN for controllable artistic style transfer that allows for addition +of a temporal regularizer at testing time to remove the flickering +artefacts. The above method comes closest in terms of providing +some consistency control at test time for NST-based methods. +However, they cannot handle classical stylization techniques. +Stylization by example [BCK∗13, JST∗19, TFK∗20, FKL∗21] +caters to both (classical and neural) paradigms via priors involving +keyframe-based warping but can only be applied as an offline +process. We aim to propose a generic solution which is agnostic +to the type of stylization and provides online performance and +interactive consistency-control. +3. Method +3.1. Temporal Consistency Enforcement +Given an input video stream ...It−1, It, It+1,... and its per-frame +processed version ...Pt−1, Pt, Pt+1,..., we seek to find a tempo- +rally consistent output ...Ot−1, Ot, Ot+1 .... Our method is ag- +nostic to the stylization technique f applied to each frame, where +Pt = f(It). However, it is necessary for f to not introduce signifi- +cant shape or content inconsistencies between consecutive frames, +as the changes in the stylized frames should correspond to the op- +tical flow (calculated based on the content). We initialize the con- +sistent output for the first frame as its per-frame processed result +i.e., O1 = P1. To obtain the output for subsequent frames (Ot at any +given instance t) we require only a snippet of input (It−1,It,It+1) +and processed streams (Pt−1,Pt,Pt+1), and the consistent output at +the previous instance Ot−1. For enforcing consistency, we solve the +following gradient-domain optimization scheme: +E(Ot) = +� +Ω +� +||∇Ot −∇Pt||2 +� +�� +� +data ++ wc||Ot −At||2 +� +�� +� +smoothness +� +dΩ. +(1) +where Ω represents the image domain. The data term in this opti- +mization enforces similarity with the per-frame processed result Pt +in the gradient-domain. Thus, high-frequency details are taken from +Pt and the smoothness term enforces temporal-consistency where +low-frequency content is taken from the image At. The optimiza- +tion formulation in Eqn. 1 is commonly known as screened Pois- +son equation and has been successfully employed for various image +Table 2: Constituent elements of smoothness term in Eqn. 1 +for different methods. Here, ws and Td refers to saliency-based +weights and temporally-denoised image respectively, introduced by +Shekhar et al. +Method +Weight +Consistent Image +Ours +wc +At +Boneell et al. [BTS∗15] +wp +Γ(Ot−1) +Shekhar et al. [SST∗19] +ws +Td +editing applications [BCCZ08,BZCC10]. In the context of consis- +tent video filtering, it was first used by Bonneel et al. [BTS∗15] +followed by Shekhar et al. [SST∗19] (Tab. 2). However, our nov- +elty is the way in which we construct our smoothness term which, +unlike previous approaches, considers both global and local consis- +tency aspects. Our novel smoothness term is able to better preserve +the color and textures in the stylized output while providing both +short-term and long-term temporal consistency. +Local Consistency. For enforcing temporal consistency at a local +level, we use optic-flow to warp neighboring per-frame processed +results to the current time instance t. This is perfomred by comput- +ing an adaptive combination of (1) warped previous per-frame pro- +cessed image Γ(Pt−1), (2) warped next per-frame processed image +Γ(Pt+1), and (3) the current per-frame processed image Pt, where +Γ is the warping function. By including both backward and for- +ward warping in our formulation, we are able to significantly re- +duce artefacts due to occlusion and flow inaccuracies. The linear +combination of (1), (2), and (3) gives us a locally consistent ver- +sion Lt where, +Lt = (1−(wp+wn))·Pt + wp·Γ(Pt−1) + wn·Γ(Pt+1). +(2) +The weights wp and wn capture the inaccuracies in the warping of +previous and next frames respectively and are defined as follows: +wp = exp +� +−α||It −Γ(It−1)||2� +and +wn = exp +� +−α||It −Γ(It+1)||2� +. +(3) +In order to also incorporate contribution from Pt, we clamp the +weights wp and wn as follows: +� +wp +� += k1 and +� +wn +� += k2, where +k1 and k2 are two constants. The locally consistent image sequence +given by Lt has improved temporal consistency over the per-frame +processed output, however, it still has visible flickering artifacts. +Thus, the reduction in flickering due to warping of only one tempo- +ral neighbor is not sufficient. To further improve consistency, one +can warp more neighboring frames around the current time instance +t. As we increase the temporal window-size for such an adaptive +combination it has a denoising effect leading to further reduction in +flickering. The temporal denoising for enforcing consistency, per- +formed by Shekhar et al. [SST∗19] can be considered as an specific +example of the above scenario. However, for interactive stylization +warping more frames to the current instance is not feasible due to +time constraint. Moreover, in case of video streams we do not have +frames to warp from the forward temporal direction. +Global Consistency. In order to overcome this limitation, exist- +ing approaches [BTS∗15, LHW∗18] adopt a global approach. For +global consistency, one can consider the previous stabilized output +Ot−1 and enforce similarity with its warped version Gt where, +Gt = Γ(Ot−1). +(4) +To enforce only global temporal smoothness, we replace At with Gt +in Eqn. 1. Further, in order to compensate for optic-flow inaccura- +cies, the smoothness term is weighted using wp (i.e., wc = wp) in +Eqn. 1. However, considering only global consistency for flicker +reduction leads to loss of stylization and local temporal varia- +tions in the final output. Moreover, in this case any warping-error +(due to flow-inaccuracies) or noise (as part of stylization process) +submitted to 200x. + +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +5 +keeps getting propagated to future frames. Due to the above fac- +tors, such an approach only gives plausible results where the gra- +dients of the original video are similar to the gradients of the pro- +cessed video. The above does not hold for the task of stylization +where stylistic elements such as brush strokes, textures or stroke +textons [ZGWX05], in general, can vary largely between frames +even for small changes in gradient. +Combining Global and Local Consistency. For preserving local +temporal variations (in terms of look and feel) while significantly +reducing the flickering artifacts, we linearly combine globally and +locally consistent images Gt and Lt respectively, +At = wp·Gt + (1−wp)·Lt. +(5) +We use the adaptively combined image At as our reference for +consistency while enforcing temporal smoothness in Eqn. 1. The +� +wp +� +can be increased to increase the influence of global-temporal +smoothness and vice versa. Further, the influence of the smoothness +term is controlled by per-pixel consistency weights wc. We would +like to invoke the smoothness term only when the warping accuracy +is sufficiently high. To this end, we construct a warped version of +the input image similar to Lt as, +AIt = (1−(wp+wn))·It + wp·Γ(It−1) + wn·Γ(It+1). +(6) +Only when the input image It is similar to AIt, the smoothness term +is invoked. To measure this similarity, we use the weight wc, +wc = λ·exp +� +−α||It −AIt|| +2� +. +(7) +The parameter λ is used to scale up or down the weight wc. +Consistency Control Modes. The above adaptive combination +of local and global consistency provides two different ways of +consistency-control in the final output. By increasing +� +wp +� +we can +increase the proportion of global consistency in the adaptively com- +bined image At and vice versa. On the other hand the optimization +parameter λ dictates how close the output Ot will be to the adap- +tively combined image At. Thus, the level of consistency in the +final output can be controlled in two different ways: (1) by set- +ting up the limit of parameter wp, i.e., +� +wp +� +or (2) by scaling the +weight parameter λ. For lower values of +� +wp +� +(Fig. 6b), the consis- +tency enforced is negligible and the final result resembles the per- +frame processed output (Fig. 6f). However, for higher values we +start observing noisy ghosting artefacts (Fig. 6e). The lower values +of +� +wp +� +translates to using only global consistency which results in +accumulation of flow inaccuracies visualized as ghosting artefacts. +Similarly, for lower values of λ (Fig. 6g), the final result is visually +similar to the per-frame processed output (Fig. 6f). However, for +higher values the optimization becomes unstable resulting in noisy +optimization-based artefacts. (Fig. 6j). +Optimization Solver. The energy terms in Eqn. 1 are smooth and +convex in nature, which allows a straightforward energy minimiza- +tion with respect to Ot. To this end, we employ an iterative ap- +proach thus avoiding – storage of a large matrix in memory and +further estimating its inverse. Moreover, an iterative approach al- +lows us to stop the solver once we have achieved visually plau- +sible results. An iterative update Otk+1 is obtained by employing +43 + (a) (b) (c) +Refinement +Flow +Estimation +Modules +Feature +Extraction +Input Frames +Output Flow +Figure 3: Modification of the PWC-Net [SYLK18] architecture for +real-time performance. We apply following network compression +steps: (a) Replace DenseNet connections with light ones, (b) Re- +duce the number of flow estimators, and (c) Replace dense connec- +tions in the refinement module with separable convolutions. +Stochastic Gradient Descent (SGD) with momentum [Qia99], +Otk+1 = Otk −η∇E(Otk)+κ(Otk −Otk−1). +(8) +where η and κ are the step size parameters, ∇E is the energy gradi- +ent with respect to Ot, and k is the iteration count. For most of our +experiments, η = 0.15 and κ = 0.2 yield plausible results. We con- +sider the trade-off between performance vs. accuracy as a stopping +criteria and do not compute energy residue for this purpose. To ob- +tain a consistent output while having interactive performance, we +empirically determine 150 iterations to be sufficient. The optimiza- +tion is stable for the given parameter settings and early stopping is +only employed for computational gain. +An integral aspect common to both our local and global consis- +tency is the warping function Γ. Apart from the number of solver +iterations, for interactive performance the above warping should +also happen at a fast rate – which in turn necessitates fast optic- +flow estimation. +3.2. Lite Optic-Flow Network +We aim to obtain a flow network capable of running at high-speed +on consumer hardware with reasonable accuracy. To this end, we +start by selecting an existing CNN-based optical flow estimation +technique, based on accuracy vs. run-time analysis. After the se- +lection of a base network, we perform further optimization steps to +increase the performance as outlined in Fig. 3. +Base Network Selection for Compression. In Fig. 4, we com- +pare several well-known optical methods to find a base network +candidate that best matches our runtime/accuracy requirements. +We employ the following models for this: FlowNet 2.0 [IMS∗17], +SpyNet [RB17], LiteFlowNet2 [HTL20], PWCNet [SYLK18], +ARFlow [LZH∗20], VCN [YR19], RAFT [TD20] and finally +GMA [JCL∗21] (state-of-the-art in terms of EPE-based accuracy). +Our experiments are carried out on a Nvidia RTX 2070 GPU, +which we deem to be a good representative of a current mid-to +submitted to 200x. + +6 +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +0px +2px +4px +6px +8px +0 +10 +20 +30 +40 +flownet2 +spynet +pwcnet +arflow +liteflownet2 +vcn +raft +gma +Sintelfinal-test EPE (lower=better) +FPS (higher=better) +Figure 4: Accuracy vs. run-time performance of existing methods +measured on Sintel Final (Test set) [BWSB12]. The Endpoint Er- +ror (EPE) metric measures Euclidean distance (in pixels) between +ground-truth and predicted optical flow vectors. +higher-end consumer GPU. Under a constraint of interactive perfor- +mance on consumer hardware, LiteFlowNet2 [HTL20] and PWC- +Net [SYLK18] offer the best trade-off between run-time perfor- +mance and accuracy (Fig. 4). LiteFlowNet2 [HTL20] is already an +optimized version of FlowNet 2.0 [IMS∗17], in comparison PWC- +Net [SYLK18] has more potential for optimization/compression. +Moreover, recently it has been shown that PWC-Net can achieve +similar accuracy to RAFT when trained on a large-scale synthetic +dataset [SVH∗21] and that PWC-Net achieves favourable trade-offs +vs. other state-of-the-art methods when selecting for runtime per- +formance or higher image resolutions [SHR∗22]. Hence, we select +PWC-Net for further compression. +Optimized Network Architecture. We start with the base archi- +tecture of PWC-Net. As the first compression step we reduce the +computationally expensive DenseNet [HLvdMW17] connections +in the flow estimators to retain connections only in the last two +layers ("-light" in Fig. 5b). Similar to LiteFlowNet2 [HTL20], we +remove the fifth flow estimator – operating on the highest resolu- +tion – as it heavily trades off run-time for only marginal increase +in accuracy (compare "4light" vs "5light" in Fig. 5b). We replace +the standard convolutions in the refinement by depthwise separable +convolutions [HZC∗17] ("-sepref" in Fig. 5b). Moreover, we also +explore reducing the number of channels [HZC∗17], but find that +reducing channels results in a worse trade-off as compared to other +optimizations. +Training. For +training, +we +follow +the +original +PWC- +Net +[SYLK18] +schedule. +However, +we +find +that +weight- +ing +the +multi-scale +losses +equally, +instead +of +exponen- +tially [SYLK18, HTL18, HTL20, YR19], improves accuracy. For +our experiments on the desktop system, we use PyTorch [PGM∗19] +and take inspiration from the implementation by Niklaus [Nik18]. +Similar to PWC-Net [SYLK18], we train our mobile architecture +on the training dataset schedule FlyingChairs [FDI∗15] → Fly- +ingThings3D [MIH∗16]→ Sintel [BWSB12]. In the supplementary +material, we provide training settings for each stage in detail. We +Table 3: Runtime performance in milliseconds per frame. We mea- +sure the total processing time (without disk IO) and the individual +stages for a mid-tier GPU (Nvidia GTX 1080Ti) and a higher-end +GPU (Nvidia RTX 3090), results are averaged over 100 runs. +Task +Optical flow +Stabilization +Total +↓ Res. / GPU +1080Ti 3090 +1080Ti 3090 +1080Ti 3090 +1920×1080 px +66.8 +40.0 +184.1 +42.7 +250.8 +82.7 +1280×720 px +31.3 +19.7 +86.5 +21.1 +117.8 +40.8 +640×480 px +12.6 +6.2 +20.6 +6.3 +33.2 +12.5 +employ a multi-scale loss [SYLK18] applied to each flow estimator +and optimize using the AdamW optimizer [LH19] with β1 = 0.09, +β2 = 0.99, and l2 weight regularization with trade-off γ = 0.0004. +Furthermore, extensive dataset augmentation is applied to prevent +model overfitting. We refer to the supplementary material for more +details. +Our Final Model. We analyze various optimization options and +chose “our-4light-sepref” as our final model for desktop systems +as it provides the best trade-off between accuracy vs. run-time. As +depicted in Fig. 5a, our method improves run-time performance of +PWC-Net from 30 FPS to 85 FPS – a speed-up of factor 2.8. For +Sintel training data the accuracy drops by ≈ 0.5px in EPE terms, +however for test data the drop in accuracy is significant where the fi- +nal EPE is 7.43. Nevertheless, the accuracy is sufficient enough for +enforcing warping-based consistency. To validate our design deci- +sions, we conduct an extensive ablation study in which we vary the +architectural and training choices – please see the supplementary +for details. Furthermore, we tune our architecture for optical flow +calculation on mobile devices using channel pruning and quantiza- +tion, which we also detail in the supplementary material. Here, we +improve run-time performance from 2.8 FPS to 24 FPS (iPad Pro +2020), and 1.5 FPS to 13 FPS (iPad Air) – an improvement of factor +8. Next to showing the general applicability of optical flow CNNs +on mobile devices, this demonstrates that real-time on-device sta- +bilization of videos using our presented approach will become fea- +sible with a further moderate increase in mobile GPU computing +power. A fast optic-flow based warping enables our framework to +interactively control the degree of consistency and generate visu- +ally plausible results. +4. Experimental Results +4.1. Implementation Details +All our experiments were performed on an consumer PC with an +AMD Ryzen 1920X 12-Core CPU, 48 GB of RAM, and a Nvidia +GTX 1080Ti and RTX 3090 graphics cards with VRAMs of 11 +GB and 24 GB respectively. We implement a real-time video- +consistency framework in C++, using ONNXRuntime for cross- +platform acceleration of our lite optical-flow network and imple- +ment the stabilization code using Nvidia CUDA (v11.4). In Tab. 3, +we measure the runtime performance of our system. We find that +an incoming stream of frames can be stabilized at real-time perfor- +mance for VGA resolution even on low- and mid-tier GPUs and +higher-tier GPUs (such as a RTX 3090) can stabilize HD at com- +mon video frame rates (approx. 24 FPS) and full-HD resolutions at +submitted to 200x. + +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +7 +2px +3px +4px +5px +0 +20 +40 +60 +80 +100 +flownet2 +liteflownet2 +pwcnet +our-5light +our-5light-5sep +our-5light-2sep +our-4light-1sep +our-5light-c50 +our-5light-c75 +our-4light +our-4light-sepref +(a) Sintelfinal-train EPE (lower=better) +FPS (higher=better) +Modifier +Description +Default +-Nlight +N light [LZH∗20] flow esti- +mators. +5 dense [SYLK18] +-Msep +last M flow estimators use +depthwise separable convo- +lutions [HZC∗17]. +standard convs. +-sepref +refinement +uses +depth- +wise +separable +convolu- +tions [HZC∗17]. +standard convs. +-cP +use P% of channels. +100% +(b) Legend of our CNN variants. +Figure 5: Accuracy vs. run-time performance of our CNN variants on desktop, measured on Sintel Final (Train) [BWSB12]. Optimization +steps that lead to significant improvement in run-time are connected by a line. Our architectural modifications to PWC-Net [SYLK18] are +detailed on the right, e.g., our-4light-sepref denotes a 4 light flow estimators and refinement using depthwise separable convolutions. +(a) Input +(b) +� +wp +� += 0.3 +(c) +� +wp +� += 0.5 +(d) +� +wp +� += 0.7 +(e) +� +wp +� += 0.9 +(f) Processed +(g) λ = 0.1 +(h) λ = 1.0 +(i) λ = 5.0 +(j) λ = 7.06 +Figure 6: The level of consistency in the final output can be controlled via parameters +� +wp +� +and λ. Here we show how the final result vary +by increasing these, for lower values the consistency is negligible and the results (Fig. 6b and Fig. 6g) visually look similar to the per-frame +processed output (Fig. 6b). For higher values we start observing artefacts due to ghosting and/or optimization (Fig. 6e and Fig. 6j). +interactive frame rates (> 10 FPS) for different parameter settings +(Tab. 3). +4.2. Parameter Settings +Initially, we tune the parameters of our consistency framework to- +wards achieving a low warping error (Tab. 5). We refer to this set- +ting as Ours-objective with the following parameter values k1 = +k2 = 0.3, α = 10 × 103, and λ = 0.7. However, we observed that +even though the warping error indicated a good temporal stabil- +ity, subjectively flickering and artefacts were noticeable. Unlike ex- +isting approaches, our framework allows for interactive parameter +adjustment. Thus, a parameter set that subjectively produces well- +stabilized results on a broad range of tasks and videos was obtained +experimentally. As our final version, we use the values of k1 = 0.3, +k2 = 0.5, α = 6.5 × 103, and λ = 2.0 to generate all the images in +the paper and the videos provided in the supplementary. We fur- +ther compare Ours-objective settings with our final version as part +of our user study to validate our parameter choices. The consistent +outputs obtained using the above parameter settings are compared +against state of the art approaches thereby showcasing its efficacy. +4.3. Consistent Outputs +We use videos from DAVIS [PPTM∗16] dataset and other open +source videos (taken from [Vid] and [Pex]) for comparison. For per- +frame stylization, we employ the following stylization techniques: +Fast NST [JAFF16], WCT [LFY∗17], and CycleGAN [ZPIE17]. +The results for the method of Lai et al. and Bonneel et al. on videos +taken from DAVIS [PPTM∗16] and Videvo ( [Vid]) are borrowed +from the results dataset provided by Lai et al. . For other videos +we employ the source code provided by the authors to generate +submitted to 200x. + +8 +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +132 +128 +127 +39 +43 +44 +0 +20 +40 +60 +80 +100 +120 +140 +Lai +Bonneel +Ours-obj. +Others +Ours +Figure 7: Statistics of the user study results on removal of temporal +flickering from per-frame stylized videos. For 19 participants and +9 different videos we compare our method against Bonneel et al. , +Lai et al. , and Ours-objective through a total of 171 randomized +A/B tests. +the results. We compare our consistent outputs with that of Bon- +neel et al. [BTS∗15] and Lai et al. [LHW∗18] in Fig. 8. Among +the three competing methods Bonneel et al. is the least effective in +preserving the underlying style for the final output (compare sec- +ond column with the fourth one in Fig. 8). Hyper-parameter tun- +ing in the above method (with only global consistency) can pro- +vide a certain degree of consistency-control. However, by employ- +ing both global and local consistency we achieve finer consistency- +control while being similar to the per-frame-processed result. For +the method of Lai et al. , we observe some color bleeding or dark- +ening in the output frames (compare second column with the third +one in Fig. 8). In comparison we are able to preserve the style, color +and textures, while being consistent (Fig. 7). +4.4. Optic Flow Results +We visualize optical flow on frames from the Sintel [BWSB12] +dataset in Fig. 9 and compare to state-of-the-art methods. All de- +picted methods have been fine-tuned on Sintel. We find that our +optimized method has more blurry motion boundaries and misses +to estimate certain details accurately (e.g., the hand in the first +row, however, PWCNet also fails at this), but still captures over- +all motion direction of objects correctly with a smooth flow field. +Fig. 10 shows results for real-world videos on the DAVIS dataset +[PTPC∗17] (no ground-truth flow available). We find that some +real-world image phenomena, such as complex/ambiguous occlu- +sions (e.g., bus behind tree) are not well-handled by state-of-the-art +methods like RAFT [TD20] or PWC-Net [SYLK18], and thus re- +sults are degraded for our optimized method as well. Besides the +stronger blurred motion boundaries, we find that our network gen- +erally performs well and is also robust for real-world videos. +5. Evaluation +5.1. Quantitative +Following Lai et al. [LHW∗18], we measure the similarity between +per-frame processed output and stabilized results, and the temporal +warping error between consecutive stabilized frames. +For the fomer, we report the similarity in form of the SSIM met- +ric in Tab. 4. We achieve significantly higher similarity scores than +the methods of Bonneel et al. [BTS∗15] and Lai et al. [LHW∗18]. +Following [BTS∗15] and [LHW∗18], we also measure the tempo- +ral warping error between a frame Vt and the warped consecutive +frame ˆVt+1, defined as: +Ewarp (Vt,Vt+1) = +1 +∑N +i=1 M(i) +t +N +∑ +i=1 +M(i) +t +���V (i) +t +− ˆV (i) +t+1 +��� +1 , +(9) +where Mt ∈ {0,1} is a non-occlusion mask [LHW∗18,RDB18], in- +dicating non-occluded regions. The warped frame ˆVt+1 is obtained +by calculating the optical flow (using GMA [JCL∗21]) between +frames Vt,Vt+1, and applying a backwards warping to frame Vt+1. +We compute Ewarp for every frame of a video and then average to +obtain the warping error of a video Ewarp(V). In Tab. 5 we report the +average warping error per dataset (see the supplementary for a per- +task breakdown). We find that the warping error is slightly higher +than that of Bonneel et al. [BTS∗15] and Lai et al. [LHW∗18]. +However, as Lai et al. [LHW∗18] notes, results with high temporal +stability (expressed by a low warping error) can also be achieved +via temporally smoothing the video, which can be seen in vari- +ous results of Bonneel et al. [BTS∗15]. Our qualitative results in +form of a user study Sec. 5.2 further substantiate the divide between +warping error (as a stability metric) and perceived stability. +5.2. Qualitative +For qualitative evaluation we perform a subjective user study where +we ask participants to compare the temporally-consistent result ob- +tained using our method with that of Lai et al. , Bonneel et al. , +and Ours-objective – a different parameter setting of ours. We use +9 different videos for this purpose: 3 from DAVIS [PPTM∗16], 3 +from Videvo [Vid], and 3 from Pexels [Pex] datasets respectively. +For each of the above video we stylize them using either the Fast +NST [JAFF16] (in the styles of udnie, rain-princess, and mosaic) +or WCT [LFY∗17] (in the styles of wave and antimono) or Cycle- +GAN (in the styles of photo2vangogh and photo2ukiyoe). For each +sample, we show the input video and its per-frame stylized version +on the top row of user-study interface for inference. In the bottom +row we show two different version of the temporally stabilized out- +put where one of them is ours. We ask the participants to select +the output which best preserves: (i) temporally consistency and (ii) +similarity with the per-frame processed video. For 9 videos and 3 +other competing methods each user sees a total of 27 blind A/B +tests which are shown in a randomized order to each participant. +In total, 19 persons (3 female and 16 male) within the ages of 22 +to 43 years participated in the study. Fig. 7 shows that our method +surpasses all others by a large margin. It was interesting to observe +that for certain cases the method of Bonneel et al. which degrades +the processed style significantly was still preferred by users over +others due to its high consistency quality. +submitted to 200x. + +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +9 +(a) Input +(b) Processed +(c) Ours +(d) Lai et al. [LHW∗18] +(e) Bonneel et al. [BTS∗15] +Figure 8: Comparing our results with Lai et al. [LHW∗18] and Bonneel et al. [BTS∗15] for three different video sequences: Cow (top two +rows), Farming (mid two rows), and Woman (last two rows). Note how the consistent output for Lai et al. and Bonneel et al. look different +from the corresponding per-frame processed results. +5.3. Using other Optical Flows +We also tested other optical flow methods within our pipeline +which were either faster [KTDVG16] or more accurate [TD20]. +For the fast optical method by Kroeger et al. [KTDVG16](DIS) +the final output is less consistent than ours in both objective and +subjective metrics. Using DIS for our stabilization, the average +warp-error over DAVIS is 0.05 (vs. 0.046 ours) and perceptual- +similarity with the per-frame processed result is 0.9 in SSIM terms +(vs. 0.923 ours). Visually, DIS-stabilized results show significantly +more flickering, validating our design choice for the optical-flow. +A much more accurate optic flow is given by the method of +Teed et al. [TD20] (RAFT) at the cost of slow computation. The +stabilized results obtained using RAFT look visually indistinguish- +able to the one obtained using our flow; the average warp-error over +DAVIS is 0.045, the perceptual-similarity is 0.923. +6. Discussion +Our approach takes a video pair as an input: (i) the original and +(ii) its per-frame stylized version. We assume that the stylization +is based on the input image-gradients and appears as variations in +the form of colors and/or textures. Thereby, we employ the origi- +nal video as a guide for enforcing consistency. However, for text- +guided generative arts such as recent diffusion model-based ap- +proaches [RDN∗22, RBL∗22] the stylized frames are often only +weakly correlated with the original input, we cannot handle such +cases. +For the evaluation we mainly use CNN-based stylization tech- +niques. However our approach can also handle classical stylization +approaches [KCWI13], we show few such examples in the supple- +mentary. Our local-consistency component comprising of convex +combination of temporal neighbors can be seen as crude form of +submitted to 200x. + +10 +S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +(a) Frame Overlay +(b) Ground-truth +(c) RAFT [TD20] +(d) PWC-Net [SYLK18] +(e) Ours +Figure 9: Optical flow estimated using the synthetic Sintel dataset [BWSB12]. +(a) Frame Overlay +(b) RAFT [TD20] +(c) PWC-Net [SYLK18] +(d) Ours +Figure 10: Optical flow estimated for the real-world dataset DAVIS [PTPC∗17]. +local temporal denoising. Previously it has been shown that tem- +poral denoising is effective in enforcing consistency [SST∗19]. We +conjecture that efficient temporal-denoising combined with flow- +based warping can further improve temporal stabilization not only +for stylization but also for other tasks. +We start with the assumption that temporal flickering is not com- +pletely undesirable for the task of stylization and thus we pro- +vide interactive consistency control. However, during the subjec- +tive user study we observed that participants had different toler- +ance levels for flickering in the foreground as compared to that in +the background. As part of future work, one can use depth-based +or saliency-based masks to vary the consistency control parameters +spatially for a more visually pleasing result. +Limitation: Our approach tends to have ghosting artifacts for +fast moving objects where the object motion between consecutive +frames is large (Fig. 11). The above can be reduced by reducing +the value of +� +wp +� +, however such a reduction also reduces consis- +(a) +� +wp +� += 0.5 +(b) +� +wp +� += 0.1 +Figure 11: The ghosting artifacts on the rear wheel of the scooter +is significant in the final output for +� +wp +� += 0.5, however it reduces +significantly for +� +wp +� += 0.1. +tency in the final output. We argue that since we provide interactive +control of parameters the above trade off between artifacts vs. con- +sistency will not hinder its usability significantly. +submitted to 200x. + +EPE: 0.000EPE: 0.000EPE: 0.629EPE: 0.171EPE: 1.283EPE: 0.339EPE: 2.091EPE: 0.520S. Shekhar et al. / Interactive Control over Temporal Consistency while Stylizing Video Streams +11 +Table 4: Quantitative evaluation on perceptual distance using SSIM (higher = more similar to per-frame processed result). +DAVIS +VIDEVO +Task +[BTS∗15] +[LHW∗18] +Ours +[BTS∗15] +[LHW∗18] +Ours +CycleGAN/photo2ukiyoe [ZPIE17] +0.693 +0.781 +0.978 +0.626 +0.743 +0.980 +CycleGAN/photo2vangogh [ZPIE17] +0.707 +0.792 +0.961 +0.679 +0.789 +0.965 +fast-neural-style/rain-princess [JAFF16] +0.553 +0.799 +0.921 +0.491 +0.796 +0.920 +fast-neural-style/udnie [JAFF16] +0.597 +0.785 +0.956 +0.579 +0.747 +0.959 +WCT/antimonocromatismo [LFY∗17] +0.389 +0.811 +0.915 +0.388 +0.761 +0.914 +WCT/asheville [LFY∗17] +0.329 +0.801 +0.904 +0.348 +0.771 +0.901 +WCT/candy [LFY∗17] +0.289 +0.763 +0.882 +0.310 +0.738 +0.885 +WCT/feathers [LFY∗17] +0.418 +0.863 +0.891 +0.415 +0.848 +0.888 +WCT/sketch [LFY∗17] +0.370 +0.845 +0.923 +0.370 +0.833 +0.922 +WCT/wave [LFY∗17] +0.358 +0.700 +0.902 +0.352 +0.637 +0.899 +Average +0.470 +0.794 +0.923 +0.456 +0.766 +0.923 +Table 5: Flow warping error average over tasks shown in Tab. 4. +A per-task breakdown is shown in the supplementary. Note that +the slightly higher warping error (lower is better) of our method is +subjectively not noticeable as we show in a user study. +Dataset +Vp +[BTS∗15] +[LHW∗18] +Ours +DAVIS +0.056 +0.034 +0.040 +0.046 +VIDEVO +0.051 +0.036 +0.036 +0.042 +7. Conclusions +We propose an approach that makes per-frame stylized videos tem- +porally coherent irrespective of the underlying stylization applied +on individual frames. At this, we introduce a novel temporal con- +sistency prior which combines both local and global consistency +aspects. We maintain similarity with the per-frame processed result +by minimizing the difference in the gradient-domain. Unlike previ- +ous approaches we provide interactive consistency control by com- +puting optic-flow on the incoming video stream with only sufficient +accuracy but at high speed. Fats optic-flow inference is achieved +by developing a lightweight flow network architecture based on +PWC-Net. The entire optimization solving is GPU-based and runs +at real-time frame-rates for HD resolution. We showcase that our +temporally consistent output is preferred over the output of com- +peting methods by conducting a user study. As part of future work +we would like to employ learning-based temporal denoising to fur- +ther improve quality of results. Moreover, we would like to ex- +plore the usage of depth-based and saliency-based masks to spa- +tially vary consistency parameters according to perceptual princi- +ples. We hope that our design paradigm of interactive consistency +control will potentially make per-frame video stylization more user +friendly. +References +[BCCZ08] +BHAT P., CURLESS B., COHEN M., ZITNICK C. L.: Fourier +analysis of the 2d screened poisson equation for gradient domain prob- +lems. 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C., GOOCH B.: +Real-time +video abstraction. +ACM Trans. Graph. 25, 3 (jul 2006), 1221–1226. +doi:10.1145/1141911.1142018. 3 +[YCC17] +YAO C.-H., CHANG C.-Y., CHIEN S.-Y.: +Occlusion- +aware video temporal consistency. +In Proceedings of the 25th +ACM International Conference on Multimedia (New York, NY, USA, +2017), MM ’17, Association for Computing Machinery, p. 777–785. +URL: https://doi.org/10.1145/3123266.3123363, doi: +10.1145/3123266.3123363. 1, 2, 3 +[YR19] +YANG +G., +RAMANAN +D.: +Volumetric Correspondence +Networks for Optical Flow. +Curran Associates Inc., Red Hook, NY, +USA, 2019. URL: https://dl.acm.org/doi/pdf/10.5555/ +3454287.3454359. 5, 6 +[ZGWX05] +ZHU S.-C., GUO C.-E., WANG Y., XU Z.: What are tex- +tons? International Journal of Computer Vision 62, 1 (2005), 121–143. +5 +[ZPIE17] +ZHU J.-Y., PARK T., ISOLA P., EFROS A. A.: +Unpaired +image-to-image translation using cycle-consistent adversarial networks. +In 2017 IEEE International Conference on Computer Vision (ICCV) +(2017), pp. 2242–2251. doi:10.1109/ICCV.2017.244. 7, 11 +submitted to 200x. + diff --git a/5dAyT4oBgHgl3EQf2fkR/content/tmp_files/load_file.txt b/5dAyT4oBgHgl3EQf2fkR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..172aa45f5127256d629e31ab9f8d9da1411ea86f --- /dev/null +++ b/5dAyT4oBgHgl3EQf2fkR/content/tmp_files/load_file.txt @@ -0,0 +1,1439 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf,len=1438 +page_content='’0x / N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (Guest Editors) Volume 0 (200x), Number 0 Interactive Control over Temporal Consistency while Stylizing Video Streams Sumit Shekhar1∗ , Max Reimann1∗ , Moritz Hilscher1, Amir Semmo1,2 , Jürgen Döllner1, and Matthias Trapp1 1Hasso Plattner Institute for Digital Engineering, University of Potsdam, Germany 2Digital Masterpieces GmbH, Germany (“*” denotes equal contribution) Abstract With the advent of Neural Style Transfer (NST), stylizing an image has become quite popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A convenient way for extending stylization techniques to videos is by applying them on a per-frame basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, such per-frame application usually lacks temporal consistency expressed by undesirable flickering artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Most of the existing approaches for enforcing temporal consistency suffers from one or more of the following drawbacks: They (1) are only suitable for a limited range of techniques, (2) typically do not support live processing as they require the complete video as input, (3) cannot provide consistency for the task of stylization, or (4) do not provide interactive consistency-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Note that existing consistent video-filtering approaches aim to completely remove flickering artifacts and thus do not respect any specific consistency-control aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For stylization tasks, however, consistency-control is an essential requirement where a certain amount of flickering can add to the artistic look and feel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, making this control interactive is paramount from a usability perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To achieve the above requirements, we propose an approach that can stylize video streams while providing interactive consistency-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For achieving interactive performance, we develop a lite optical-flow network that operates at 80 Frames per second (FPS) on desktop systems with sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Further, we employ an adaptive combination of local and global consistent features and enable interactive selection between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' By objective and subjective evaluation, we show that our method is superior to state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' CCS Concepts Computing methodologies ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', Image-based rendering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Non-photorealistic rendering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Image processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Introduction For thousands of years, paintings have served as a tool for vi- sual communication and expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, it was not until the late 20th century that computers were used to simulate paint- ings [Hae90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In the course of following decades, the field of artistic stylization [KCWI13] has significantly developed and extended by NSTs [SID17,JYF∗20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Even though a large number of image styl- ization techniques exist, extending these to video remains challeng- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A major obstacle in this regard is the enforcement of temporal coherence between stylized video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' With the proliferation of video streaming applications, stylizing video streams has become popular, however, the requirements of low-latency processing add additional challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Most of the existing methods, to address the above, can be classified into one of the following four categories: Style Specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A common approach is to develop a specific method for a particular artistic style and exploit its characteris- tics for temporal coherency [BNTS07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Such methods work ef- fectively for the specific target style, however, do not generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Many of these specialized approaches have been discussed by Bénard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTC13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Coherent Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Another class of techniques adopt and transform a generic, temporally-coherent noise function to yield a visually plausible stylized output [BLV∗10, KP11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Compared to target- based coherence enforcement [BNTS07], these are applicable to a wider range of techniques but are limited for scenarios with rapid temporal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Stylization by Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' More recently, authors have adopted a stylization-by-example approach to support a wide range of styl- ization techniques [BCK∗13, JST∗19, TFK∗20, FKL∗21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' How- ever, this approach requires the paring of the complete video and keyframe marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thus, by design it is not applicable to video streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Consistent Video Filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' One can also enable stylization of video streams using consistent video filtering techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Exist- ing approaches are either not well-suited for Image-based Artis- tic Rendering (IB-AR) [BTS∗15,YCC17] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1) or do not pro- vide interactive consistency control [LHW∗18,TDKP21], which submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='00750v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='GR] 2 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams Table 1: Comparing existing consistent video filtering methods with ours with regards to consistency-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Here, the color green denotes the aspect which is favourable to interactive consistency-control while the color red denotes otherwise (“NA” stands for Not-Applicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [YCC17] Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [SST∗19] Thiomonier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [TDKP21] Ours Requires pre-processing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' No Yes No Yes No No Provides consistency-control at inference time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Yes No No Yes No Yes Is the consistency-control interactive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' No NA NA Yes NA Yes (a) Input (b) Processed (c) Ours (d) Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] (e) Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] Figure 1: For the top-row: first two columns depicts (a) input and (b) processed result for frame-24, column three to five depict the correspond- ing consistent output using (c) Ours (d) Lai’s, and (e) Bonneel’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For the mid-row: depict the corresponding results for frame-80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For the bottom-row: we show the Temporal Slice Image (TSI) for the entire video sequence depicting long-term temporal similarity with the per-frame processed output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Note, that our method is able to preserve the look and feel of the per-frame processed result in comparison to the method of Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' which suffers from color bleeding artifacts while the stylized textures are lost for the output of Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Please see the supplementary material for video results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' is an essential requirement for artistic rendering [FLJ∗14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Cur- rently, the only method that provides interactive consistency- control is limited to offline processing and requires pre- processing [SST∗19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We aim to develop a temporal-consistency enforcement ap- proach for artistic stylization techniques that provides (1) interac- tive consistency-control and (2) online processing to facilitate the application to video streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A determining factor towards the slow performance of ex- isting online and interactive consistent video filtering tech- nique [BTS∗15] is the costly step of optic-flow computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Pre- vious works using learning-based methods are able to achieve a considerable accuracy for optic-flow estimation [TD20, JCL∗21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, we argue that such a high accuracy is not particularly necessary to enforce temporal consistency for artistic stylization tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To validate our conjecture, we conduct a user study, wherein the participants prefer the final consistent video output generated using our flow network as compared to that being obtained us- ing State-of-the-art (SOTA) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We define artistic styl- ization as the adaptation of colors, textures, and strokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' While our approach is effective for most image-based stylization tech- niques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', NSTs, algorithmic filtering), it is not able to han- dle significant shape or content inconsistencies between frames in- troduced by semantically-driven image synthesis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', image-to- image diffusion-based models [RBL∗22]), as flow-based warping is insufficient to enforce consistency in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In contrast to accuracy, little attention has been paid to improve the run-time performance of optic-flow estimation, but which is essential for online-interactive editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To this end, we develop a lite optic-flow neural network that runs at a high-speed (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 80 FPS on mid-tier desktop GPUs) while maintaining sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The compact network is also deployable on mobile de- vices (iPhones and iPads) where it runs at interactive frame rates (24 FPS on iPad Pro 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We use the optic-flow output from the above network to enforce warping-based consistency at interactive frame rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, we construct an adaptive consistency prior which allows for global and local temporal-consistency control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To summarize we present the following contributions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A novel approach for making per per-frame stylized videos tem- porally consistent via adaptive combination of local and global consistency features which allows for interactive consistency- control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A lite optic-flow network, to achieve interactive performance, that runs at 80 FPS on a mid-tier desktop PC and at 24 FPS on a mobile device while achieving reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Background & Related Work Consistent Video Filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LWA∗12] propose a solution to enforce temporal consistency for a large-class of optimization-based problems via iterative filtering along the mo- tion path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [DBZY15] address the problem of temporal inconsistency for enhancement algorithms by dividing individual video frames into multiple regions and performing a region-based spatio-temporal optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] was the submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams 3 𝐼𝑡−1 𝐼𝑡−1 𝐼𝑡𝐼𝑡 𝐼𝑡+1 𝐼𝑡+1 𝑃𝑡 𝑃𝑡 𝑃𝑡−1 𝑃𝑡−1 𝑃𝑡+1 𝑃𝑡+1 𝑂𝑡−1 𝑂𝑡−1 𝑤𝑝 𝑤𝑛 ϴ𝑙 ϴ𝑙 Linear combination ϴ𝑔 ϴ𝑔 Linear combination 𝐴𝑡 𝐴𝑡 Optimization Solving 𝑂𝑡 𝑂𝑡 𝑤𝑝 Use 𝑤𝑝 and 𝑤𝑛 for combining 1 2 3 4 5 Estimate 𝑤𝑐 for optimization Figure 2: Schematic overview of our approach: (1) We start by calculating the warping weights wp and wn using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (2) The computed weights are used to linearly combine Pt, Pt−1, and Pt+1 to obtain the locally consistent image Lt, see Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (3) To obtain the globally consistent version Gt we warp the output at previous time instance Ot−1 as depicted in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (4) The local and global consistent images, Lt and Gt, are linearly combined to obtain a temporally smooth version At, see Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (5) To include high-frequency details from the per-frame processed result, At and Pt is adaptively combined via the optimization in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1 using the weights wc (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 7) to obtain the final result Ot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' first to present a generalized approach for consistent video filter- ing which is agnostic to the type of filtering applied on individ- ual video-frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The method combines gradient-based character- istics of the per-frame processed result with the warped version of the previous-frame output using a gradient-domain based optimiza- tion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [YCC17] propose a similar approach how- ever considers multiple key-frames for warping-based consistency to avoid problems due to occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Both of the approaches assume that the gradient of the processed video is similar to that of the input video and thus cannot handle artistic rendering tasks where new gradients resembling brush strokes are generated as part of the stylization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, due to slow optic-flow computa- tion they are non-interactive in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [SST∗19] employs a similar formulation as Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' , with the dif- ference of using a temporally denoised version of the current- frame for consistency guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, the temporal denois- ing requires the complete video as input making the method of- fline in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] propose the first learning- based technique in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The authors employ perceptual loss to enforce similarity with the processed frames and for con- sistency make use of short-term and long-term temporal losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thimonier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [TDKP21] employ a ping-pong loss and a cor- responding training procedure for temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Both the learning based technique are faster than their optimization-based counterpart since they do not perform optic-flow computation at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, these learning based techniques do not allow to control the degree of consistency in the final output which is vital for the task of stylization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thus, the above discussed meth- ods are either non-interactive/offline or do not provide any con- sistency control at inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our approach addresses these limitations (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Optic Flow for Consistent Filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Both Booneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' and Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' use the PatchMatch algorithm [BSFG09] for flow-based warping, however, the slow performance of PatchMatch makes them non-interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' use FlowNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 [IMS∗17] for flow- based warping to design their short-term and long-term temporal consistency losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' FlowNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 is on par with the quality of state- of-the-art classical methods, however, due to large number of pa- rameters and operations, achieves only interactive frame rates even on high-end desktop Graphical Processing Units (GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' An im- proved compact optic-flow Convolutional Neural Network (CNN) is proposed by Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [SYLK18] – PWC-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' It combines coarse-to-fine estimation with pyramidal image features, correla- tion, warping, and CNN-based estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Furthermore, a refine- ment CNN is stacked at the end to improve the final flow estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' PWC-Net is orders of magnitude smaller than FlowNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0, runs at real-time frame rates using desktop GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LZH∗20] employ their approach to train a similar architecture in an un- supervised setting and achieve reasonable accuracy – ARFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' LiteFlowNet and its successor LiteFlowNet2, both proposed by Hui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [HTL18, HTL20], have similar compact architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Further improvement in accuracy is achieved by models using iter- ative refinement, such as RAFT [TD20] and transformer modules such as GMA [JCL∗21], however they heavily trade runtime for ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Based on a runtime-accuracy comparison (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2), we select PWC-Net as a base network to develop a "Lite" flow network with improved performance for interactive consistent filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Temporal Consistency for Video Stylization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Litwinow- icz [Lit97] describes a technique to apply an impressionist effect on images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For enforcing temporal coherence, optical flow was used to transform the brush strokes from one frame to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Winnemöller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [WOG06] develop a real-time video and image abstraction framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The authors employ soft quan- tization that spreads over a larger area, thus significantly reducing temporal incoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Bousseau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BNTS07] advects texture in forward and backward direction using optical flow for coherent water-colorization of videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Numerous such specialized video- based approaches have been discussed by Bénard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTC13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The above classical IB-AR techniques approximate rendering primitives by modifying traditional image filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Most often, they use low-level image features for modeling and fail to model structures resembling a particular style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Recently, deep CNNs were successfully used to transfer high-level style attributes from a painting onto a given image [GEB16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Various methods have been proposed to extend the above for videos [HWL∗17,CLY∗17, GJAF17,RDB18,LLKY19,PP19,DTD∗21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Ruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [RDB18] submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams propose novel initialization technique and loss functions for consistent stylized output even in cases with large motion and strong occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The methods of Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [GJAF17], Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [CLY∗17], and Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [HWL∗17] enforce con- sistency via certain formulation of temporal loss and use optical flow based warping only during the training phase thus achieving fast performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Puy and Pérez [PP19] develop a flexible deep CNN for controllable artistic style transfer that allows for addition of a temporal regularizer at testing time to remove the flickering artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The above method comes closest in terms of providing some consistency control at test time for NST-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, they cannot handle classical stylization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Stylization by example [BCK∗13, JST∗19, TFK∗20, FKL∗21] caters to both (classical and neural) paradigms via priors involving keyframe-based warping but can only be applied as an offline process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We aim to propose a generic solution which is agnostic to the type of stylization and provides online performance and interactive consistency-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Temporal Consistency Enforcement Given an input video stream .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='It−1, It, It+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' and its per-frame processed version .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='Pt−1, Pt, Pt+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', we seek to find a tempo- rally consistent output .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='Ot−1, Ot, Ot+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='. Our method is ag- nostic to the stylization technique f applied to each frame, where Pt = f(It).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, it is necessary for f to not introduce signifi- cant shape or content inconsistencies between consecutive frames, as the changes in the stylized frames should correspond to the op- tical flow (calculated based on the content).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We initialize the con- sistent output for the first frame as its per-frame processed result i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', O1 = P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To obtain the output for subsequent frames (Ot at any given instance t) we require only a snippet of input (It−1,It,It+1) and processed streams (Pt−1,Pt,Pt+1), and the consistent output at the previous instance Ot−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For enforcing consistency, we solve the following gradient-domain optimization scheme: E(Ot) = � Ω � ||∇Ot −∇Pt||2 � �� � data + wc||Ot −At||2 � �� � smoothness � dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (1) where Ω represents the image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The data term in this opti- mization enforces similarity with the per-frame processed result Pt in the gradient-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thus, high-frequency details are taken from Pt and the smoothness term enforces temporal-consistency where low-frequency content is taken from the image At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The optimiza- tion formulation in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1 is commonly known as screened Pois- son equation and has been successfully employed for various image Table 2: Constituent elements of smoothness term in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1 for different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Here, ws and Td refers to saliency-based weights and temporally-denoised image respectively, introduced by Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Method Weight Consistent Image Ours wc At Boneell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] wp Γ(Ot−1) Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [SST∗19] ws Td editing applications [BCCZ08,BZCC10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In the context of consis- tent video filtering, it was first used by Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] followed by Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [SST∗19] (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, our nov- elty is the way in which we construct our smoothness term which, unlike previous approaches, considers both global and local consis- tency aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our novel smoothness term is able to better preserve the color and textures in the stylized output while providing both short-term and long-term temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Local Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For enforcing temporal consistency at a local level, we use optic-flow to warp neighboring per-frame processed results to the current time instance t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' This is perfomred by comput- ing an adaptive combination of (1) warped previous per-frame pro- cessed image Γ(Pt−1), (2) warped next per-frame processed image Γ(Pt+1), and (3) the current per-frame processed image Pt, where Γ is the warping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' By including both backward and for- ward warping in our formulation, we are able to significantly re- duce artefacts due to occlusion and flow inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The linear combination of (1), (2), and (3) gives us a locally consistent ver- sion Lt where, Lt = (1−(wp+wn))·Pt + wp·Γ(Pt−1) + wn·Γ(Pt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (2) The weights wp and wn capture the inaccuracies in the warping of previous and next frames respectively and are defined as follows: wp = exp � −α||It −Γ(It−1)||2� and wn = exp � −α||It −Γ(It+1)||2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (3) In order to also incorporate contribution from Pt, we clamp the weights wp and wn as follows: � wp � = k1 and � wn � = k2, where k1 and k2 are two constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The locally consistent image sequence given by Lt has improved temporal consistency over the per-frame processed output, however, it still has visible flickering artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thus, the reduction in flickering due to warping of only one tempo- ral neighbor is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To further improve consistency, one can warp more neighboring frames around the current time instance t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' As we increase the temporal window-size for such an adaptive combination it has a denoising effect leading to further reduction in flickering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The temporal denoising for enforcing consistency, per- formed by Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [SST∗19] can be considered as an specific example of the above scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, for interactive stylization warping more frames to the current instance is not feasible due to time constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, in case of video streams we do not have frames to warp from the forward temporal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Global Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In order to overcome this limitation, exist- ing approaches [BTS∗15, LHW∗18] adopt a global approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For global consistency, one can consider the previous stabilized output Ot−1 and enforce similarity with its warped version Gt where, Gt = Γ(Ot−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (4) To enforce only global temporal smoothness, we replace At with Gt in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Further, in order to compensate for optic-flow inaccura- cies, the smoothness term is weighted using wp (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', wc = wp) in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, considering only global consistency for flicker reduction leads to loss of stylization and local temporal varia- tions in the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, in this case any warping-error (due to flow-inaccuracies) or noise (as part of stylization process) submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams 5 keeps getting propagated to future frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Due to the above fac- tors, such an approach only gives plausible results where the gra- dients of the original video are similar to the gradients of the pro- cessed video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The above does not hold for the task of stylization where stylistic elements such as brush strokes, textures or stroke textons [ZGWX05], in general, can vary largely between frames even for small changes in gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Combining Global and Local Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For preserving local temporal variations (in terms of look and feel) while significantly reducing the flickering artifacts, we linearly combine globally and locally consistent images Gt and Lt respectively, At = wp·Gt + (1−wp)·Lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (5) We use the adaptively combined image At as our reference for consistency while enforcing temporal smoothness in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The � wp � can be increased to increase the influence of global-temporal smoothness and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Further, the influence of the smoothness term is controlled by per-pixel consistency weights wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We would like to invoke the smoothness term only when the warping accuracy is sufficiently high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To this end, we construct a warped version of the input image similar to Lt as, AIt = (1−(wp+wn))·It + wp·Γ(It−1) + wn·Γ(It+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (6) Only when the input image It is similar to AIt, the smoothness term is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To measure this similarity, we use the weight wc, wc = λ·exp � −α||It −AIt|| 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (7) The parameter λ is used to scale up or down the weight wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Consistency Control Modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The above adaptive combination of local and global consistency provides two different ways of consistency-control in the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' By increasing � wp � we can increase the proportion of global consistency in the adaptively com- bined image At and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' On the other hand the optimization parameter λ dictates how close the output Ot will be to the adap- tively combined image At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thus, the level of consistency in the final output can be controlled in two different ways: (1) by set- ting up the limit of parameter wp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', � wp � or (2) by scaling the weight parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For lower values of � wp � (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6b), the consis- tency enforced is negligible and the final result resembles the per- frame processed output (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, for higher values we start observing noisy ghosting artefacts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The lower values of � wp � translates to using only global consistency which results in accumulation of flow inaccuracies visualized as ghosting artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Similarly, for lower values of λ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6g), the final result is visually similar to the per-frame processed output (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, for higher values the optimization becomes unstable resulting in noisy optimization-based artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Optimization Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The energy terms in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1 are smooth and convex in nature, which allows a straightforward energy minimiza- tion with respect to Ot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To this end, we employ an iterative ap- proach thus avoiding – storage of a large matrix in memory and further estimating its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, an iterative approach al- lows us to stop the solver once we have achieved visually plau- sible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' An iterative update Otk+1 is obtained by employing 43 (a) (b) (c) Refinement Flow Estimation Modules Feature Extraction Input Frames Output Flow Figure 3: Modification of the PWC-Net [SYLK18] architecture for real-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We apply following network compression steps: (a) Replace DenseNet connections with light ones, (b) Re- duce the number of flow estimators, and (c) Replace dense connec- tions in the refinement module with separable convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Stochastic Gradient Descent (SGD) with momentum [Qia99], Otk+1 = Otk −η∇E(Otk)+κ(Otk −Otk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (8) where η and κ are the step size parameters, ∇E is the energy gradi- ent with respect to Ot, and k is the iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For most of our experiments, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='15 and κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2 yield plausible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We con- sider the trade-off between performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' accuracy as a stopping criteria and do not compute energy residue for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To ob- tain a consistent output while having interactive performance, we empirically determine 150 iterations to be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The optimiza- tion is stable for the given parameter settings and early stopping is only employed for computational gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' An integral aspect common to both our local and global consis- tency is the warping function Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Apart from the number of solver iterations, for interactive performance the above warping should also happen at a fast rate – which in turn necessitates fast optic- flow estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Lite Optic-Flow Network We aim to obtain a flow network capable of running at high-speed on consumer hardware with reasonable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To this end, we start by selecting an existing CNN-based optical flow estimation technique, based on accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' run-time analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' After the se- lection of a base network, we perform further optimization steps to increase the performance as outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Base Network Selection for Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4, we com- pare several well-known optical methods to find a base network candidate that best matches our runtime/accuracy requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We employ the following models for this: FlowNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 [IMS∗17], SpyNet [RB17], LiteFlowNet2 [HTL20], PWCNet [SYLK18], ARFlow [LZH∗20], VCN [YR19], RAFT [TD20] and finally GMA [JCL∗21] (state-of-the-art in terms of EPE-based accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our experiments are carried out on a Nvidia RTX 2070 GPU, which we deem to be a good representative of a current mid-to submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams 0px 2px 4px 6px 8px 0 10 20 30 40 flownet2 spynet pwcnet arflow liteflownet2 vcn raft gma Sintelfinal-test EPE (lower=better) FPS (higher=better) Figure 4: Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' run-time performance of existing methods measured on Sintel Final (Test set) [BWSB12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The Endpoint Er- ror (EPE) metric measures Euclidean distance (in pixels) between ground-truth and predicted optical flow vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' higher-end consumer GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Under a constraint of interactive perfor- mance on consumer hardware, LiteFlowNet2 [HTL20] and PWC- Net [SYLK18] offer the best trade-off between run-time perfor- mance and accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' LiteFlowNet2 [HTL20] is already an optimized version of FlowNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 [IMS∗17], in comparison PWC- Net [SYLK18] has more potential for optimization/compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, recently it has been shown that PWC-Net can achieve similar accuracy to RAFT when trained on a large-scale synthetic dataset [SVH∗21] and that PWC-Net achieves favourable trade-offs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' other state-of-the-art methods when selecting for runtime per- formance or higher image resolutions [SHR∗22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Hence, we select PWC-Net for further compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Optimized Network Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We start with the base archi- tecture of PWC-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' As the first compression step we reduce the computationally expensive DenseNet [HLvdMW17] connections in the flow estimators to retain connections only in the last two layers ("-light" in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Similar to LiteFlowNet2 [HTL20], we remove the fifth flow estimator – operating on the highest resolu- tion – as it heavily trades off run-time for only marginal increase in accuracy (compare "4light" vs "5light" in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We replace the standard convolutions in the refinement by depthwise separable convolutions [HZC∗17] ("-sepref" in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, we also explore reducing the number of channels [HZC∗17], but find that reducing channels results in a worse trade-off as compared to other optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For training, we follow the original PWC- Net [SYLK18] schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, we find that weight- ing the multi-scale losses equally, instead of exponen- tially [SYLK18, HTL18, HTL20, YR19], improves accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For our experiments on the desktop system, we use PyTorch [PGM∗19] and take inspiration from the implementation by Niklaus [Nik18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Similar to PWC-Net [SYLK18], we train our mobile architecture on the training dataset schedule FlyingChairs [FDI∗15] → Fly- ingThings3D [MIH∗16]→ Sintel [BWSB12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In the supplementary material, we provide training settings for each stage in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We Table 3: Runtime performance in milliseconds per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We mea- sure the total processing time (without disk IO) and the individual stages for a mid-tier GPU (Nvidia GTX 1080Ti) and a higher-end GPU (Nvidia RTX 3090), results are averaged over 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Task Optical flow Stabilization Total ↓ Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / GPU 1080Ti 3090 1080Ti 3090 1080Ti 3090 1920×1080 px 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='7 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='7 1280×720 px 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='8 640×480 px 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5 employ a multi-scale loss [SYLK18] applied to each flow estimator and optimize using the AdamW optimizer [LH19] with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='09, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='99, and l2 weight regularization with trade-off γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Furthermore, extensive dataset augmentation is applied to prevent model overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We refer to the supplementary material for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our Final Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We analyze various optimization options and chose “our-4light-sepref” as our final model for desktop systems as it provides the best trade-off between accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5a, our method improves run-time performance of PWC-Net from 30 FPS to 85 FPS – a speed-up of factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For Sintel training data the accuracy drops by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5px in EPE terms, however for test data the drop in accuracy is significant where the fi- nal EPE is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Nevertheless, the accuracy is sufficient enough for enforcing warping-based consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' To validate our design deci- sions, we conduct an extensive ablation study in which we vary the architectural and training choices – please see the supplementary for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Furthermore, we tune our architecture for optical flow calculation on mobile devices using channel pruning and quantiza- tion, which we also detail in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Here, we improve run-time performance from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='8 FPS to 24 FPS (iPad Pro 2020), and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5 FPS to 13 FPS (iPad Air) – an improvement of factor 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Next to showing the general applicability of optical flow CNNs on mobile devices, this demonstrates that real-time on-device sta- bilization of videos using our presented approach will become fea- sible with a further moderate increase in mobile GPU computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A fast optic-flow based warping enables our framework to interactively control the degree of consistency and generate visu- ally plausible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Experimental Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Implementation Details All our experiments were performed on an consumer PC with an AMD Ryzen 1920X 12-Core CPU, 48 GB of RAM, and a Nvidia GTX 1080Ti and RTX 3090 graphics cards with VRAMs of 11 GB and 24 GB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We implement a real-time video- consistency framework in C++, using ONNXRuntime for cross- platform acceleration of our lite optical-flow network and imple- ment the stabilization code using Nvidia CUDA (v11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3, we measure the runtime performance of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We find that an incoming stream of frames can be stabilized at real-time perfor- mance for VGA resolution even on low- and mid-tier GPUs and higher-tier GPUs (such as a RTX 3090) can stabilize HD at com- mon video frame rates (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 24 FPS) and full-HD resolutions at submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams 7 2px 3px 4px 5px 0 20 40 60 80 100 flownet2 liteflownet2 pwcnet our-5light our-5light-5sep our-5light-2sep our-4light-1sep our-5light-c50 our-5light-c75 our-4light our-4light-sepref (a) Sintelfinal-train EPE (lower=better) FPS (higher=better) Modifier Description Default Nlight N light [LZH∗20] flow esti- mators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5 dense [SYLK18] Msep last M flow estimators use depthwise separable convo- lutions [HZC∗17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' standard convs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' sepref refinement uses depth- wise separable convolu- tions [HZC∗17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' standard convs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' cP use P% of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 100% (b) Legend of our CNN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Figure 5: Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' run-time performance of our CNN variants on desktop, measured on Sintel Final (Train) [BWSB12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Optimization steps that lead to significant improvement in run-time are connected by a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our architectural modifications to PWC-Net [SYLK18] are detailed on the right, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', our-4light-sepref denotes a 4 light flow estimators and refinement using depthwise separable convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (a) Input (b) � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3 (c) � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5 (d) � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='7 (e) � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='9 (f) Processed (g) λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1 (h) λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 (i) λ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 (j) λ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='06 Figure 6: The level of consistency in the final output can be controlled via parameters � wp � and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Here we show how the final result vary by increasing these, for lower values the consistency is negligible and the results (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6g) visually look similar to the per-frame processed output (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For higher values we start observing artefacts due to ghosting and/or optimization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6e and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' interactive frame rates (> 10 FPS) for different parameter settings (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Parameter Settings Initially, we tune the parameters of our consistency framework to- wards achieving a low warping error (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We refer to this set- ting as Ours-objective with the following parameter values k1 = k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3, α = 10 × 103, and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, we observed that even though the warping error indicated a good temporal stabil- ity, subjectively flickering and artefacts were noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Unlike ex- isting approaches, our framework allows for interactive parameter adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thus, a parameter set that subjectively produces well- stabilized results on a broad range of tasks and videos was obtained experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' As our final version, we use the values of k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3, k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5, α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5 × 103, and λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='0 to generate all the images in the paper and the videos provided in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We fur- ther compare Ours-objective settings with our final version as part of our user study to validate our parameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The consistent outputs obtained using the above parameter settings are compared against state of the art approaches thereby showcasing its efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Consistent Outputs We use videos from DAVIS [PPTM∗16] dataset and other open source videos (taken from [Vid] and [Pex]) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For per- frame stylization, we employ the following stylization techniques: Fast NST [JAFF16], WCT [LFY∗17], and CycleGAN [ZPIE17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The results for the method of Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' and Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' on videos taken from DAVIS [PPTM∗16] and Videvo ( [Vid]) are borrowed from the results dataset provided by Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For other videos we employ the source code provided by the authors to generate submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams 132 128 127 39 43 44 0 20 40 60 80 100 120 140 Lai Bonneel Ours-obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Others Ours Figure 7: Statistics of the user study results on removal of temporal flickering from per-frame stylized videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For 19 participants and 9 different videos we compare our method against Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' , Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' , and Ours-objective through a total of 171 randomized A/B tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We compare our consistent outputs with that of Bon- neel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] and Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Among the three competing methods Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' is the least effective in preserving the underlying style for the final output (compare sec- ond column with the fourth one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Hyper-parameter tun- ing in the above method (with only global consistency) can pro- vide a certain degree of consistency-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, by employ- ing both global and local consistency we achieve finer consistency- control while being similar to the per-frame-processed result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For the method of Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' , we observe some color bleeding or dark- ening in the output frames (compare second column with the third one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In comparison we are able to preserve the style, color and textures, while being consistent (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Optic Flow Results We visualize optical flow on frames from the Sintel [BWSB12] dataset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 9 and compare to state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' All de- picted methods have been fine-tuned on Sintel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We find that our optimized method has more blurry motion boundaries and misses to estimate certain details accurately (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', the hand in the first row, however, PWCNet also fails at this), but still captures over- all motion direction of objects correctly with a smooth flow field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 10 shows results for real-world videos on the DAVIS dataset [PTPC∗17] (no ground-truth flow available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We find that some real-world image phenomena, such as complex/ambiguous occlu- sions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', bus behind tree) are not well-handled by state-of-the-art methods like RAFT [TD20] or PWC-Net [SYLK18], and thus re- sults are degraded for our optimized method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Besides the stronger blurred motion boundaries, we find that our network gen- erally performs well and is also robust for real-world videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Evaluation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Quantitative Following Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18], we measure the similarity between per-frame processed output and stabilized results, and the temporal warping error between consecutive stabilized frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For the fomer, we report the similarity in form of the SSIM met- ric in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We achieve significantly higher similarity scores than the methods of Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] and Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Following [BTS∗15] and [LHW∗18], we also measure the tempo- ral warping error between a frame Vt and the warped consecutive frame ˆVt+1, defined as: Ewarp (Vt,Vt+1) = 1 ∑N i=1 M(i) t N ∑ i=1 M(i) t ���V (i) t − ˆV (i) t+1 ��� 1 , (9) where Mt ∈ {0,1} is a non-occlusion mask [LHW∗18,RDB18], in- dicating non-occluded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The warped frame ˆVt+1 is obtained by calculating the optical flow (using GMA [JCL∗21]) between frames Vt,Vt+1, and applying a backwards warping to frame Vt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We compute Ewarp for every frame of a video and then average to obtain the warping error of a video Ewarp(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5 we report the average warping error per dataset (see the supplementary for a per- task breakdown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We find that the warping error is slightly higher than that of Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] and Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, as Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] notes, results with high temporal stability (expressed by a low warping error) can also be achieved via temporally smoothing the video, which can be seen in vari- ous results of Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our qualitative results in form of a user study Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2 further substantiate the divide between warping error (as a stability metric) and perceived stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Qualitative For qualitative evaluation we perform a subjective user study where we ask participants to compare the temporally-consistent result ob- tained using our method with that of Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' , Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' , and Ours-objective – a different parameter setting of ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We use 9 different videos for this purpose: 3 from DAVIS [PPTM∗16], 3 from Videvo [Vid], and 3 from Pexels [Pex] datasets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For each of the above video we stylize them using either the Fast NST [JAFF16] (in the styles of udnie, rain-princess, and mosaic) or WCT [LFY∗17] (in the styles of wave and antimono) or Cycle- GAN (in the styles of photo2vangogh and photo2ukiyoe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For each sample, we show the input video and its per-frame stylized version on the top row of user-study interface for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In the bottom row we show two different version of the temporally stabilized out- put where one of them is ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We ask the participants to select the output which best preserves: (i) temporally consistency and (ii) similarity with the per-frame processed video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For 9 videos and 3 other competing methods each user sees a total of 27 blind A/B tests which are shown in a randomized order to each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In total, 19 persons (3 female and 16 male) within the ages of 22 to 43 years participated in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 7 shows that our method surpasses all others by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' It was interesting to observe that for certain cases the method of Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' which degrades the processed style significantly was still preferred by users over others due to its high consistency quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams 9 (a) Input (b) Processed (c) Ours (d) Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] (e) Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] Figure 8: Comparing our results with Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [LHW∗18] and Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [BTS∗15] for three different video sequences: Cow (top two rows), Farming (mid two rows), and Woman (last two rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Note how the consistent output for Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' and Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' look different from the corresponding per-frame processed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Using other Optical Flows We also tested other optical flow methods within our pipeline which were either faster [KTDVG16] or more accurate [TD20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For the fast optical method by Kroeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [KTDVG16](DIS) the final output is less consistent than ours in both objective and subjective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Using DIS for our stabilization, the average warp-error over DAVIS is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='05 (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='046 ours) and perceptual- similarity with the per-frame processed result is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='9 in SSIM terms (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='923 ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Visually, DIS-stabilized results show significantly more flickering, validating our design choice for the optical-flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A much more accurate optic flow is given by the method of Teed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' [TD20] (RAFT) at the cost of slow computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The stabilized results obtained using RAFT look visually indistinguish- able to the one obtained using our flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' the average warp-error over DAVIS is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='045, the perceptual-similarity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Discussion Our approach takes a video pair as an input: (i) the original and (ii) its per-frame stylized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We assume that the stylization is based on the input image-gradients and appears as variations in the form of colors and/or textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Thereby, we employ the origi- nal video as a guide for enforcing consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, for text- guided generative arts such as recent diffusion model-based ap- proaches [RDN∗22, RBL∗22] the stylized frames are often only weakly correlated with the original input, we cannot handle such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' For the evaluation we mainly use CNN-based stylization tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However our approach can also handle classical stylization approaches [KCWI13], we show few such examples in the supple- mentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Our local-consistency component comprising of convex combination of temporal neighbors can be seen as crude form of submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' / Interactive Control over Temporal Consistency while Stylizing Video Streams (a) Frame Overlay (b) Ground-truth (c) RAFT [TD20] (d) PWC-Net [SYLK18] (e) Ours Figure 9: Optical flow estimated using the synthetic Sintel dataset [BWSB12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' (a) Frame Overlay (b) RAFT [TD20] (c) PWC-Net [SYLK18] (d) Ours Figure 10: Optical flow estimated for the real-world dataset DAVIS [PTPC∗17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' local temporal denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Previously it has been shown that tem- poral denoising is effective in enforcing consistency [SST∗19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We conjecture that efficient temporal-denoising combined with flow- based warping can further improve temporal stabilization not only for stylization but also for other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We start with the assumption that temporal flickering is not com- pletely undesirable for the task of stylization and thus we pro- vide interactive consistency control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' However, during the subjec- tive user study we observed that participants had different toler- ance levels for flickering in the foreground as compared to that in the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' As part of future work, one can use depth-based or saliency-based masks to vary the consistency control parameters spatially for a more visually pleasing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Limitation: Our approach tends to have ghosting artifacts for fast moving objects where the object motion between consecutive frames is large (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The above can be reduced by reducing the value of � wp � , however such a reduction also reduces consis- (a) � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5 (b) � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1 Figure 11: The ghosting artifacts on the rear wheel of the scooter is significant in the final output for � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='5, however it reduces significantly for � wp � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' tency in the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We argue that since we provide interactive control of parameters the above trade off between artifacts vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' con- sistency will not hinder its usability significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' EPE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='000EPE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='923 Table 5: Flow warping error average over tasks shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' A per-task breakdown is shown in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Note that the slightly higher warping error (lower is better) of our method is subjectively not noticeable as we show in a user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Dataset Vp [BTS∗15] [LHW∗18] Ours DAVIS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='046 VIDEVO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='042 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Conclusions We propose an approach that makes per-frame stylized videos tem- porally coherent irrespective of the underlying stylization applied on individual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' At this, we introduce a novel temporal con- sistency prior which combines both local and global consistency aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We maintain similarity with the per-frame processed result by minimizing the difference in the gradient-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Unlike previ- ous approaches we provide interactive consistency control by com- puting optic-flow on the incoming video stream with only sufficient accuracy but at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Fats optic-flow inference is achieved by developing a lightweight flow network architecture based on PWC-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' The entire optimization solving is GPU-based and runs at real-time frame-rates for HD resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We showcase that our temporally consistent output is preferred over the output of com- peting methods by conducting a user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' As part of future work we would like to employ learning-based temporal denoising to fur- ther improve quality of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Moreover, we would like to ex- plore the usage of depth-based and saliency-based masks to spa- tially vary consistency parameters according to perceptual princi- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' We hope that our design paradigm of interactive consistency control will potentially make per-frame video stylization more user friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' References [BCCZ08] BHAT P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', CURLESS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', COHEN M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', ZITNICK C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Fourier analysis of the 2d screened poisson equation for gradient domain prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In Computer Vision – ECCV 2008 (2008), Springer Berlin Heidel- berg, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 114–128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1007/978-3-540-88688-4_9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 4 [BCK∗13] BÉNARD P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', COLE F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', KASS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', MORDATCH I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', HEGARTY J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', SENN M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', FLEISCHER K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', PESARE D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', BREEDEN K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Stylizing animation by example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 32, 4 (jul 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1145/2461912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2461929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1, 4 [BLV∗10] BÉNARD P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', LAGAE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', VANGORP P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', LEFEBVRE S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', DRETTAKIS G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', THOLLOT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': A dynamic noise primitive for coherent stylization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Computer Graphics Forum 29, 4 (2010), 1497–1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1467-8659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='01747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1 [BNTS07] BOUSSEAU A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', NEYRET F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', THOLLOT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', SALESIN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Video watercolorization using bidirectional texture advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 26, 3 (jul 2007), 104–es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1145/1276377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1276507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1, 3 [BSFG09] BARNES C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', SHECHTMAN E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', FINKELSTEIN A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', GOLDMAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Patchmatch: A randomized correspondence algorithm for struc- tural image editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 28, 3 (jul 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1145/1531326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1531330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 3 [BTC13] BÉNARD P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', THOLLOT J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', COLLOMOSSE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Temporally Coherent Video Stylization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 257–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1007/ 978-1-4471-4519-6_13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1, 3 [BTS∗15] BONNEEL N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', TOMPKIN J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', SUNKAVALLI K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', SUN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', PARIS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', PFISTER H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Blind video temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 34, 6 (oct 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1145/2816795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2818107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 1, 2, 4, 8, 9, 11 [BWSB12] BUTLER D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', WULFF J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', STANLEY G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', BLACK M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': A naturalistic open source movie for optical flow evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In European Conference on Computer Vision (ECCV) (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 611–625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1007/978-3-642-33783-3_44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 6, 7, 8, 10 [BZCC10] BHAT P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=', ZITNICK C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=': Unpaired image-to-image translation using cycle-consistent adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' In 2017 IEEE International Conference on Computer Vision (ICCV) (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 2242–2251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='1109/ICCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content='244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} +page_content=' 7, 11 submitted to 200x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dAyT4oBgHgl3EQf2fkR/content/2301.00750v1.pdf'} diff --git a/5tFAT4oBgHgl3EQfmx2I/content/tmp_files/2301.08625v1.pdf.txt b/5tFAT4oBgHgl3EQfmx2I/content/tmp_files/2301.08625v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1c6033c55eca59b400f1c8095c47853001efc08 --- /dev/null +++ b/5tFAT4oBgHgl3EQfmx2I/content/tmp_files/2301.08625v1.pdf.txt @@ -0,0 +1,1005 @@ +arXiv:2301.08625v1 [physics.plasm-ph] 20 Jan 2023 +Interaction of thin tungsten and tantalum films with ultrashort laser pulses: +calculations from first principles +N. A. Smirnov∗ +Federal State Unitary Enterprise, Russian Federal Nuclear Center - Zababakhin +All-Russian Research Institute of Technical Physics, 456770, Snezhinsk, Russia +(Dated: January 23, 2023) +The interaction of ultrashort laser pulses with thin tungsten and tantalum films is investigated +through the full-potential band-structure calculations. Our calculations show that at relatively low +absorbed energies (the electron temperature Te≲7 kK), the lattice of tantalum undergoes noticeable +hardening. The hardening leads to the change of the tantalum complete melting threshold under +these conditions. Calculations suggest that for the isochorically heated Ta film, if such hardening +really occurs, the complete melting threshold will be at least 25% higher. It is also shown that +the body-centered cubic structures of W and Ta crystals become dynamically unstable when the +electronic subsystem is heated to sufficiently high temperatures (Te>22 kK). This lead to their +complete melting on the sub-picosecond time scale. +PACS numbers: +I. +INTRODUCTION +As shown in a number of experimental studies, the +melting of different materials after their interaction with +ultrashort (femtosecond) laser pulses have their specific +features [1–5]. +Absorption of this radiation leads to a +strongly non-equilibrium heating of the system where the +temperatures of its electronic and ionic subsystems are +very much different, Te≫Ti. This state may keep for tens +of picoseconds and even longer [5]. Under these condi- +tions, semiconductors, for example, undergo the so-called +nonthermal melting caused not by their lattice heating +due to heat transfer from hot electrons to cold ions but +by a dramatic change in the shape of the potential energy +surface and hence dynamic lattice destabilization [1–3]. +In semimetallic bismuth, the situation seems to be sim- +ilar [4]. The determining factor here is the estimate of +the electron-phonon coupling factor G, which defines the +rate of heat transfer from the electronic to ionic subsys- +tem. For bismuth, the theoretical estimates of G strongly +differ [6–8], leaving room for disputes on the presence of +nonthermal melting in this metal after interaction with +ultrashort laser pulses [9]. +On the other hand, the change of the shape of the po- +tential energy surface may also lead, under certain con- +ditions, to the hardening of irradiated crystal [10–12], +thus increasing the time of its melting and causing its +strong overheating. Despite some claims that the lattice +hardening has been experimentally observed [13], there +is still no evidence of its reliable detection in experiments +[5, 12, 14]. +The experimental work reported in Ref. [15] aimed to +explore the possibility of the nonthermal melting of tung- +sten by measuring reflectivity of the metal surface after +its irradiation. The experiments show that above a cer- +∗Electronic address: deldavis@mail.ru +tain value of absorbed excitation fluence, ablation of the +metal surface proceeds in a sub-picosecond time interval. +The revealed effect may be indicative of the ultrafast non- +thermal melting because in the normal thermal scenario +of ablation, the characteristic times of this process must +be much higher than those obtained in experiment [15]. +Ab initio calculations [16] show that the heating of +the electronic subsystem of tungsten to Te above 20 kK +may lead to a structural transition from bcc to fcc phase. +The transition is also caused by the abrupt change in the +shape of the potential energy surface, leading to fcc sta- +bilization at high values of Te [16]. In its turn, the bcc +structure may lose dynamic stability under these condi- +tions. It is however difficult to detect this transition in +experiment because of the possibility of sub-picosecond +nonthermal melting. Just this was shown in molecular +dynamics (MD) calculations [17] where the interaction +of femtosecond laser pulses with thin tungsten film was +investigated. The nuclei of the new fcc phase were only +able to form mainly on the surface of the film before the +sample melted during about 0.8 ps. On whole, MD re- +sults [17] suggest that the detection probability for the +nonthermal melting of tungsten is much higher than for +the structural transition predicted in Ref. [16]. +As mentioned above, an important factor of detecting +nonthermal phenomena in metals is the electron-phonon +coupling factor G. Its values for metals are usually high +[18], meaning that the nonthermal character of processes +that occur after irradiation can hardly be recognized. +There are different approaches to the theoretical deter- +mination of G (see, for example, [12, 18, 19]). In our re- +search we will follow methodology described in Ref. [12], +but also discuss results obtained with other approaches. +This paper studies the interaction of femtosecond laser +pulses with thin (a few tens of nanometers thick) tung- +sten and tantalum films. The physical quantities required +for calculations with a two-temperature model [20] were +obtained from first principles. The issues discussed in- +clude the processes involved in the nonthermal melting + +2 +of the metals and the possibility of detecting tantalum +lattice hardening at moderate absorbed energies. +Our +results are compared with available experimental data +and other calculations. +II. +CALCULATION METHOD +In this work, the temperature evolution of electronic +and ionic subsystems with time after irradiation by ul- +trashort laser pulses is determined using a well-known +two-temperature model [20]. Since the thin (∼10 nm) +films of W and Ta are considered, the two-temperature +model equations can be written as +Ce(Te)∂Te +∂t = −(Te − Ti)G(Te) + S(t), +(1) +Ci(Ti)∂Ti +∂t = (Te − Ti)G(Te), +(2) +where S(t) is the time dependent radiation source func- +tion [17], Ce(Te) and Ci(Ti) are electron and lattice heat +capacities, and G(Te) is the electron-phonon coupling fac- +tor. Here we neglect lattice (κi) and electron (κe) ther- +mal conductivities because, on the one hand, κe≫κi in +our case, and on the other hand, in thin foils, ballistic +electrons bring the electronic subsystem to thermody- +namic equilibrium over a time about a pulse duration τp +[21, 22]. So, no significant gradients in temperature oc- +cur in the target. The method to calculate Ce, Ci, and G +as functions of electron and ion temperatures from first +principles is described in rather detail in Ref. [12]. Here +we only provide the key formula for the electron-phonon +coupling factor. It reads as +G(Te) = +2πℏ +(Tl − Te) +∞ +� +0 +ΩdΩ +∞ +� +−∞ +N(ε)α2F(ε, Ω) +× S(ε, ε + ℏΩ)dε. +(3) +where N(ε) is the electronic density of states (DOS), +α2F(ε,Ω) is the electron-phonon spectral function, ε +and ℏΩ are, respectively, electron and phonon energies, +S(ε,ε + ℏΩ)=[fe(ε)-fe(ε + ℏΩ)][n(ℏΩ,Ti)-n(ℏΩ,Te)] with +fe standing for the Fermi distribution function and n for +the Bose-Einstein distribution function. +Another formula which is often used to determine +G(Te) has some simplifications as compared to (3) and +reads as [18] +G(Te) = πℏkBλ⟨ω2⟩ +N(EF ) +∞ +� +−∞ +N 2(ε) +� +−∂fe +∂ε +� +dε. +(4) +Here λ is the electron-phonon mass enhancement param- +eter, ⟨Ω⟩2 is the second moment of the phonon spectrum +[23], and EF is the Fermi energy. Formula 4 is derived un- +der the assumption that in the interaction with phonon, +the scattering probability matrix elements is independent +of the initial {k, i} and final {k′, j} electronic states. The +authors of Ref. [18] determined the values of λ and ⟨Ω⟩2 +from experimental evaluation, not from first-principles +calculations. +One more way to calculate G(Te) is based on the calcu- +lation of the electron-ion collision integral Ie−i +nm with the +use of an approximate tight-binding model to calculate +the band structure, combined with MD simulation [19]. +The expression for Ie−i +nm is written as +Ie−i +nm = 2π +ℏ |Me−i(εn, εm)|2 +� +fe(εn)[2 − fe(εm)] − fe(εm)[2 − fe(εn)]e−∆ε/Ti; +for n>m +fe(εm)[2 − fe(εn)]e−∆ε/Ti − fe(εn)[2 − fe(εm)]; +otherwise , +(5) +where ∆ε=εn − εm is the energy difference between two +states, and Me−i is the electron-ion scattering matrix el- +ement. The electron-phonon coupling factor can be writ- +ten as +G(Te) = +1 +V (Te − Ti) +� +n,m +εmIe−i +nm , +(6) +here V is the specific volume. It should be noted here +that our method for determining G(Te) (by formula (3)) +does not use any experimentally determined parameters +or approximations which simplify the scattering proba- +bility matrix element, as it is done in Ref. [18], or serious +simplifications related to particle interactions in the sys- +tem, as it is done in the tight-binding model [19]. +In this work, first-principles calculations were done +with the all-electron full-potential linear muffin-tin or- + +3 +0 +2 +4 +6 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0 +2 +4 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +PDOS (arb. units) +Frequency (THz) +W +Frequency (THz) +Ta +FIG. 1: Tungsten and tantalum phonon spectra at the equi- +librium experimental specific volume from calculations done +in this work for zero temperature (red lines) and from exper- +iment at room temperature [30] (circles connected by a line). +bital method (FP-LMTO) [24]. We consider here pro- +cesses at a constant specific volume, i.e. +the isochoric +heating of targets. +Within the scope of density func- +tional theory the FP-LMTO method calculates the elec- +tron structure, internal and free energies, phonon spec- +trum and other material properties [12, 24–26]. Phonon +spectrum and electron-phonon spectral function calcu- +lations for the metals of interest were done with lin- +ear response theory implemented in the FP-LMTO code +[24, 25]. +Integration over the Brillouin zone was done +with an improved tetrahedron method [27]. Meshes in +k-space corresponded to equidistant spacing 30×30×30. +For integration over the q-points of the phonon spectrum, +a 10×10×10 mesh appeared quite sufficient (see [26] for +more details on meshes). The cutoff energy for repre- +senting the basis functions as a set of plane waves in the +interstitial region was taken to be 900 eV. The basis set +included MT-orbitals with moments to lb +max=5. Charge +density and potential expansions in terms of spherical +harmonics were done to lw +max=7. The internal FP-LMTO +parameters such as the linearization energy, tail energies, +and the radius of the MT-sphere were chosen using an +approach similar to that one used in Ref. [28]. +The valence electrons in our calculations were 5s, 5p, +4f, 5d, and 6s. For better comparison with calculations +by other authors, the exchange-correlation potential was +chosen to be similar to that one used in Ref. [17], i.e., +PBE [29]. This functional reproduces well the different +properties of tungsten and tantalum. For example, the +equilibrium volume V0 from calculation differs by no more +than 2% from experiment for both the metals. Figure 1 +shows the phonon densities of states (PDOS) from calcu- +lation in comparison with experimental data [30]. They +are seen to be in quite a good agreement. +The entropy of the electronic subsystem was deter- +0 +15 +30 +45 +-0.6 +-0.3 +0.0 +0.3 +0.6 +0.9 +0 +15 +30 +45 + + W + T +e +=1 kK + T +e +=10 kK + T +e +=20 kK + + +N (states/Ry/atom) + Ta + + +N (states/Ry/atom) +E- + (Ry) +FIG. 2: Electronic DOS for W (top) and Ta (bottom) at equi- +librium specific volume and zero temperature (black lines). +The green, blue and red lines are the Fermi distribution func- +tions at different electron temperatures. +mined as +Se(Te) = −kB +� ∞ +−∞ +dεN(ε)[feln(fe) + (1 − fe)ln(1 − fe)]. +(7) +With the known entropy Se(Te) and internal energy +Ee(Te) of electrons, it is easy to obtain the free energy +Fe=Ee − TeSe of the electron gas. +The phonon spectrum of tungsten and tantalum was +determined within quasiharmonic approximation [12]. +The melting temperature Tm of crystal W and Ta versus +electron temperature was estimated in the same manner +as it was done in Ref. [31] with the well performing Lin- +demann criterion. +III. +RESULTS +Let’s first compare the electronic structures of tungsten +and tantalum. Figure 2 shows their electronic densities +of states versus energy at V =V0 and T =0 calculated in +this work. It is seen that the chemical potential µ which +coincides with the Fermi energy at zero temperature is +near the minimum of the DOS for tungsten, while for +tantalum, the density of states at ε=µ is much higher +compared to W. For Ta, the Fermi level is near the peak +of the DOS. Compared to tantalum, the electronic struc- +ture of tungsten is very much depleted in states in the +vicinity of µ. Calculations show that as Te grows to ∼15 +kK, the values of N(µ) increase for tungsten and decrease +for tantalum. This causes certain differences in the be- +havior of these metals at elevating electron temperatures. +Now consider how the free energy of electrons depends +on the lattice parameter c/a (i.e., the Bain path) at dif- +ferent temperatures Te. +Figures 3 and 4 show results + +4 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +-8 +-4 +0 +4 +8 +12 + W + T +e +=8.7 kK + T +e +=14.5 kK + T +e +=29 kK +fcc +F +e +-F +0 + (mRy/atom) +c/a +bcc +FIG. 3: Free electron energy versus lattice parameter c/a at +different Te for tungsten (V =V0). The vertical lines show the +values of c/a which correspond to its bcc and fcc structures. +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +-5 +0 +5 +10 +15 +20 + Ta + + +F +e +-F +0 + (mRy/atom) +c/a + T +e +=1 kK + T +e +=5.8 kK + T +e +=17.4 kK + T +e +=34.8 kK +bcc +fcc +FIG. 4: Free electron energy versus lattice parameter c/a at +different Te for tantalum (V =V0). The vertical lines show the +values of c/a which correspond to its bcc and fcc structures. +obtained for W and Ta, respectively. In both metals, the +fcc structure is seen to be dynamically unstable at low +electron temperatures. With the increasing temperature +it stabilizes and at Te>15 kK it becomes thermodynam- +ically more preferable than bcc. It is seen that tantalum +behaves very much like tungsten but requires somewhat +higher temperatures for stabilization of the fcc structure. +On the other hand, with the increasing Te the bcc struc- +ture becomes dynamically unstable both in tungsten and +in tantalum. These changes must lead to a bcc→fcc tran- +sition when the electronic subsystem is heated. As how- +ever mentioned in paper [17], in such conditions their +melting is more probable. On whole, our calculations for +tungsten agree well with results presented in Ref. [16]. +One more feature of tantalum should be noted here. It +is seen from Fig. 4 that there exists a limited interval of +temperatures at relatively low values of Te (see Te=5.8 +0 +2 +4 +6 +0.0 +0.2 +0.4 +0.6 +0.8 +0 +2 +4 +6 +0.0 +0.4 +0.8 +1.2 + T +e +=300 K + T +e +=5.8 kK + T +e +=11.6 kK +PDOS (arb. units) +Frequency (THz) +W +Frequency (THz) +Ta +FIG. 5: Phonon densities of states in tungsten (left) and tan- +talum (right) at different electron temperatures (V =V0). +kK), where the bcc lattice hardens. The free energy curve +runs steeper near the minimum corresponding to the bcc +phase. This feature is absent in tungsten. Figure 5 shows +the densities of phonon states for W and Ta we calculated +in this work for different electron temperatures. It is seen +that with the increasing Te tungsten gradually softens +and its phonon frequencies reduce. The phonon frequen- +cies of tantalum first increase with the growing Te and +cause bcc lattice hardening. Then the tendency changes +– the high-frequency part of the spectrum goes on to +harden, while the low-frequency part begins to soften re- +ducing its frequencies (see Fig. 5, Te=11.6 kK). At Te +above 20 kK the bcc structure in both metals loses its +dynamic stability. It happens at about 22 kK in tung- +sten and 29 kK in tantalum. The hardening of the Ta +lattice at relatively low electron temperatures leads to a +sudden effect we will consider later. +Figures 6 and 7 show the electron-phonon coupling fac- +tor G as a function of electron temperature at V =V0, +calculated in this work for tungsten and tantalum, re- +spectively. The dependences G(Te) are provided for bcc +and fcc structures in their stability regions. +The val- +ues of G for the structures are seen to be close to each +other and it is quite possible to approximate our results +by a continuous line. The figures also show data from +low-temperature experiments [32–34]. For tungsten, our +results are seen to agree quite well with experiment. For +tantalum, experimental data from Ref. [34] provides only +the lower boundary of G, which does not contradict our +calculations. Figures 6 and 7 also show results from some +other calculations. It is seen that compared to our re- +sults, calculations by Lin et al. [18] for W give overesti- +mated values of G for the increasing temperature (Fig. 6). +Such a behavior has earlier been observed in other metals +[12] and can be related to the more correct account for the +energy dependence of α2F(ε,Ω) in formula (3). In turn, +the values of G(Te) from Ref. [19] are much lower than +our results and the experimental data available. +Note +that the presence of adjustable parameters in the calcu- +lation method may reduce the accuracy of results if they + +5 +0 +10 +20 +30 +40 +0 +3 +6 +9 +12 +fcc + + +G (10 +17 + W/m +3 +/K) +T +e + (kK) +bcc + W +FIG. 6: Electron-phonon coupling factor versus Te for tung- +sten from our calculation (solid, dashed lines for bcc and fcc, +respectively), from calculations reported in papers [18] (dot- +ted line) and [19] (dashed-dotted line), and from experiments +[32] and [33] (the circle and the triangle, respectively). The +vertical line shows the approximate value of Te above which +the fcc phase becomes more energetically favorable than bcc. +0 +10 +20 +30 +40 +0 +2 +4 +6 +8 +10 + + +G (10 +17 + W/m +3 +/K) +T +e + (kK) + Ta +bcc +fcc +FIG. 7: Electron-phonon coupling factor versus Te for tan- +talum from our calculations using formula (3) (solid, dashed +lines for bcc and fcc, respectively) and by a formula (4) (dot- +ted line). Other calculations: dashed-dotted line - Ref. [19], +dashed-dotted-dotted line - Ref. [34] by a formula from +Ref. [18] (see the text). The triangle shows the lower bound- +ary of G from experiment [34]. The vertical line shows the +approximate value of Te above which the fcc phase is more +energetically preferable than bcc. +are adjusted to conditions (for example, at T =0) different +from what we are having here. +For tantalum (fig. 7), our calculations by expression (4) +(the dotted line) had one distinction from those reported +in paper [18]: the values of λ and ⟨Ω⟩2 were determined +from first-principles calculations rather than from experi- +mental evaluation. It is seen that in this case, approaches +[18] and [12] give close values for G(Te), the differences +0 +4 +8 +12 +16 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 + W +I / I +0 +t (ps) +FIG. 8: Intensity of diffraction peak (211) versus time for +tungsten for absorbed energy density 0.8 MJ/kg from our +calculation (the solid line), calculations with a constant G +[33] (the dashed line), calculations with G(Te) from Ref. [18] +(the dashed-dotted line), and measurements [33] (circles). +are minimal. In Ref. [34], the electron-phonon coupling +factor was also calculated with formula (4) but with the +electronic DOS determined from MD calculations. But +here deviations from our results come, first of all, from +the underestimated parameter λ. +The authors of [34] +used the empirical value from Ref. [23], λ=0.65. Our cal- +culations from first principles gave λ=0.88 in the case of +tantalum. For tungsten, the difference between the em- +pirical [23] and calculated values of λ is not so large; they +agree within ∼3%. +Let’s consider the accuracy of our calculations in com- +parison with other experimental results. The authors of +paper [33] measured how evolved the intensity of the Laue +diffraction peak (211) after a 30-nm-thick tungsten film +deposited on a silicon nitride substrate was irradiated by +400-nm laser pulses with τp=130 fs. The absorbed energy +density Eabs was about 0.8 MJ/kg. Figure 8 compares +experimental data with calculations performed in three +variants (see [12] for calculation details). In addition to +our computation with use of formula (4), it shows cal- +culations with G(Te) taken from Ref. [18] and with con- +stant G=2 · 1017 W/m3/K and ΘD=312 K [33]. The re- +sults obtained with expression (3) are seen to agree quite +well with experiment. The use of G(Te) from Ref. [18] +slightly worsens the agreement and the calculation with +the constant G markedly underestimates the change of +the diffraction peak intensity at times below 10 ps. +Figure 9 presents ion temperature versus electron tem- +perature for tungsten, calculated by solving equations +(1)-(2). +We reproduced experimental conditions from +Ref. [33] but did calculations for several values of Eabs. +The possibility of the bcc→fcc transition was not consid- +ered because ultrafast melting was here more probable +[17]. Figure 9 also shows the melting temperature of W +versus Te, obtained in this work and by Murphy et al. + +6 +0 +5 +10 +15 +20 +25 +30 +0 +1 +2 +3 +4 +5 +6 +T +m +(T +e +) +0.91 MJ/kg +2.77 MJ/kg + W +T +i + (kK) +T +e + (kK) +0.8 MJ/kg +T +m +0 +FIG. 9: Calculated evolution of electron and ion tempera- +tures (isochoric heating) after irradiation of the 30-nm-thick +tungsten film by a 130-fs pulse for different absorbed energy +densities (dashed, dashed-dotted, and dashed-dotted-dotted +lines). The solid line shows the melting temperature Tm as a +function of Te from our calculation, the circles show Tm(Te) +from Ref. [17] (non-isochoric conditions), and the dotted line +shows the normal melting temperature of W. +[17] from MD calculations. Remind that our Tm(Te) was +calculated with the Lindemann criterion. As seen from +Fig. 9, the melting temperature of tungsten decreases +with the increasing Te due to lattice softening (Fig. 5). +The resulted dependence Tm(Te) agrees rather well with +data from Ref. [17] despite the essentially different ap- +proaches to its determination. Some discrepancy comes +from the fact that our calculation corresponded to the +isochore V =V0, while in MD simulation [17], the sample +could expand along the axis normal to the target surface. +In paper [33], a threshold value Em +abs required for the +complete melting of tungsten was determined. For the +conditions of that experiment, it was found to be 0.9 +MJ/kg. Our calculations give a very close value of 0.91 +MJ/kg (details of calculation can be found in paper [12]). +Complete melting occurs after the temperature Tm is +reached and the lattice gets sufficient heat to overcome +the latent heat of fusion, ∆Hm [35]. The absorbed en- +ergy density of 0.8 MJ/kg is not enough to completely +melt the target [33]. It is seen from Fig. 9 that at high +Eabs (>2.5 MJ/kg) the lattice temperature Ti reaches Tm +even earlier than Ti(Te) reaches its maximum. At high +Te, the melting temperature of tungsten becomes much +lower than the normal melting temperature determined +at ambient pressure, T 0 +m≈3.7 kK. MD calculations and +analytic equations of state [36, 37], including that one +for tungsten, suggest that the heat of fusion changes un- +der the action of external conditions and it will reduce as +Tm decreases. This will also influence the time of melt- +ing. Usually, Te reaches a maximum after irradiation by +ultrashort pulses at a time of about a few τp. +There- +fore at sufficiently high Eabs (>2.5 MJ/kg) tungsten will +0 +5 +10 +15 +20 +25 +30 +35 +0 +1 +2 +3 +4 +5 +6 +7 +1.12 MJ/kg +3.2 MJ/kg + Ta +1 MJ/kg +T +i + (kK) +T +e + (kK) +T +m +0 +T +m +(T +e +) +FIG. 10: Calculated evolution of electron and ion temper- +atures (isochoric heating) after irradiation of a 30-nm-thick +tantalum film by a 130-fs-pulse for different absorbed energy +densities (dashed, dashed-dotted, and dashed-dotted-dotted +lines). The solid line shows Tm versus Te from our calculation +and the dotted line shows the normal melting temperature of +Ta. +melt during sub-picosecond times which is also proved by +calculations [17]. +Now consider tantalum. Figure 10 demonstrates the +Ti(Te) dependence for Ta similarly to tungsten. Irradia- +tion conditions and target thickness are the same as for +W. It is seen that the melting curve Tm(Te) reaches a +maximum approximately at Te=7.3 kK due to the hard- +ening of the Ta crystal lattice at these temperatures, as +mentioned earlier (see Fig. 5). Unlike gold, whose melt- +ing temperature begins to increase only at Te>15 kK (re- +maining almost constant at lower Te) [12], for tantalum +this growth of Tm starts right after the electron temper- +ature increases. +At Te higher than 7.3 kK, its lattice +begins to gradually soften. Like tungsten, tantalum at +sufficiently high values of Eabs (>3 MJ/kg) must melt on +the sub-picosecond time scale due to the loss of dynamic +stability by its lattice (Fig. 10). We do not consider the +bcc→fcc transition here also. The high electron-phonon +coupling factor of tantalum signals a higher probability +of its ultrafast melting. However, the existence of a max- +imum of Tm(Te) at relatively low electron temperatures +gives an interesting effect. If such hardening really oc- +curs, it should lead to an increase in the melting thresh- +old Em +abs for Ta metal. As shown in calculations, Em +abs +will be at least 25% higher. For tantalum normal melt- +ing temperature, T 0 +m=3.29 kK, the threshold value �Em +abs +equals 0.74 MJ/kg. If the crystal lattice hardens, then, +under isochoric heating, an absorbed energy density of +∼1.12 MJ/kg is required for complete melting. For non- +isochoric conditions, the threshold may be lower, about +0.93 MJ/kg. However, the value is still rather far from +normal �Em +abs=0.74 MJ/kg and can be determined quite +reliably in experiment (see, for example, [5]). In addi- + +7 +tion, the growth of Tm make the latent heat of fusion +higher which will also delay the complete melting. +A similar maximum of Tm(Te) at relatively low heating +(Te∼5 kK) is also present in platinum [12]. As shown by +calculations from first principles, its electronic structure +is also characterized by a high electronic density of states +N(µ) on the Fermi level [18], which strongly reduces with +the increasing Te. Our calculations show that the effect +of lattice hardening is a bit lower here and the melting +threshold increases by about 18%. +But since �Em +abs for +platinum at the normal melting temperature T 0 +m is quite +small (∼0.39 MJ/kg), the detection of its increase in ex- +periment may be limited by experimental accuracy. +IV. +CONCLUSIONS +The paper studied the interaction of femtosecond laser +pulses with thin tungsten and tantalum films through cal- +culations from first principles. Calculated results shows +the body-centered cubic structure of both the metals to +lose its dynamic stability at rather high electron tem- +peratures. This effect must lead to their melting on the +sub-picosecond time scale when the electronic subsystem +is heated above 22 kK. 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Matter 32, 435403 (2020). + diff --git a/5tFAT4oBgHgl3EQfmx2I/content/tmp_files/load_file.txt b/5tFAT4oBgHgl3EQfmx2I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7380f9991096e1e075bbfdab1a6ed0f67f32cf7 --- /dev/null +++ b/5tFAT4oBgHgl3EQfmx2I/content/tmp_files/load_file.txt @@ -0,0 +1,718 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf,len=717 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='08625v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='plasm-ph] 20 Jan 2023 Interaction of thin tungsten and tantalum films with ultrashort laser pulses: calculations from first principles N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Smirnov∗ Federal State Unitary Enterprise, Russian Federal Nuclear Center - Zababakhin All-Russian Research Institute of Technical Physics, 456770, Snezhinsk, Russia (Dated: January 23, 2023) The interaction of ultrashort laser pulses with thin tungsten and tantalum films is investigated through the full-potential band-structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Our calculations show that at relatively low absorbed energies (the electron temperature Te≲7 kK), the lattice of tantalum undergoes noticeable hardening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The hardening leads to the change of the tantalum complete melting threshold under these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Calculations suggest that for the isochorically heated Ta film, if such hardening really occurs, the complete melting threshold will be at least 25% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is also shown that the body-centered cubic structures of W and Ta crystals become dynamically unstable when the electronic subsystem is heated to sufficiently high temperatures (Te>22 kK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This lead to their complete melting on the sub-picosecond time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' PACS numbers: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' INTRODUCTION As shown in a number of experimental studies, the melting of different materials after their interaction with ultrashort (femtosecond) laser pulses have their specific features [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Absorption of this radiation leads to a strongly non-equilibrium heating of the system where the temperatures of its electronic and ionic subsystems are very much different, Te≫Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This state may keep for tens of picoseconds and even longer [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Under these condi- tions, semiconductors, for example, undergo the so-called nonthermal melting caused not by their lattice heating due to heat transfer from hot electrons to cold ions but by a dramatic change in the shape of the potential energy surface and hence dynamic lattice destabilization [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In semimetallic bismuth, the situation seems to be sim- ilar [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The determining factor here is the estimate of the electron-phonon coupling factor G, which defines the rate of heat transfer from the electronic to ionic subsys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For bismuth, the theoretical estimates of G strongly differ [6–8], leaving room for disputes on the presence of nonthermal melting in this metal after interaction with ultrashort laser pulses [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' On the other hand, the change of the shape of the po- tential energy surface may also lead, under certain con- ditions, to the hardening of irradiated crystal [10–12], thus increasing the time of its melting and causing its strong overheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Despite some claims that the lattice hardening has been experimentally observed [13], there is still no evidence of its reliable detection in experiments [5, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The experimental work reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [15] aimed to explore the possibility of the nonthermal melting of tung- sten by measuring reflectivity of the metal surface after its irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The experiments show that above a cer- ∗Electronic address: deldavis@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='ru tain value of absorbed excitation fluence, ablation of the metal surface proceeds in a sub-picosecond time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The revealed effect may be indicative of the ultrafast non- thermal melting because in the normal thermal scenario of ablation, the characteristic times of this process must be much higher than those obtained in experiment [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Ab initio calculations [16] show that the heating of the electronic subsystem of tungsten to Te above 20 kK may lead to a structural transition from bcc to fcc phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The transition is also caused by the abrupt change in the shape of the potential energy surface, leading to fcc sta- bilization at high values of Te [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In its turn, the bcc structure may lose dynamic stability under these condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is however difficult to detect this transition in experiment because of the possibility of sub-picosecond nonthermal melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Just this was shown in molecular dynamics (MD) calculations [17] where the interaction of femtosecond laser pulses with thin tungsten film was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The nuclei of the new fcc phase were only able to form mainly on the surface of the film before the sample melted during about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' On whole, MD re- sults [17] suggest that the detection probability for the nonthermal melting of tungsten is much higher than for the structural transition predicted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' As mentioned above, an important factor of detecting nonthermal phenomena in metals is the electron-phonon coupling factor G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Its values for metals are usually high [18], meaning that the nonthermal character of processes that occur after irradiation can hardly be recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' There are different approaches to the theoretical deter- mination of G (see, for example, [12, 18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In our re- search we will follow methodology described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [12], but also discuss results obtained with other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This paper studies the interaction of femtosecond laser pulses with thin (a few tens of nanometers thick) tung- sten and tantalum films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The physical quantities required for calculations with a two-temperature model [20] were obtained from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The issues discussed in- clude the processes involved in the nonthermal melting 2 of the metals and the possibility of detecting tantalum lattice hardening at moderate absorbed energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Our results are compared with available experimental data and other calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' CALCULATION METHOD In this work, the temperature evolution of electronic and ionic subsystems with time after irradiation by ul- trashort laser pulses is determined using a well-known two-temperature model [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Since the thin (∼10 nm) films of W and Ta are considered, the two-temperature model equations can be written as Ce(Te)∂Te ∂t = −(Te − Ti)G(Te) + S(t), (1) Ci(Ti)∂Ti ∂t = (Te − Ti)G(Te), (2) where S(t) is the time dependent radiation source func- tion [17], Ce(Te) and Ci(Ti) are electron and lattice heat capacities, and G(Te) is the electron-phonon coupling fac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Here we neglect lattice (κi) and electron (κe) ther- mal conductivities because, on the one hand, κe≫κi in our case, and on the other hand, in thin foils, ballistic electrons bring the electronic subsystem to thermody- namic equilibrium over a time about a pulse duration τp [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' So, no significant gradients in temperature oc- cur in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The method to calculate Ce, Ci, and G as functions of electron and ion temperatures from first principles is described in rather detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Here we only provide the key formula for the electron-phonon coupling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It reads as G(Te) = 2πℏ (Tl − Te) ∞ � 0 ΩdΩ ∞ � −∞ N(ε)α2F(ε, Ω) × S(ε, ε + ℏΩ)dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' (3) where N(ε) is the electronic density of states (DOS), α2F(ε,Ω) is the electron-phonon spectral function, ε and ℏΩ are, respectively, electron and phonon energies, S(ε,ε + ℏΩ)=[fe(ε)-fe(ε + ℏΩ)][n(ℏΩ,Ti)-n(ℏΩ,Te)] with fe standing for the Fermi distribution function and n for the Bose-Einstein distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Another formula which is often used to determine G(Te) has some simplifications as compared to (3) and reads as [18] G(Te) = πℏkBλ⟨ω2⟩ N(EF ) ∞ � −∞ N 2(ε) � −∂fe ∂ε � dε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' (4) Here λ is the electron-phonon mass enhancement param- eter, ⟨Ω⟩2 is the second moment of the phonon spectrum [23], and EF is the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Formula 4 is derived un- der the assumption that in the interaction with phonon, the scattering probability matrix elements is independent of the initial {k, i} and final {k′, j} electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18] determined the values of λ and ⟨Ω⟩2 from experimental evaluation, not from first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' One more way to calculate G(Te) is based on the calcu- lation of the electron-ion collision integral Ie−i nm with the use of an approximate tight-binding model to calculate the band structure, combined with MD simulation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The expression for Ie−i nm is written as Ie−i nm = 2π ℏ |Me−i(εn, εm)|2 � fe(εn)[2 − fe(εm)] − fe(εm)[2 − fe(εn)]e−∆ε/Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' for n>m fe(εm)[2 − fe(εn)]e−∆ε/Ti − fe(εn)[2 − fe(εm)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' otherwise , (5) where ∆ε=εn − εm is the energy difference between two states, and Me−i is the electron-ion scattering matrix el- ement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The electron-phonon coupling factor can be writ- ten as G(Te) = 1 V (Te − Ti) � n,m εmIe−i nm , (6) here V is the specific volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It should be noted here that our method for determining G(Te) (by formula (3)) does not use any experimentally determined parameters or approximations which simplify the scattering proba- bility matrix element, as it is done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18], or serious simplifications related to particle interactions in the sys- tem, as it is done in the tight-binding model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In this work, first-principles calculations were done with the all-electron full-potential linear muffin-tin or- 3 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='7 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 PDOS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' units) Frequency (THz) W Frequency (THz) Ta FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 1: Tungsten and tantalum phonon spectra at the equi- librium experimental specific volume from calculations done in this work for zero temperature (red lines) and from exper- iment at room temperature [30] (circles connected by a line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' bital method (FP-LMTO) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' We consider here pro- cesses at a constant specific volume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' the isochoric heating of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Within the scope of density func- tional theory the FP-LMTO method calculates the elec- tron structure, internal and free energies, phonon spec- trum and other material properties [12, 24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Phonon spectrum and electron-phonon spectral function calcu- lations for the metals of interest were done with lin- ear response theory implemented in the FP-LMTO code [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Integration over the Brillouin zone was done with an improved tetrahedron method [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Meshes in k-space corresponded to equidistant spacing 30×30×30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For integration over the q-points of the phonon spectrum, a 10×10×10 mesh appeared quite sufficient (see [26] for more details on meshes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The cutoff energy for repre- senting the basis functions as a set of plane waves in the interstitial region was taken to be 900 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The basis set included MT-orbitals with moments to lb max=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Charge density and potential expansions in terms of spherical harmonics were done to lw max=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The internal FP-LMTO parameters such as the linearization energy, tail energies, and the radius of the MT-sphere were chosen using an approach similar to that one used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The valence electrons in our calculations were 5s, 5p, 4f, 5d, and 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For better comparison with calculations by other authors, the exchange-correlation potential was chosen to be similar to that one used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [17], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=', PBE [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This functional reproduces well the different properties of tungsten and tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For example, the equilibrium volume V0 from calculation differs by no more than 2% from experiment for both the metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 1 shows the phonon densities of states (PDOS) from calcu- lation in comparison with experimental data [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' They are seen to be in quite a good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The entropy of the electronic subsystem was deter- 0 15 30 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='9 0 15 30 45 W T e =1 kK T e =10 kK T e =20 kK N (states/Ry/atom) Ta N (states/Ry/atom) E- (Ry) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 2: Electronic DOS for W (top) and Ta (bottom) at equi- librium specific volume and zero temperature (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The green, blue and red lines are the Fermi distribution func- tions at different electron temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' mined as Se(Te) = −kB � ∞ −∞ dεN(ε)[feln(fe) + (1 − fe)ln(1 − fe)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' (7) With the known entropy Se(Te) and internal energy Ee(Te) of electrons, it is easy to obtain the free energy Fe=Ee − TeSe of the electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The phonon spectrum of tungsten and tantalum was determined within quasiharmonic approximation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The melting temperature Tm of crystal W and Ta versus electron temperature was estimated in the same manner as it was done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [31] with the well performing Lin- demann criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' RESULTS Let’s first compare the electronic structures of tungsten and tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 2 shows their electronic densities of states versus energy at V =V0 and T =0 calculated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen that the chemical potential µ which coincides with the Fermi energy at zero temperature is near the minimum of the DOS for tungsten, while for tantalum, the density of states at ε=µ is much higher compared to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For Ta, the Fermi level is near the peak of the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Compared to tantalum, the electronic struc- ture of tungsten is very much depleted in states in the vicinity of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Calculations show that as Te grows to ∼15 kK, the values of N(µ) increase for tungsten and decrease for tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This causes certain differences in the be- havior of these metals at elevating electron temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Now consider how the free energy of electrons depends on the lattice parameter c/a (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=', the Bain path) at dif- ferent temperatures Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figures 3 and 4 show results 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='5 8 4 0 4 8 12 W T e =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='7 kK T e =14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='5 kK T e =29 kK fcc F e F 0 (mRy/atom) c/a bcc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 3: Free electron energy versus lattice parameter c/a at different Te for tungsten (V =V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The vertical lines show the values of c/a which correspond to its bcc and fcc structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='5 5 0 5 10 15 20 Ta F e F 0 (mRy/atom) c/a T e =1 kK T e =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 kK T e =17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 kK T e =34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 kK bcc fcc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 4: Free electron energy versus lattice parameter c/a at different Te for tantalum (V =V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The vertical lines show the values of c/a which correspond to its bcc and fcc structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' obtained for W and Ta, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In both metals, the fcc structure is seen to be dynamically unstable at low electron temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' With the increasing temperature it stabilizes and at Te>15 kK it becomes thermodynam- ically more preferable than bcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen that tantalum behaves very much like tungsten but requires somewhat higher temperatures for stabilization of the fcc structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' On the other hand, with the increasing Te the bcc struc- ture becomes dynamically unstable both in tungsten and in tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' These changes must lead to a bcc→fcc tran- sition when the electronic subsystem is heated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' As how- ever mentioned in paper [17], in such conditions their melting is more probable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' On whole, our calculations for tungsten agree well with results presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' One more feature of tantalum should be noted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 4 that there exists a limited interval of temperatures at relatively low values of Te (see Te=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 T e =300 K T e =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 kK T e =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 kK PDOS (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' units) Frequency (THz) W Frequency (THz) Ta FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 5: Phonon densities of states in tungsten (left) and tan- talum (right) at different electron temperatures (V =V0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' kK), where the bcc lattice hardens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The free energy curve runs steeper near the minimum corresponding to the bcc phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This feature is absent in tungsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 5 shows the densities of phonon states for W and Ta we calculated in this work for different electron temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen that with the increasing Te tungsten gradually softens and its phonon frequencies reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The phonon frequen- cies of tantalum first increase with the growing Te and cause bcc lattice hardening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Then the tendency changes – the high-frequency part of the spectrum goes on to harden, while the low-frequency part begins to soften re- ducing its frequencies (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 5, Te=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 kK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' At Te above 20 kK the bcc structure in both metals loses its dynamic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It happens at about 22 kK in tung- sten and 29 kK in tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The hardening of the Ta lattice at relatively low electron temperatures leads to a sudden effect we will consider later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figures 6 and 7 show the electron-phonon coupling fac- tor G as a function of electron temperature at V =V0, calculated in this work for tungsten and tantalum, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The dependences G(Te) are provided for bcc and fcc structures in their stability regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The val- ues of G for the structures are seen to be close to each other and it is quite possible to approximate our results by a continuous line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The figures also show data from low-temperature experiments [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For tungsten, our results are seen to agree quite well with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For tantalum, experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [34] provides only the lower boundary of G, which does not contradict our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figures 6 and 7 also show results from some other calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen that compared to our re- sults, calculations by Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18] for W give overesti- mated values of G for the increasing temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Such a behavior has earlier been observed in other metals [12] and can be related to the more correct account for the energy dependence of α2F(ε,Ω) in formula (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In turn, the values of G(Te) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [19] are much lower than our results and the experimental data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Note that the presence of adjustable parameters in the calcu- lation method may reduce the accuracy of results if they 5 0 10 20 30 40 0 3 6 9 12 fcc G (10 17 W/m 3 /K) T e (kK) bcc W FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 6: Electron-phonon coupling factor versus Te for tung- sten from our calculation (solid, dashed lines for bcc and fcc, respectively), from calculations reported in papers [18] (dot- ted line) and [19] (dashed-dotted line), and from experiments [32] and [33] (the circle and the triangle, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The vertical line shows the approximate value of Te above which the fcc phase becomes more energetically favorable than bcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 0 10 20 30 40 0 2 4 6 8 10 G (10 17 W/m 3 /K) T e (kK) Ta bcc fcc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 7: Electron-phonon coupling factor versus Te for tan- talum from our calculations using formula (3) (solid, dashed lines for bcc and fcc, respectively) and by a formula (4) (dot- ted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Other calculations: dashed-dotted line - Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [19], dashed-dotted-dotted line - Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [34] by a formula from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18] (see the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The triangle shows the lower bound- ary of G from experiment [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The vertical line shows the approximate value of Te above which the fcc phase is more energetically preferable than bcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' are adjusted to conditions (for example, at T =0) different from what we are having here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For tantalum (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 7), our calculations by expression (4) (the dotted line) had one distinction from those reported in paper [18]: the values of λ and ⟨Ω⟩2 were determined from first-principles calculations rather than from experi- mental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen that in this case, approaches [18] and [12] give close values for G(Te), the differences 0 4 8 12 16 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 W I / I 0 t (ps) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 8: Intensity of diffraction peak (211) versus time for tungsten for absorbed energy density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 MJ/kg from our calculation (the solid line), calculations with a constant G [33] (the dashed line), calculations with G(Te) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18] (the dashed-dotted line), and measurements [33] (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [34], the electron-phonon coupling factor was also calculated with formula (4) but with the electronic DOS determined from MD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' But here deviations from our results come, first of all, from the underestimated parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The authors of [34] used the empirical value from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [23], λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Our cal- culations from first principles gave λ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='88 in the case of tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For tungsten, the difference between the em- pirical [23] and calculated values of λ is not so large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' they agree within ∼3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Let’s consider the accuracy of our calculations in com- parison with other experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The authors of paper [33] measured how evolved the intensity of the Laue diffraction peak (211) after a 30-nm-thick tungsten film deposited on a silicon nitride substrate was irradiated by 400-nm laser pulses with τp=130 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The absorbed energy density Eabs was about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 MJ/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 8 compares experimental data with calculations performed in three variants (see [12] for calculation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In addition to our computation with use of formula (4), it shows cal- culations with G(Te) taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18] and with con- stant G=2 · 1017 W/m3/K and ΘD=312 K [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The re- sults obtained with expression (3) are seen to agree quite well with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The use of G(Te) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [18] slightly worsens the agreement and the calculation with the constant G markedly underestimates the change of the diffraction peak intensity at times below 10 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 9 presents ion temperature versus electron tem- perature for tungsten, calculated by solving equations (1)-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' We reproduced experimental conditions from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [33] but did calculations for several values of Eabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The possibility of the bcc→fcc transition was not consid- ered because ultrafast melting was here more probable [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 9 also shows the melting temperature of W versus Te, obtained in this work and by Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 6 0 5 10 15 20 25 30 0 1 2 3 4 5 6 T m (T e ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='91 MJ/kg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='77 MJ/kg W T i (kK) T e (kK) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 MJ/kg T m 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 9: Calculated evolution of electron and ion tempera- tures (isochoric heating) after irradiation of the 30-nm-thick tungsten film by a 130-fs pulse for different absorbed energy densities (dashed, dashed-dotted, and dashed-dotted-dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The solid line shows the melting temperature Tm as a function of Te from our calculation, the circles show Tm(Te) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [17] (non-isochoric conditions), and the dotted line shows the normal melting temperature of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [17] from MD calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Remind that our Tm(Te) was calculated with the Lindemann criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' As seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 9, the melting temperature of tungsten decreases with the increasing Te due to lattice softening (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The resulted dependence Tm(Te) agrees rather well with data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [17] despite the essentially different ap- proaches to its determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Some discrepancy comes from the fact that our calculation corresponded to the isochore V =V0, while in MD simulation [17], the sample could expand along the axis normal to the target surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In paper [33], a threshold value Em abs required for the complete melting of tungsten was determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For the conditions of that experiment, it was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='9 MJ/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Our calculations give a very close value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='91 MJ/kg (details of calculation can be found in paper [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Complete melting occurs after the temperature Tm is reached and the lattice gets sufficient heat to overcome the latent heat of fusion, ∆Hm [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The absorbed en- ergy density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='8 MJ/kg is not enough to completely melt the target [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 9 that at high Eabs (>2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='5 MJ/kg) the lattice temperature Ti reaches Tm even earlier than Ti(Te) reaches its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' At high Te, the melting temperature of tungsten becomes much lower than the normal melting temperature determined at ambient pressure, T 0 m≈3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='7 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' MD calculations and analytic equations of state [36, 37], including that one for tungsten, suggest that the heat of fusion changes un- der the action of external conditions and it will reduce as Tm decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This will also influence the time of melt- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Usually, Te reaches a maximum after irradiation by ultrashort pulses at a time of about a few τp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' There- fore at sufficiently high Eabs (>2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='5 MJ/kg) tungsten will 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='12 MJ/kg 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='2 MJ/kg Ta 1 MJ/kg T i (kK) T e (kK) T m 0 T m (T e ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 10: Calculated evolution of electron and ion temper- atures (isochoric heating) after irradiation of a 30-nm-thick tantalum film by a 130-fs-pulse for different absorbed energy densities (dashed, dashed-dotted, and dashed-dotted-dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The solid line shows Tm versus Te from our calculation and the dotted line shows the normal melting temperature of Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' melt during sub-picosecond times which is also proved by calculations [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Now consider tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Figure 10 demonstrates the Ti(Te) dependence for Ta similarly to tungsten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Irradia- tion conditions and target thickness are the same as for W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is seen that the melting curve Tm(Te) reaches a maximum approximately at Te=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 kK due to the hard- ening of the Ta crystal lattice at these temperatures, as mentioned earlier (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Unlike gold, whose melt- ing temperature begins to increase only at Te>15 kK (re- maining almost constant at lower Te) [12], for tantalum this growth of Tm starts right after the electron temper- ature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' At Te higher than 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='3 kK, its lattice begins to gradually soften.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Like tungsten, tantalum at sufficiently high values of Eabs (>3 MJ/kg) must melt on the sub-picosecond time scale due to the loss of dynamic stability by its lattice (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' We do not consider the bcc→fcc transition here also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The high electron-phonon coupling factor of tantalum signals a higher probability of its ultrafast melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' However, the existence of a max- imum of Tm(Te) at relatively low electron temperatures gives an interesting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' If such hardening really oc- curs, it should lead to an increase in the melting thresh- old Em abs for Ta metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' As shown in calculations, Em abs will be at least 25% higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For tantalum normal melt- ing temperature, T 0 m=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='29 kK, the threshold value �Em abs equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='74 MJ/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' If the crystal lattice hardens, then, under isochoric heating, an absorbed energy density of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='12 MJ/kg is required for complete melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' For non- isochoric conditions, the threshold may be lower, about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='93 MJ/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' However, the value is still rather far from normal �Em abs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='74 MJ/kg and can be determined quite reliably in experiment (see, for example, [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In addi- 7 tion, the growth of Tm make the latent heat of fusion higher which will also delay the complete melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' A similar maximum of Tm(Te) at relatively low heating (Te∼5 kK) is also present in platinum [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' As shown by calculations from first principles, its electronic structure is also characterized by a high electronic density of states N(µ) on the Fermi level [18], which strongly reduces with the increasing Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Our calculations show that the effect of lattice hardening is a bit lower here and the melting threshold increases by about 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' But since �Em abs for platinum at the normal melting temperature T 0 m is quite small (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content='39 MJ/kg), the detection of its increase in ex- periment may be limited by experimental accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' CONCLUSIONS The paper studied the interaction of femtosecond laser pulses with thin tungsten and tantalum films through cal- culations from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Calculated results shows the body-centered cubic structure of both the metals to lose its dynamic stability at rather high electron tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' This effect must lead to their melting on the sub-picosecond time scale when the electronic subsystem is heated above 22 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' It is also demonstrated that the metals have rather high values of the electron-phonon coupling factor (∼ several units per 1017 W/m3/K) at electron temperatures from room temperature to ∼45 kK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' In addition, unlike tungsten, the crystal lattice of tantalum hardens at relatively low values of Te (≲7 kK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' The hardening changes the value of the complete melt- ing threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' Our calculations show that the melting threshold will be at least 25% higher if hardening re- ally occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' We suppose that this effect for tantalum can be detected quite reliably by modern experimental techniques used to study the interaction of matter with ultrashort laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tFAT4oBgHgl3EQfmx2I/content/2301.08625v1.pdf'} +page_content=' [1] C.' 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Training such models us- +ing standard 3D representations, like voxels or +point clouds, is challenging and requires com- +plex tools for proper color rendering. In order to +overcome this limitation, Neural Radiance Fields +(NeRFs) offer a state-of-the-art quality in synthe- +sizing novel views of complex 3D scenes from a +small subset of 2D images. +In the paper, we propose a generative model +called HyperNeRFGAN, which uses hypernet- +works paradigm to produce 3D objects repre- +sented by NeRF. Our GAN architecture leverages +a hypernetwork paradigm to transfer gaussian +noise into weights of NeRF model. The model +is further used to render 2D novel views, and a +classical 2D discriminator is utilized for training +the entire GAN-based structure. Our architecture +produces 2D images, but we use 3D-aware NeRF +representation, which forces the model to produce +correct 3D objects. The advantage of the model +over existing approaches is that it produces a ded- +icated NeRF representation for the object without +sharing some global parameters of the rendering +component. We show the superiority of our ap- +proach compared to reference baselines on three +challenging datasets from various domains. +1. Introduction +Generative Adversarial Nets (GANs) (Goodfellow et al., +2014) allow us to generate high-quality 2D images (Yu +et al., 2017; Karras et al., 2017; 2019; 2020; Struski et al., +2022). On the other hand, maintaining similar quality for +*Equal contribution +1Faculty of Mathematics and Computer +Science, Jagiellonian University 6 Lojasiewicza Street, 30-348 +Krak´ow, Poland 2Department of Artificial Intelligence, Univer- +sity of Science and Technology Wyb. Wyspianskiego 27, 50-370, +Wrocław, Poland. Correspondence to: Przemysław Spurek . +Figure 1. HyperNeRFGAN architecture leverages a hypernetwork +paradigm to transfer gaussian noise into weights of NeRF model. +After that, we render 2D novel views by NeRF and use a classical +2D discriminator. Our architecture produces 2D images, but we +use 3D-aware NeRF representation, which forces the model to +produce correct 3D objects. +3D objects is challenging. It is mainly caused by using +3D representations like voxels and point clouds that require +massive deep architectures and have problems with proper +color rendering. +We can solve this problem by operating directly on 2D +image space. We expect our approach to extract informa- +tion from unlabeled 2D views to obtain 3D shapes. To +obtain such effects, we can use Neural Radiance Fields +(NeRFs) (Mildenhall et al., 2021), which allow synthesizing +novel views of complex 3D scenes from a small subset of +2D images. Based on the relations between those base im- +ages and computer graphics principles, such as ray tracing, +this neural network model can render high-quality images +of 3D objects from previously unseen viewpoints. +Unfortunately, it is not trivial how to use NeRF represen- +tation with GAN-type architecture. The most challenging +problem is connected with the conditioning mechanism (Re- +bain et al., 2022) dedicated to NeRF. Therefore, most models +arXiv:2301.11631v1 [cs.CV] 27 Jan 2023 + +[x,y,z] +Weights +Generator +NeRE +Training Data +True +Discriminator +FalseHyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +Figure 2. Comparison of HyperNeRFGAN and HoloGAN, GRAF, π-GAN on CARLA. We obtain similar results to π-GAN, but we have +a better value of FID score, see Tab 2. +use SIREN (Sitzmann et al., 2020) instead of NeRF, where +we can naturally add conditioning. But the quality of the 3D +object is slice worst than in NeRF. In GRAF (Schwarz et al., +2020) and π-GAN (Chan et al., 2021), authors propose a +model which uses SIREN and a conditioning mechanism to +produce implicit representation. Such solutions give promis- +ing results, but it is not trivial how to use NeRF instead of +SIREN in such solutions. In Fig. 2 we present a qualita- +tive comparison between our model, GRAF (Schwarz et al., +2020) and π-GAN (Chan et al., 2021). As we can see, our +model can model the transparency of glass. +In the paper, we propose a generative model called HyperN- +eRFGAN1, which combines the hypernetworks paradigm +and NeRF representation. Hypernetworks, introduced in +(Ha et al., 2016), are defined as neural models that generate +weights for a separate target network solving a specific task. +Our GAN-based model leverages a hypernetwork paradigm +to transfer gaussian noise into weights of NeRF (see Fig. 1). +After that, we render 2D novel views by NeRF and use a +classical 2D discriminator to train the entire GAN-based +structure in implicit form. Our architecture produces 2D +images, but we use 3D-aware NeRF representation, which +forces the model to produce correct 3D objects. +Our contributions to this paper include the following: +• We introduce the NeRF-based implicit GAN architec- +ture - the first GAN model for generating high-quality +3D NeRF representations. +• We show that utilizing the hypernetwork paradigm for +NeRFs leads to a better quality of 3D representations +1The source code is available at: https://github.com/ +gmum/HyperNeRFGAN +than SIREN-based architectures. +• Our model allows 3D-aware image synthesis from un- +supervised 2D images. +2. Related Work +Neural representations and rendering +3D objects can +be represented by using many different approaches, includ- +ing voxel grids (Choy et al., 2016), octrees (H¨ane et al., +2017), multi-view images (Arsalan Soltani et al., 2017; Liu +et al., 2022), point clouds (Achlioptas et al., 2018; Shu +et al., 2022; Yang et al., 2022), geometry images (Sinha +et al., 2016), deformable meshes (Girdhar et al., 2016), and +part-based structural graphs (Li et al., 2017). +The above representations are discreet, which causes some +problems in real-life applications. Contrary, we can repre- +sent 3D objects as a continuous function (Dupont et al., +2022). In practice implicit occupancy (Chen & Zhang, +2019; Mescheder et al., 2019; Peng et al., 2020), distance +field (Michalkiewicz et al., 2019; Park et al., 2019) and +surface parametrization (Yang et al., 2019; Spurek et al., +2020; 2022; Cai et al., 2020) models use a neural network to +parameterize a 3D object. In such a case, we do not have a +fixed number of voxels, points, or vertices, but we represent +shapes as a continuous function. +These models are limited by their requirement of access to +ground truth 3D geometry. Implicit neural representations +(NIR) have been proposed to solve such a problem. Such +architectures can reconstruct 3D structures from multi-view +2D images (Mildenhall et al., 2021; Niemeyer et al., 2020; +Tewari et al., 2020). +The two most important methods are NeRF (Mildenhall + +HoloGAN +GRAF +pi-GANHyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +et al., 2021) and SIREN (Sitzmann et al., 2020). NeRF +uses volume rendering (Kajiya & Von Herzen, 1984) for +reconstructing a 3D scene using neural radiance and den- +sity fields to synthesize novel views. SIREN replaced the +popular ReLU activation function with sine functions with +modulated frequencies. Most NeRF and SIREN-based meth- +ods focus on a single 3D object or scene. In practice, we +overfit individual objects or scenes. In our paper, we focus +on generating 3D models represented by NeRF. +Single-View Supervised 3D-Aware GANs +Generative +Adversarial Nets (GANs) (Goodfellow et al., 2014) allow +for the generation of high-quality images (Yu et al., 2017; +Karras et al., 2017; 2019; 2020; Struski et al., 2022). How- +ever, GANs operate on 2D images and ignore the 3D nature +of our physical world. Therefore, it is important to use 3D +structures of objects to generate images and 3D objects. +The first approach for 3D-aware image syntheses like Visual +Object Networks (Zhu et al., 2018) and PrGANs (Gadelha +et al., 2017) first generating a voxelized 3D shape using a +3D-GAN (Wu et al., 2016) and then projecting it into 2D. +HoloGAN (Nguyen-Phuoc et al., 2019) and Block- +GAN (Nguyen-Phuoc et al., 2020) work in a similar fusion +but use implicit 3D representation for modeling 3D represen- +tation of the world. Unfortunately, using an explicit volume +representation has constrained their resolution (Lunz et al., +2020). In (Szab´o et al., 2019), authors propose using meshes +to represent 3D geometry. On the other hand, in (Liao et al., +2020) uses collections of primitives for image synthesis. +In GRAF (Schwarz et al., 2020) and π-GAN (Chan et al., +2021), authors use implicit neural radiance fields for 3D- +aware image and geometry generation. In our work, we +use NeRF instead of SIREN and hypernetwork paradigm +instead of a conditioning procedure. +Authors use a shading-guided pipeline in ShadeGAN (Pan +et al., 2021), and in GOF (Xu et al., 2021), they gradu- +ally shrink the sampling region of each camera ray. GI- +RAFFE (Niemeyer & Geiger, 2021) we first generate low- +resolution feature maps. In the second step, we passed +representation to a 2D CNN to generate outputs at a higher +resolution. +In StyleSDF (Or-El et al., 2022), authors merge an SDF- +based 3D representation with a StyleGAN2 for image gen- +eration. In (Chan et al., 2022), authors use StyleGAN2 +generator and tri-plane representation of 3D objects. Such +methods outperform other methods in the quality of gener- +ated objects but are extremely hard to train. +Hypernetworks + generative modeling +Combining hy- +pernetworks and generative models is not new. In (Ratzlaff +& Fuxin, 2019; Henning et al., 2018) authors built GANs +to generate parameters of a neural network dedicated to +regression or classification tasks. HyperVAE (Nguyen et al., +2020) is designated to encode any target distribution by +producing generative model parameters given distribution +samples. HCNAF (Oechsle et al., 2019) is a hypernetwork +that produces parameters for a conditional autoregressive +flow model (Kingma et al., 2016; Oord et al., 2018; Huang +et al., 2018). In (Skorokhodov et al., 2021) authors proposed +INR-GAN (Skorokhodov et al., 2020) uses a hypernetwork +to produce a continuous representation of images. The hy- +pernetwork can modify the shared weights by the low-cost +mechanism of factorized multiplicative modulation. +3. HyperNeRFGAN: Hypernetwork for +Generating NeRF representions +In this section, we present HyperNeRFGAN - a novel gener- +ative model for 3D objects. The main idea of the proposed +approach is that generator serves as a hypernetwork (Ha +et al., 2016) and transforms the noise vector sampled from +the known distribution to the weights of the target model. +Compared to previous works (Skorokhodov et al., 2020), +the target model is represented by NeRF (Mildenhall et al., +2021) 3D representation of the object. Consequently, it is +possible to generate many images of the object from various +perspectives in a controllable manner. Moreover, thanks to +the NeRF-based image rendering, the discriminator operates +on 2D images generated from multiple perspectives, com- +pared to GAN-based models fed by complex 3D structures. +In this section, we first briefly discuss the basic concepts +used in our approach, and further, we focus on the architec- +ture and training details. +Hypernetwork +Hypernetworks, introduced in (Ha et al., +2016), are defined as neural models that predict weights +for a different target network designed to solve a specific +task. This approach reduces the number of trainable param- +eters compared to standard methods that inject additional +information into the target model using a single embedding. +A significant reduction of the size of the target model can +be achieved since it is not sharing the global weights, but +they are returned by the hypernetwork. Making an analogy +between Hypernetworks and generative models, the authors +of (Sheikh et al., 2017), use this mechanism to generate +a diverse set of target networks approximating the same +function. +Hypernetworks are widely used in many domains, including +few-shot problems (Sendera et al., 2023) or probabilistic +regression scenarios (Zieba et al., 2020). Various methods +also use them to produce a continuous representation of 3D +objects (Spurek et al., 2020; 2022). For instance, Hyper- +Cloud (Spurek et al., 2020) represents a 3D point cloud as a +classical MLP that serves as a target model and transforms + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +Figure 3. Elements generated by model train on three classes of ShapeNet (car, plane, chairs). +points from a uniform distribution on the gaussian ball to the +point clouds that represent the desired shape. , In (Spurek +et al., 2022), the target model is represented by a Continuous +Normalizing Flow (Grathwohl et al., 2018), the generative +model that creates the point cloud from the assumed base +distribution in 3D space. +GAN +is a framework for training deep generative models +using a minimax game. The goal is to learn a generator +distribution PG(x) that matches the real data distribution +Pdata(x). GAN learns a generator network G that produces +samples from the generator distribution PG by transform- +ing a noise variable z ∼ Pnoise(z) (usually Gaussian noise +N(0, I)) into a sample G(z). The generator learns by play- +ing against an adversarial discriminator network D aiming +to distinguish between samples from the true data distri- +bution Pdata and the generator’s distribution PG. More +formally, the minimax game is given by the following ex- +pression: +minG maxD[V (D, G) = +Ex∼Pdata[log D(x)] + Ez∼Pnoise[log(1 − D(G(z)))]]. +The main advantage over other models is producing sharp +images indistinguishable from real ones. GANs are impres- +sive regarding the visual quality of images sampled from the +model, but the training process is often challenging and un- +stable. This phenomenon is caused by direct optimization of +the training objective is intractable, and the model is usually +trained by optimizing the parameters of the discriminator +and generator in alternating steps. +In recent years, many researchers focused on modifying +the vanilla GAN procedure to improve the stability of the +training process. Some modifications were based on chang- +ing the objective function to Wasserstein distance (WGAN) +(Arjovsky et al., 2017), restrictions on the gradient penal- +ties (Gulrajani et al., 2017; Kodali et al., 2017), Spectral +Normalization (Miyato et al., 2018), or imbalanced learning +rate for generator and discriminator(Gulrajani et al., 2017; + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +Figure 4. Linear interpolation examples generated with models trained on images of cars, planes, and chairs from ShapeNet (three first +lines) and CARL data set (last two rows). +Miyato et al., 2018). The model architectures were also +more deeply explored by utilizing self-attention mechanisms +SAGAN (Zhang et al., 2018), and progressively growing +ProGAN (Karras et al., 2017) and style-gan architectures +StyleGAN (Karras et al., 2019). +INR-GAN +Implicit Neural Representation GAN (Sko- +rokhodov et al., 2020) is a variant of the GAN-based model +that utilizes hypernetworks to generate parameters for the +target model instead of direct image generation. The tar- +get model, represented by simple MLP, returns the color in +RGB format for a given pixel location. The model is very +close architecturally to StyleGAN2 (Karras et al., 2020) and +has clear advantages over the direct approach, mainly be- +cause using INR-GAN enables generating images without +assuming the arbitrarily given resolution. +NeRF representation of 3D objects +NeRFs (Mildenhall +et al., 2021) represent a scene using a fully-connected archi- +tecture. As the input, NeRF takes a 5D coordinate (spatial +location x = (x, y, z) and viewing direction d = (θ, ψ)) +and it outputs an emitted color c = (r, g, b) and volume +density σ. +A vanilla NeRF uses a set of images for training. In such a +scenario, we produce many rays passing through the image +and a 3D object represented by a neural network. NeRF +approximates this 3D object with a MLP network: +FΘ : (x, d) → (c, σ), +and optimizes its weights Θ to map each input 5D coordinate +to its corresponding volume density and directional emitted +color. +The loss of NeRF is inspired by classical volume rendering +(Kajiya & Von Herzen, 1984). We render the color of all +rays passing through the scene. The volume density σ(x) +can be interpreted as the differential probability of a ray. The +expected color C(r) of camera ray r(t) = o + td (where o +is ray origin and d is direction) can be computed with an +integral. +In practice, this continuous integral is numerically estimated +using a quadrature. We use a stratified sampling approach +where we partition our ray [tn, tf] into N evenly-spaced +bins and then draw one sample uniformly at random from +within each bin: +ti ∼ U[tn + i − 1 +N +(tf − tn), tn + i +N (tf − tn)]. +We use these samples to estimate C(r) with the quadrature +rule discussed in the volume rendering review by Max (Max, +1995): +ˆC(r) = +N +� +i=1 +Ti(1 − exp(−σiδi))ci, +where T(t) = exp +� +�− +i−1 +� +j=1 +σiδi +� +� , +where δi = ti+1 − ti is the distance between adjacent sam- +ples. This function for calculating ˆC(r) from the set of +(ci, σi) values is trivially differentiable. +We then use the volume rendering procedure to render the +color of each ray from both sets of samples. Contrary to the +baseline NeRF (Mildenhall et al., 2021), where two ”coarse” + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +and ”fine” models were simultaneously trained, we use only +the ”coarse” architecture. +3.1. HyperNeRFGAN +In this work, we propose a novel GAN architecture, HyperN- +eRFGAN, for generating 3D representations. The proposed +approach utilizes INR-GAN, the implicit approach for gen- +erating samples. We postulate using the NeRF model as a +target network compared to standard INR-GAN architec- +ture, which uses the MLP model to create the output image. +Thanks to that approach, the generator creates a specific +3D representation of the scene or object by delivering the +specific NeRF parameters. +The architecture of our model is provided in Fig. 1. The +generator G takes the sample from the assumed base dis- +tribution (Gaussian) and returns the set of parameters Θ. +These parameters are further used inside the NeRF model +FΘ to transform the spatial location x = (x, y, z) to emitted +color c = (r, g, b) and volume density σ. Instead of stan- +dard linear architecture, FΘ uses factorized multiplicative +modulation (FMM) layers. +The FMM layer with input of size nin and output of size +nout can be defined as: +y = W ⊙ (A × B) · xin + b = ˜W · xin + b, +where W and b are matrices that share the parameters across +3D representations, and A, B are two modulation matrices +of shapes nout × k, k × nin respectively, created by the +generator. The parameter k controls the rank of A × B. +Higher values of k increase the expressiveness of the FMM +layer but also increase the amount of memory required by +the hypernetwork. We always use k = 10. +The INR model FΘ is a simplified version of the baseline +NeRF. To make training less computationally expensive, +we do not optimize two networks as the original NeRF. We +reject the bigger ”fine” network and only employ the smaller +“coarse” network. Additionally, we reduce the size of the +”coarse” network by decreasing the number of channels in +each hidden layer from 256 to 128. In some experiments, +we also decrease the number of layers from 8 to 4. +We differ from the baseline NeRF in one more aspect, as we +don’t use the viewing direction. That’s because the images +used for training don’t have view-dependent features like +reflections. Even though the viewing direction is not used +in our architecture, there is no reason why it couldn’t be +enabled for datasets that would benefit from it. +Our NeRF is a single MLP, which takes only the spatial +location as input: +FΘ : x → (c, σ), +In this work, we utilize the StyleGAN2 architecture, follow- +ing the design patterns of INR-GAN. The entire model is +trained using the StyleGanv2 objective in a similar way as +in INR-GAN. In each training iteration, the noise vector is +sampled and transformed using generator G to obtain the +weights of the target NeRF model FΘ. The target model is +further used to render 2D images from various angles. The +generated 2D images further serve as fake images for the +discriminator D. The role of the generator G is to create the +3D representation that enables to render 2D images that will +fool the discriminator. The discriminator aims to distinguish +between fake renders and authentic 2D images from the data +distribution. +4. Experiments +In this section, we first evaluate the quality of generating 3D +objects by HyperNeRFGAN. We use a data set containing +2D images of 3D objects obtained from ShapeNet (Zimny +et al., 2022). The data set contains 50 images of each ele- +ment from the plane, chair, and car classes. It is the most +suitable data set for our purpose since each object has a few +images of each element. Then we use CARLA (Dosovitskiy +et al., 2017), which contains images of cars. In such a case, +we have only one image per object, but still, we have photos +of objects from all sides. We can produce full 3D objects, +which can be used in VR or augmented reality. In the end, +we use classical CelebA (Liu et al., 2015) dataset, which +contains faces. From a 3D generation point of view, it is +challenging since we only have fronts of faces. In practice, +3D based generative model can be used to 3D-aware image +synthesis (Chan et al., 2022). +4.1. 3D object generation from ShapeNet +In our first experiments, we use a ShapeNet base data set +containing 50 images of each element from the plane, chair, +and car classes. Such representation is perfect for training +3D models since each element has been seen from many +views. The data was taken from (Zimny et al., 2022), where +authors train an autoencoder-based generative model. In +Fig. 3, we present objects generated from our model. In +Fig. 4, we also present linear interpolation of objects. As +we can see, objects are of very good quality, see Tab 1. +ShapeNet +cars +planes +chairs +Points2NeRF +82.1 +239 +129.3 +HyperNeRFGAN +29.6 +33.4 +22.0 +Table 1. Competition of HyperNeRFGAN and autoencoder based +model by using FID. Competition between GAN and autoencoder +and GAN is difficult. But we can obtain a better FID score. + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +Figure 5. Examples from a model trained on CARLA. +Figure 6. Meshes generated with a model trained on CARLA +dataset and with models trained on planes and chairs from +ShapeNet. +4.2. 3D object generation from CARLA data set +In the second experiment, we compare our model on +CARLA dataset with other GAN-based models: Holo- +GAN (Nguyen-Phuoc et al., 2019), GRAF (Schwarz et al., +2020) and π-GAN (Chan et al., 2021). CARLA (Dosovit- +skiy et al., 2017) contains images of cars. We have only +one image per object, but still, we have photos of objects +from all sides. Consequently, full 3D objects can be used +in VR or augmented reality. The visual comparison we +present in Fig. 2. As illustrated, we can effectively model +the transparency of glass in cars, see Fig. 5. In Tab. 2 we +present a numerical comparison of the Frechet Inception +Distance (FID), Kernel Inception Distance (KID), and In- +ception Score (IS). As can be seen, we obtain better results +than the π-GAN model. +In the case of NeRF representation, we can produce meshes, +see Fig. 6. +CARL +FID +KID +IS +HoloGAN +67.5 +3.95 +3.52 +GRAF +41.7 +2.43 +3.60 +π−GAN +29.2 +1.36 +4.27 +HyperNeRFGAN +20.5 +0.78 +4.20 +Table 2. FID, KID mean×100, and IS for CARLA dataset. +4.3. 3D-aware image synthesis from CelebA +In our third experiment, we further compare the same mod- +els as in the second experiment by changing the setup to +face generation. For this task, we utilize the CelebA (Liu +et al., 2015) dataset, which contains 200,000 high-resolution +face images of 10,000 different celebrities. We crop the im- +ages from the top of the hair to the bottom of the chin and +resize them to 128x128 resolution as π-GAN authors did. +We present quantitative results in Tab. 3. As you can notice, +HyperNeRFGAN and π-GAN achieve similar performance, +which can also be seen in Fig. 7. +5. Conclusions +In this work, we presented a novel approach to generating +NeRF representation from 2D images. Our model leverages + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +CelebA +FID +KID +IS +HoloGAN +39.7 +2.91 +1.89 +GRAF +41.1 +2.29 +2.34 +π−GAN +14.7 +0.39 +2.62 +HyperNeRFGAN +15.04 +0.66 +2.63 +Table 3. FID, KID mean×100, and IS for CelebA dataset. +HyperNeRFGAN +π-GAN +Figure 7. Comparison between HyperNeRFGAN (first 3 columns) +and π-GAN uncurated generated faces +a hypernetwork paradigm and NeRF representation of the +3D scene. HyperNeRFGAN take a Gaussian noise and +return the weights of a NeRF network that reconstructs 3D +objects from 2D images. In training, we use only unlabeled +images and a StyleGan2 discriminator. Such representation +gives several advantages over the existing approaches. First +of all, we can use NeRF instead of SIREN representation +in the GAN type algorithm. Secondly, our model is simple +and can be effectively trained on 3D objects. Finally, our +model directly produces NeRF objects without sharing some +global parameters of the rendering component. +Limitations +The main limitation of HyperNeRFGAN is +the fact that we use only 2D images instead any knowledge +about 3D object representation. In future work, we plan to +add some information about the structure of 3D meshes. +References +Achlioptas, P., Diamanti, O., Mitliagkas, I., and Guibas, L. +Learning representations and generative models for 3d +point clouds. In International conference on machine +learning, pp. 40–49. PMLR, 2018. +Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein gen- +erative adversarial networks. In International conference +on machine learning, pp. 214–223, 2017. +Arsalan Soltani, A., Huang, H., Wu, J., Kulkarni, T. D., and +Tenenbaum, J. B. Synthesizing 3d shapes via modeling +multi-view depth maps and silhouettes with deep genera- +tive networks. 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Additional qualitative results of HyperNeRFGAN. +Figure 8. Linear interpolation between latent codes with model trained on CARLA. + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +Figure 9. Elements generated by model trained on cars from ShapeNet. + +HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN +Figure 10. Elements generated by model trained on planes from ShapeNet. + +X +X +Yx \ No newline at end of file diff --git a/J9FJT4oBgHgl3EQfwy0B/content/tmp_files/load_file.txt b/J9FJT4oBgHgl3EQfwy0B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..83b685d6f82a3f12f4646728bf9fe23b3b10c545 --- /dev/null +++ b/J9FJT4oBgHgl3EQfwy0B/content/tmp_files/load_file.txt @@ -0,0 +1,968 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf,len=967 +page_content='HyperNeRFGAN: Hypernetwork approach to 3D NeRF GAN Adam Kania 1 Artur Kasymov 1 Maciej Zieba 2 Przemysław Spurek 1 Abstract Recently, generative models for 3D objects are gaining much popularity in VR and augmented reality applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Training such models us- ing standard 3D representations, like voxels or point clouds, is challenging and requires com- plex tools for proper color rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' In order to overcome this limitation, Neural Radiance Fields (NeRFs) offer a state-of-the-art quality in synthe- sizing novel views of complex 3D scenes from a small subset of 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' In the paper, we propose a generative model called HyperNeRFGAN, which uses hypernet- works paradigm to produce 3D objects repre- sented by NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Our GAN architecture leverages a hypernetwork paradigm to transfer gaussian noise into weights of NeRF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' The model is further used to render 2D novel views, and a classical 2D discriminator is utilized for training the entire GAN-based structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Our architecture produces 2D images, but we use 3D-aware NeRF representation, which forces the model to produce correct 3D objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' The advantage of the model over existing approaches is that it produces a ded- icated NeRF representation for the object without sharing some global parameters of the rendering component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' We show the superiority of our ap- proach compared to reference baselines on three challenging datasets from various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Introduction Generative Adversarial Nets (GANs) (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=', 2014) allow us to generate high-quality 2D images (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Struski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' On the other hand, maintaining similar quality for Equal contribution 1Faculty of Mathematics and Computer Science, Jagiellonian University 6 Lojasiewicza Street, 30-348 Krak´ow, Poland 2Department of Artificial Intelligence, Univer- sity of Science and Technology Wyb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Wyspianskiego 27, 50-370, Wrocław, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9FJT4oBgHgl3EQfwy0B/content/2301.11631v1.pdf'} +page_content=' Correspondence to: Przemysław Spurek 0, the open ball B(x, ε) := +� +y ∈ Γ | ¯d(x, y) < ε +� +is not +isometric to (−ε, ε) ⊂ R. If x ∈ Γ is an essential vertex, then for any model +(G, l) of Γ and any edge e ∈ E(G) we have x /∈ int (e), and so, the set of essential +vertices of Γ is a subset of E(G) for any model (G, l) of Γ. Since G is a finite +graph, Γ has only finitely many essential vertices. +4 + +Lemma 2.2 Let Γ be a metric graph, E the set of essential vertices of Γ, and +S a finite nonempty subset of Γ. Then, the set S is a vertex set of Γ if and only +if E ⊆ S. +Proof. Suppose that ∅ ̸= S is a vertex set in Γ. Then, S induces a model (G, l) +of Γ where S = V (G) and, so +Γ \ S = Γ \ V (G) ≡ +� +e∈E(G) +(0, l(e)). +If E ∩ (Γ \ S) ̸= ∅ then there exists x ∈ E and an edge e ∈ E(G) such that x ∈ +int (e) which contradicts x being an essential vertex. Therefore, E ∩ (Γ \ S) = ∅ +and E ⊆ S. Now, assume that E ⊆ S. If E = ∅ then Γ is isometric to a circle, +and so, any non-empty finite subset of Γ is a vertex set. Suppose that E ̸= ∅. Let +(G, l) be a model of Γ, and V , E be the set of vertices, edges of G respectively. +Then, the set V = � +e∈E ∂(e), where ∂(e) is the boundary set of e ⊂ Γ, is a +vertex set of Γ. As E is the set of essential vertices, and V is a vertex set, it +follows, from what we have shown above, that E ⊆ V . Now, if E = V , then E is +a vertex set. Assume that E ⊊ V . We know that the set V \ E is always finite. +If this is a one-element set i.e., V \ E = {x1}, then there exist unique edges +e1, f1 ∈ E, e1 ̸= f1 such that x1 is a common endpoint of e1 and f1. Then, we +obtain that +Γ \ E = (Γ \ V ) ∪ {x1} ≡ +� +e∈E +(0, l(e)) ∪ {x1} +≡ +� +e∈E +e̸=e1,f1 +(0, l(e)) ⊔ (0, l(e1) + l(f1)) +which implies that E is a vertex set. If V \ E = {x1, x2}, then there exist unique +edges ei, fi ∈ E with ei ̸= fi such that xi is a common endpoint of ei and fi +for i = 1, 2. In the case when one of e1 and e2 is equal to one of f1 and f2, say, +f1 = e2, we have that +Γ \ E ≡ +� +e∈E +e̸=e1,e2,f2 +(0, l(e)) ⊔ (0, l(e1) + l(e2) + l(f2)). +If both e1 and e2 are different to both f1 and f2, then +Γ \ E ≡ +� +e∈E +e̸=e1,e2,f1,f2 +(0, l(e)) ⊔ (0, l(e1) + l(e2)) ⊔ (0, l(f1) + l(f2)) +and therefore, E is a vertex set. Similarly we get we get that E is a vertex set if +V \ E = {x1, x2, . . . , xn}, Thus, Γ \ E is isometric to a disjoint union of finitely +many open real intervals. Since Γ \ S ⊂ Γ \ E, we have that Γ \ S is also is +isometric to a disjoint union of finitely many open intervals, and therefore, S is +a vertex set. □ +5 + +A metric graph is said to be a metric loop if it is isometric to a circle. If +Γ is not a metric loop, then E ̸= ∅ is a vertex set of Γ. The model (GE, lE) +induced by the essential vertex set E is called the essential model of Γ. From +Lemma 2.1, the essential model (GE, lE) is minimal in the sense that any other +model of Γ can be obtained by a sequence of edge subdivisions of GE. Thus, all +models are refinements of the essential model. In addition, this implies that the +valence of a point x ∈ Γ defined as the valence of x in GS for S a vertex set of +Γ and x ∈ S, is well-defined notion. The valence of the point x ∈ Γ is denoted +by val(x). +2.2 +Harmonic maps and tropical morphisms. +Definition 2.3 +Let Γ1 and Γ2 be metric graphs with loopless models (G1, l1) +and (G2, l2) respectively, where E(G1) = {e1} and E(G2) = {e2}. +A map +ϕ : Γ1 → Γ2 is said to be linear if there exist isometries ρ1 : Γ1 → [0, l1(e1)] and +ρ2 : Γ2 → [0, l2(e2)] such that the map ρ2 ◦ ϕ ◦ ρ−1 +1 +: [0, l1(e1)] → [0, l2(e2)] is an +affine linear map. +Definition 2.4 Let Γ1 and Γ2 be two metric graphs. A continuous map ϕ : +Γ1 → Γ2 is said to be piecewise linear if there exist loopless models (G1, l1) and +(G2, l2) of Γ1 and Γ2 respectively, such that for any edge e1 ∈ E(G1) there exists +an edge e2 ∈ E(G2) such that ϕ(e1) ⊆ e2 and ϕ|e1 : e1 → e2 is a linear map. +Let ϕ : Γ1 → Γ2 be a piecewise linear map of metric graphs, v ∈ Γ1 and +w := ϕ(v). Let (G1, l1) (resp., (G2, l2)) be loopless models of Γ1 (resp., Γ2) +such that for all e1 ∈ E(G1) there exists e2 ∈ E(G2) such that ϕ(e1) ⊆ e2, +ϕ|e1 : e1 → e2 is a linear map, and assume that v ∈ V (G1) and w ∈ V (G2). +Fix a direction ⃗w at w (i.e., a ’unit vector’ starting at w with direction of a +path emanating from w), and let e2 ∈ E(G2) such that w is an endpoint of e2 +and e2 is in the direction ⃗w. Let {ev1, ev2, . . . , evr} ⊆ E(G1) be the set of edges +emanating from v. Without loss of generality, assume that +{ev1, ev2, . . . , evs} = {evj | ϕ(evj) ⊆ e2, j = 1, 2, . . . , r} +for some s such that 0 ⩽ s ⩽ r. Then, ϕ|evj : evj → e is a linear map for +j = 1, 2, . . . , s because of the choice of models (G1, l1) and (G2, l2). Denote by +mϕ, ⃗w(v) the sum of slopes of these linear maps ϕ|evj, j = 1, 2, . . ., s. i.e., +mϕ, ⃗w(v) = +s +� +j=1 +slope (ρ ◦ ϕ ◦ ρ−1 +vj ) +where ρ : e2 → [0, l2(e2)] and ρvj : evj → [0, l1(evj)] are the chosen isometries +with unique parametrizations ρ(w) = ρvj(v) = 0 for i = 1, 2, . . . , s i.e., that map +initial endpoints of e2, evj, j = 1, 2, . . . , s to 0. This definition of the slope of +the linear maps ϕ|evj, j = 1, 2, . . . , s, and their sum mϕ, ⃗w(v) is independent of +the choice of such models (G1, l1) and (G2, l2). +6 + +Definition 2.5 A continuous map ϕ : Γ1 → Γ2 is said to be a harmonic map +of metric graphs if it is piecewise linear with integer slopes and satisfies the +harmonicity condition: For any point v ∈ Γ and any two directions ⃗w1, ⃗w2 +emanating from w := ϕ(v) we have mϕ, ⃗ +w1(v) = mϕ, ⃗ +w2(v). +Let ϕ : Γ1 → Γ2 be a harmonic map and v ∈ Γ. Then, mϕ(v) := mϕ, ⃗ +w1(v) = +mϕ, ⃗ +w2(v) for any two directions ⃗w1, ⃗w2 emanating from ϕ(v) is said to be the +local degree of ϕ at v. The degree of a non-constant harmonic map ϕ : Γ1 → Γ2 +is defined to be the sum of all local degrees of ϕ at the pre-images under ϕ of +any point w ∈ Γ′ i.e., +deg ϕ := +� +v∈Γ,ϕ(v)=w +mϕ(w) +for any w ∈ Γ′. The degree of ϕ is independent of the choice of w. (see Section +2.4, Kag18]). +Definition 2.6 A non-constant harmonic map ϕ : Γ → Γ′ of metric graphs is +said to be a tropical morphism between metric graphs if the slopes of ϕ along +the edges of linearity are nonzero and the following inequality +(k − 2) ⩾ mϕ(v) · (l − 2) +holds for all points v ∈ Γ, where k is the valence of v, and l is the valence of +w := ϕ(v). The above inequality is known as the Riemann-Hurwitz condition. +2.3 +Tree gonality. +Let Γ be a metric graph, T a metric tree, and let v ∈ Γ, w ∈ T be two points +such that val(w) = 1. Denote by Γ′ the quotient space of Γ ⊔ T with respect +to the equivalence relation ∼ that identifies v with w. The metric space Γ′ is +a metric graph, and we say that Γ′ is obtained by grafting the metric tree T +onto the point v ∈ Γ. In this article, we allow the inverse operation of grafting +a metric tree onto a point of a metric graph, and we call it deleting a metric +tree onto a point of the metric graph. +Definition 2.7 +A tropical modification of a metric graph Γ is another metric +graph Γ′ that is obtained by grafting or deleting a finite number of metric trees +onto points of Γ. +Given a tropical modification Γ′ of Γ and a tropical morphism ϕ : Γ′ → T of +metric graphs, then there exists a tropical modification Γ′′ (resp. T ′) of Γ′ (resp. +T ) respectively and a tropical morphism ϕ′ : Γ′′ → T ′ that extends ϕ and has +the same degree as ϕ (CD18]). The following definition is the key definition in +this article. +Definition 2.8 The tree gonality of a metric graph Γ, denoted by tgon(Γ), is +defined as the minimum of degrees of all tropical morphisms from any tropical +modification of Γ to any metric tree. +7 + +In order to study tree gonality and tropical morphisms of metric graphs, we +consider the equivalence relation on metric graphs under tropical modification +called tropical equivalence. Metric graphs under tropical equivalence are said to +be tropically equivalent. +First, we recall the notions of contracting and deleting an edge of a graph. +Let G = (V, E, ∂) be a graph and e ∈ E with ∂(e) = {v, w}. Contracting G +at the edge e ∈ E yields the graph G1 = (V1, E1, ∂1) where V1 := V/ ∼ where +∼ identifies v with w, E1 := E \ {e} and ∂1 : E1 → P(V1) given as follows: +for e′ ∈ E1 such that ∂(e′) = {v′, w′} we define ∂1(e′) = {p(v′), p(w′)}, where +p : V → V1 is the quotient map. Deleting the edge e ∈ E yields the graph +G′ := (V, E \ {e} , ∂|E\{e}). +Next, we work with the notion of dangling edges which is due to DV19]. +Note that we regard a singleton graph (a graph without an edge) as a tree. +Definition 2.9 +Let G be a connected graph. An edge e ∈ E(G) is said to be +dangling if deleting e gives a graph with two connected components and one of +them is a tree. +Let Γ be a metric graph with model (G, l). Assume that g(Γ) ≥ 2. Denote +by ˜G the graph obtained by successively contracting the dangling edges of G, +and let ˜l be a length map on ˜G given as the restriction of l on E( ˜G). Let ˜Γ +be metric graph which is the realization of ( ˜G, ˜l). Then, the metric graph Γ +is a tropical modification of ˜Γ, and note that by construction, ˜Γ satisfies the +following property: ˜Γ is the unique metric graph tropically equivalent to Γ whose +essential model (E, lE) has valency at least 3 i.e., every vertex point has valence +at least three. +2.4 +Hyperelliptic metric graphs. +We first recall the basic theory of divisors on metric graphs (Cha13], BN07]). +Let Γ be a metric graph. An element of the free abelian group Div(Γ) generated +by points of Γ is called a divisor on Γ. If +D = +� +v∈Γ +D(v) · v +is a divisor in Γ, then define the degree of D to be +deg(D) := +� +x∈Γ +D(v) ∈ Z +Denote by Div0(Γ) the subgroup of divisors of degree 0. A function f : Γ → R +is called rational function on Γ if it is continuous, piecewise-linear with integer +slopes along its domains of linearity. We denote by Rat(Γ) the set of rational +functions on Γ. For f ∈ Rat(Γ) and a point v in Γ, the sum of the outgoing +8 + +slopes of f at v is denoted by ordv(f). This sum is 0 except for all but finitely +many points of Γ, and therefore, +div(f) := +� +v∈Γ +ordv(f) +is a divisor on Γ. The set of principal divisors on Γ is defined to be Prin(Γ) := +{div(f) | f ∈ Rat(Γ)}. Note that Prin(Γ) is a subgroup of Div0(Γ). Two divisors +D and D′ are said to be linearly equivalent, and we write D ∼ D′, if D − D′ ∈ +Prin(Γ). A divisor D = � +v∈Γ D(v) · v ∈ Div(Γ) is said to be effective, and we +write D ⩾ 0, if D(v) ⩾ 0 for all v ∈ Γ. Denote by Divk ++(Γ) the set of all effective +divisors with degree k. For a divisor D ∈ Div(Γ) a complete linear system |D| +is defined to be |D| := {D′ ∈ Div(Γ) | D′ ⩾ 0, D′ ∼ D} . The rank of a divisor +D is defined to be −1 if |D| = ∅, and +max +� +k ∈ Z | ∀D′ ∈ Divk ++(Γ) we have |D − D′| ̸= ∅ +� +if |D| ̸= ∅. +The rank of the divisor D is simply denoted by r(D). +In the +literature, there exists a notion of a hyperelliptic metric graph. For example +in Cha13], a metric graph Γ is said to be hyperelliptic if there exists a divisor +D ∈ Div(Γ) such that deg(D) = 2 and r(D) = 1. +In this article, we give +a definiton of hyperelliptic metric graphs in terms of tropical morphisms and +their tree gonality and which is different to the one given in Cha13]. +Definition 2.10 A metric graph Γ is said to be hyperelliptic if there exists a +tropical morphism from Γ to a metric tree with degree tgon(Γ). +One of our goals in this article is to investigate genus 3 nonhyperelliptic metric +graphs Γ with tree gonality 3. Note that if Γ is hyperelliptic in the sense of +Kawaguchi-Yamaki (KY15]) that does not imply that Γ is hyperelliptic in our +sense. For example, the metric graph Γ in Figure 25 is hyperelliptic in the sense +of Kawaguchi-Yamaki but is not hyperelliptic in our sense. This is because the +harmonic map coming from the unique hyperelliptic involution ι (Theorem 3.5, +KY15]) does not satisfy the Riemann-Hurwitz condition. +9 + +3 +Construction of tropical morphisms +The main result in this article is the constructive solution given to the Problem +1 stated below. Before we do that, we give the following lemma, which will be +useful to construct tropical morphisms. +Lemma 3.1 +Let Γ = (G, l), T = (H, m) be two metric graphs where H does +not have multiple edges and ψ : V (G) → V (H) a map on the set of vertices. +Suppose that for any v, w ∈ V (G) that are the endpoints of some non-loop edge +e ∈ E(G), we have ψ(v) = ψ(w), or ψ(v) and ψ(w) are endpoints of some edge +e′ ∈ H. Then, there exists a unique continuous map ϕ : Γ → T such that +ϕ|V (G) = ψ and ϕ is linear over each edge e in G. +Proof. +If e ∈ E(G) is an edge with endpoints v, w such that ψ(v) = ψ(w), then +take ϕe : e → T to be the constant map on e with image ψ(v) = ψ(w). In the +case when e ∈ E(G) is an edge with endpoints v, w such that ψ(v) and ψ(w) are +endpoints of some edge e′ ∈ H, then choose ϕe : e → e′ to be the linear map +with slope m(e′)/l(e). Now, we take ϕ : Γ → T to be the unique continuous +map such that ϕ|e = ϕe for all edges e ∈ E(G). □ +Problem 1. Let Γ be a genus 3 metric graph with tree gonality 3 which is not +hyperelliptic. Construct a tropical modification Γ′ of Γ, a metric tree T and a +tropical morphism ϕ : Γ′ → T of degree 3. +Solution of Problem 1. Consider genus 3 nonhyperelliptic metric graphs with +tree gonality 3 up to tropical equivalence. +There is a complete list (up to +tropical equivalence) of genus 3 metric graphs (Figure 4, Cin15]), and also a +complete list of genus 3 hyperelliptic metric graphs (the tropical hyperelliptic +curves of genus 3 with unmarked vertices in Figure 2, Cha13]). Note that there +is a hyperelliptic metric graph in the latter list, namely the one in Figure 25, +which is not hyperelliptic in our sense. Based on this, now it is enough to make +the constructions for the tropically equivalent metric graphs Γ whose essential +model (G, l) has valency at least 3. They are depicted in Figures 1,5,7,9,.. . ,25. +We divide the constructions into four cases depending on the number bridges +(edges of a connected graph whose deletion increases its number of connected +components) that the essential model (G, l) possesses. +Case 1. If the metric graph Γ has no bridges, then Γ is one of the metric +graphs given in Figure 1, 5, 7, 9, 11, or 13. +Solution of Case 1. +Case 1.1. Consider the metric graph Γ whose essential model (G, l) is +given in Figure 1, where the graph G = (V, E, ∂) is given by V = {v1, v2, v3, v4}, +E = {e1, e2, . . . , e6}, and ∂(e1) = {v1, v2}, ∂(e2) = {v1, v3}, ∂(e3) = {v4, v1}, +∂(e5) = {v2, v3}, and ∂(e6) = {v2, v4}. The length map l on E is defined by +assigning e1 �→ a, e2 �→ b, e3 �→ c, e4 �→ d, e5 �→ e and e6 �→ f, where a, b, c, d, e, +and f are real positive numbers. +10 + +v1 +v2 +v3 +v4 +a +e +d +c +f +b +Curve +Figure 1. The essential model (G, l) of Γ +Choose any vertex, say v1 ∈ V (G), and without loss of generality, assume +that c ⩽ b ⩽ a. +It is enough to consider the following three subcases: (A) +c < b ⩽ a, (B) c = b < a, and (C) c = b = a. We give the constructions for each +subcase separately as follows. +Case 1.1.A. Let (G1, l1) be another model of Γ given in Figure 1.1, where +the graph G1 = (V1, E1, ∂1) is obtained by subdividing the edges ei ∈ E into +e′ +i, e′′ +i , e′′′ +i (i = 1, 2), and ej ∈ E into e′ +j, e′′ +j (j = 4, 5, 6) with orientation given by: +∂1(e′ +1) = {v1, v6} +∂1(e′′ +1) = {v6, v4} +∂1(e′′′ +1 ) = {v5, v2} +∂1(e′′ +2) = {v8, v7} +∂1(e′′′ +2 ) = {v7, v3} +∂1(e′ +4) = {v4, v9} +∂1(e′′ +4) = {v9, v3} +∂1(e′ +5) = {v2, v10} +∂1(e′′ +5) = {v3, v10} +∂1(e′ +2) = {v1, v8} +∂1(e′′ +6) = {v4, v11} +∂1(e′ +6) = {v2, v11} +and length map l1, which is equal to l on E \ {e1, e2, e4, e5, e6}, whereas on +{e1, e2, e4, e5, e6} it is equal to +l1(e′ +1) = l1(e′′ +1) = (a − c)/2 +l1(e′ +4) = l1(e′′ +4) = d/2 +l1(e′ +2) = l1(e′′ +2) = (b − c)/2 +l1(e′ +5) = l1(e′′ +5) = e/2 +l1(e′ +6) = l1(e′′′ +1 ) = l1(e′′′ +2 ) = c +l1(e′′ +6) = f/2. +T = (T ′, t′) +Γ′ = (G′, l′) +v5 +v2 +c +v3 +c +v1 +v4 +c +w1 +w0 +w6 +w8 +w10 +w11 +w9 +d +e +f +v9 +v11 +v10 +ϕ +Curve +v7 +b − c +a − c +v8 +v6 +Figure 1.1. The model (G1, l1) of Γ +11 + +Let Γ′ be the the tropical modification of Γ with model (G′, l′) in Figure 1.2, +where the graph G′ is given by its vertex set V (G′) = V1 ∪ {v′ +6, v′ +8, v′ +9, v′ +10, v′ +11}, +and edge set E(G′) = E1 ∪{v2v′ +9, v3v′ +11, v4v′ +10, v5v′ +8, v7v′ +6}. The length map l′ on +G′ is defined by l′ = l1 on E1, and +l′(v2v′ +9) = d +2 +l′(v3v′ +11) = f +2 +l′(v4v′ +10) = e +2 +l′(v5v′ +8) = b − c +2 +l′(v7v′ +6) = a − c +2 +. +T = (T ′, t′) +Γ′ = (G′, l′) +v5 +v2 +v3 +v1 +v4 +w1 +w0 +w6 +w8 +w10 +w11 +w9 +v9 +v11 +v10 +v′ +11 +v′ +10 +v′ +9 +ϕ +Curve +v7 +v8 +v6 +v′ +8 +v′ +6 +Figure 1.2. The model (G′, l′) of Γ′ +Let T be the metric tree with model (T ′, t′) in Figure 1.3, where the tree +T ′ is given with its vertex set V (T ′) = {w0, w1, w6, w8, w9, w10, w11}, and edge +set E(T ′) = {w0w1, w0w9, w0w10, w0w11, w1w6, w1w8}, whereas the length map +t′ on T is defined by +t′(w0w9) = d +2 +t′(w0w10) = e +2 +t′(w0w11) = f +2 +t′(w0w1) = c +t′(w1w8) = b − c +2 +t′(w1w6) = a − c +2 +. +12 + +T = (T ′, t′) +Γ′ = (G′, l′) +v5 +v2 +c +v3 +c +v1 +v4 +c +w1 +w0 +w6 +w8 +w10 +w11 +w9 +d +e +f +v′ +9 +v11 +v10 +ϕ +Curve +v7 +b − c +a − c +v8 +v6 +Figure 1.3. The model (T ′, t′) of T +Let ψ : V (G′) → V (T ) the map on the set of vertices given by v2, v3, v4 �→ +w0, v1, v7, v5 �→ w1, and vi, v′ +i �→ wi for i = 6, 8, 9, 10, 11. Then, the map ψ +satisfies the condition in Lemma 3.1, and so, there exist a unique continuous +map ϕ : Γ′ → T such that ϕ|V (G′) = ψ, and ϕ is linear on each edge e′ ∈ E(G′) +with slope t′(e)/l′(e′), where e = ϕ(e′) ∈ E(T ) with endpoints ψ(v) and ψ(w). +The map ϕ given in Figure 2. By construction, the models (G′, l′) and (T ′, t′) +satisfy the condition in Definition 2.4, and therefore, ϕ is a piecewise linear +function. From our choice of length maps t′, l′, the slope of ϕ|e′ is equal to 1 for +all edges e′. Thus, ϕ has non-zero integer slopes along its edges of linearity. It +is remaining to show that the map ϕ satisfies (i) the harmonicity condition and +(ii) the Riemann-Hurtswitz condition on every point v ∈ Γ′. +(i) Assume that v ∈ Γ +′ is a vertex point, say v = v1 ∈ V (G′). Then, for all +the directions ⃗w at ϕ(v) = w1, we have that mϕ, ⃗w(v1) = 1. We check that +the harmonicity condition holds on every other vertex point in a similar +fashion, and this checking process terminates because the vertex set is +finite. Whenever v is not a vertex point, say v ∈ int(e) for some edge +e ∈ E(G′), we have that ϕ(v) ∈ int(e′) where e′ = ϕ(e). Consider the +new vertex sets on Γ′ and T by adding v and w respectively. There are +only two directions ⃗w1 and ⃗w2 at ϕ(v) because val(ϕ(v)) = 2. The slopes +of ϕ at v with directions ⃗w1 and ⃗w2 at ϕ(v) are equal to the slope of the +same linear map ϕ|e i.e., mϕ, ⃗w1(v) = mϕ, ⃗w2(v), and so, we get that ϕ is +a harmonic map. Its degree is 3 because for a fixed w ∈ T , say w1, the +degree of ϕ is given by +deg(ϕ) = +� +v∈Γ′,ϕ(v)=w1 +mϕ(v) += mϕ(v1) + mϕ(v7) + mϕ(v5) += 3. +(ii) Assume that v ∈ Γ′ is a vertex point, say v = v8 ∈ V (G′). Then, mϕ(v8) = +2, val(v8) = 2, and val(ϕ(v8)) = 1. Therefore, +(val(v8) − 2) − mϕ(v8) · +� +val (ϕ(v8)) − 2 +� += 2 > 0. +Similarly, we check that the Riemann-Hurwitz condition holds on every +other vertex point. Now, assume that v is not a vertex point. Consider +13 + +the new vertex sets on Γ′ and T (just like in part (i)) by adding v and +w respectively. Then, we have that val(v) = val(ϕ(v)) = 2, and so the +Riemann-Hurwitz condition holds. +From (i) and (ii), we obtain that the map ϕ : Γ′ → T is a tropical morphism +of metric graphs of degree 3, and so, the solution for the Case 1.1.A is finished. +T +Γ′ +v2 +c +v3 +c +v1 +v4 +c +c +a−c +2 +b−c +2 +e +2 +f +2 +d +2 +d +e +f +f +2 +e +2 +d +2 +ϕ +Curve +b − c +a − c +b−c +2 +a−c +2 +Figure 2. The tropical morphism ϕ : Γ′ → T +Remark 3.1 Let ϕ : Γ′ → T be non-constant piecewise linear map with +nonzero integer slopes (as in Case 1.1.A), where the models (G′, l′), (T ′, t′) of Γ′, +T respectively, are taken so that the condition in the Definition 2.4 is satisfied. +In order to show that ϕ satisfies the harmonicity and the Riemann-Hurwitz +condition on Γ′, it is enough to check those conditions on vertex points. This is +due to the parts (i) and (ii) above. +Case 1.1.B. Let Γ′ +1 be the tropical modification of Γ with model (G′ +1, l′ +1), +where the graph G′ +1 is obtained by contracting the edges v1v8, v8v7, and v5v′ +8 +of G′ in Figure 1.2. Let T1 be the metric tree with model (T ′ +1, t′ +1), where the +tree T ′ +1 is obtained by contracting the edge w1w8 of T ′ in Figure 1.3. Next, let +ψ1 : V (G′ +1) → V (T1) the map on the set of vertices given by v2, v3, v4 �→ w0, +v1, v5 �→ w1, and vi, v′ +i �→ wi for i = 6, 9, 10, 11. +This map ψ satisfies the +condition in Lemma 3.1, and so, there exist a unique continuous map ϕ1 : Γ′ +1 → +T1, given in Figure 3, such that ϕ1|V (G′ +1) = ψ1 and ϕ1 is linear on each edge +e′ ∈ E(G′ +1) with slope t′ +1(e)/l′ +1(e′), where e = ϕ1(e′) ∈ E(T1) with endpoints +ψ1(v) and ψ1(w). Following the reasoning in (i) and (ii), we get that ϕ1 is a +tropical map of degree 3, and thus, the solution of Case 1.1.B is done. +14 + +T1 +Γ′ +1 +v2 +c +v3 +v1 +v4 +c +c +a−c +2 +e +2 +f +2 +d +2 +d +e +f +f +2 +e +2 +d +2 +ϕ1 +Curve +a − c +a−c +2 +c +Figure 3. The tropical map ϕ1 : Γ′ +1 → T1 +T ′ +2 +Γ′ +2 +v2 +v3 +v1 +v4 +c +c +e +2 +f +2 +d +2 +d +e +f +f +2 +e +2 +d +2 +ϕ2 +Curve +c +c +Figure 4. The tropical map ϕ2 : Γ′ +2 → T2 +Case 1.1.C. Let Γ′ +2 be the tropical modification of Γ with model (G′ +2, l′ +2), +where G′ +2 is obtained by contracting the edges v1v6, v6v5, v7v′ +6, v1v8, v8v7 and +v5v′ +8 of the graph G′ as in Figure 1.2. Let T2 be the metric tree with model +(T ′ +2, t′ +2), where the tree T ′ +2 is obtained by contracting the edges w1w6, w1w8 +of the tree T ′ as in Figure 1.3. Next, let ψ2 : V (G′ +2) → V (T2) the map on +the set of vertices given by v2, v3, v4 �→ w0, v1 �→ w1 and vi, v′ +i �→ wi for +i = 9, 10, 11. +The function ψ2 satisfies the condition in Lemma 3.1 and so, +there exist a unique continuous map ϕ2 : Γ′ +2 → T2, given in Figure 4, such that +ϕ2|V (G′ +2) = ψ2 and ϕ2 is linear on each edge e′ ∈ E(G′ +2) with slope t′ +2(e)/l′ +2(e′), +15 + +where e = ϕ2(e′) ∈ E(T2) with endpoints ψ2(v) and ψ2(w). +Following the +reasoning in (i) and (ii), we conclude that ϕ2 is a tropical map of degree 3, and +therefore, the solution of Case 1.1.C is finished. +Case 1.2. +Consider the metric graph Γ with essential model (G, l) in +Figure 5. The graph G is given by its vertex set V (G) = {v1, v2, v3, v4}, and +edge set E(G) = {v1v2, v3v4, e1, e2, e3, e4}, where e1, e2 (resp., e3, e4) are two +edges with endpoints v1, v4 (resp., v2, v3). The length map l : E(G) → (0, ∞) +is defined by assigning v1v2 �→ a, v3v4 �→ b, e1 �→ c, e2 �→ d, e3 �→ e and e4 �→ f +where a, b, c, d, e, and f are real positive numbers such that a < b. Note that if +a = b, then Γ is a hyperelliptic metric graph. +v1 +v2 +v4 +v3 +a +e +b +d +c +f +Curve +Figure 5. The essential model (G, l) of Γ +Let (G1, l1) be another model of Γ as in Figure 5.1. +The graph G1 is +obtained from G by subdividing the following edges: v3v4 ∈ E(G) into v3v6, +v6v5, v5v4; e1 ∈ E(G) into v1v7, v7v4; e2 ∈ E(G) into v1v8, v8v4; e3 ∈ E(G) into +v2v9, v9v3, and e4 ∈ E(G) into v2v10, v10v3, such that +l1(v3v6) = l1(v6v5) = b − a +2 +l1(v1v7) = l1(v7v4) = c +2 +l1(v1v8) = l1(v8v4) = d +2 +l1(v4v5) = a +l1(v2v9) = l1(v9v3) = e +2 +l1(v2v10) = l1(v10v3) = f +2 . +16 + +Γ′ +T +v1 +v2 +a +v3 +v4 +v5 +a +v9 +e +v10 +f +a +v6 b − a +v8 +d +v7 +c +d +2 +c +2 +f +2 +e +2 +b−a +2 +ϕ +Curve +Figure 5.1. The model (G1, l1) of Γ +Let Γ′ be the tropical modification of Γ with model (G′, l′) in Figure 5.2, +where the graph G′ is given with its vertex set V (G′) = V (G1)∪{v′ +6, v′ +7, . . . , v′ +11}, +and edge set E(G′) = {v2v′ +6, v3v′ +11, v′ +11v′ +7, v′ +11v′ +8, v5v′ +9, v5v′ +10}∪E(G1). The length +map l′ on G′ is given by l′ = l1 on E(G1), and +l′(v1v7) = l′(v7v4) = l′(v11v′ +7) = c +2 +l′(v5v′ +10) = l′(v2v10) = l′(v10, v3) = f +2 +l′(v11v′ +8) = l′(v1v8) = l′(v8v4) = d +2 +l′(v2v′ +6) = l′(v3v6) = l′(v6v5) = b − a +2 +l′(v5v′ +9) = l′(v2v9) = l′(v9, v3) = e +2 +l′(v1v2) = l′(v4v5) = l′(v3v11) = a. +Γ′ +T +v1 +v2 +a +v3 +a +v4 +v5 +a +v9 +e +v10 +f +a +v6 b − a +v8 +d +v7 +c +d +2 +c +2 +v′ +8 +v′ +7 +f +2 +e +2 +b−a +2 +v′ +6 +b−a +2 +v′ +10 +f +2 +v′ +9 +e +2 +ϕ +Curve +Figure 5.2. The model (G′, l′) of Γ′ +Choose T to be the metric tree with model (T ′, t′) in Figure 5.3, where the +tree T ′ is given by its vertex set V (T ′) = {w1, w2, w6, w7, . . . , w10}, and edge +set E(T ′) = {w1w2, w1w7, w1w8, w2w6, w2w9, w2w10}. The length map t′ on T ′ +17 + +is given by +t′(w2w1) = a +t′(w2w6) = b − a +2 +t′(w1w7) = c +2 +t′(w1w8) = d +2 +t′(w2w9) = e +2 +t′(w2w10) = f +2 . +Γ′ +T +v1 +v2 +a +v3 +v4 +v5 +a +v9 +e +v10 +f +w1 +w2 +a +v6 b − a +v8 +d +v7 +c +w8 +d +2 +w7 +c +2 +w10 +f +2 +w9 +e +2 +w6 +b−a +2 +ϕ +Curve +Figure 5.3. The model (T ′, t′) of T +Let ψ : V (G′) → V (T ) the map on the set of vertices given by v1, v4, v′ +11 �→ +w1, v2, v3, v5 �→ w2, and vi, v′ +i �→ wi for i = 6, 8, 9, 10, 11. +The function ψ +satisfies the condition in Lemma 3.1, and so, there exist a unique continuous +map ϕ : Γ′ → T , shown in Figure 6, such that ϕ|V (G′) = ψ, and ϕ is linear +on each edge e′ ∈ E(G′) with slope t′(e)/l′(e′), where e = ϕ(e′) ∈ E(T ) with +endpoints ψ(v) and ψ(w). The tropical morphism ϕ : Γ′ → T is of degree 3 +essentially because of the reasoning in (i) and (ii). +Remark 3.2 The constructions of tropical morphisms of the remaining metric +graphs are done similarly as for the metric graph in the Case 1.1.A. In order to +avoid tedious writing, we give the construction of a model, a tropical modifica- +tion, a metric tree, and a tropical morphism, using only figures from now on. +The vertices labeled with a small × are the ’midpoints’ of the edges i.e., when +subdividing an edge e into e1 and e2 then both lengths of e1 and e2 are equal +to the half of the length of edge e. +18 + +Γ′ +T +v1 +v2 +a +v3 +a +v4 +a +e +f +a +b − a +d +c +d +2 +c +2 +f +2 +e +2 +b−a +2 +b−a +2 +f +2 +e +2 +ϕ +Curve +Figure 6. The tropical morphism ϕ : Γ′ → T +v1 +v4 +v3 +d +e +c +f +b +Curve +Figure 7. The essential model (G, l) of Γ1 +Case 1.3. Consider the metric graph Γ1 with essential model (G, l) in Figure +7, where b, c, d, e, and f are real positive numbers. The model (G1, l1) which is +obtained by subdividing (G, l) is shown in Figure 7.1. The tropical modification +Γ′ +1, the metric tree T1 with models (G′ +1, l′ +1), (T ′ +1, t′ +1) is given in Figure 7.2, 7.3, +respectively. The construction of the tropical morphism ϕ1 : Γ′ +1 → T1 of degree +3 is depicted in Figure 8. +19 + +v1 +v3 +v4 +v6 +b +v9 +e +v10 +f +v7 +c +v8 +d +w1 +w7 +c +2 +w8 +d +2 +w9 +e +2 +w10 +f +2 +w6 +b +2 +Curve +Figure 7.1. The model (G1, l1) of Γ1 +Γ′ +1 +T1 +ϕ1 +v1 +v3 +v4 +v6 +b +v9 +e +v10 +f +v7 +c +v8 +d +w1 +w7 +c +2 +w8 +d +2 +w9 +e +2 +w10 +f +2 +v′ +9 +e +2 +v′ +10 +f +2 +w3 +b +2 +v′ +6 +b +2 +v′ +8 +d +2 +v′ +7 +c +2 +Curve +Figure 7.2. The model (G′ +1, l′ +1) of Γ′ +1 +v1 +v3 +v4 +v6 +b +v9 +e +v10 +f +v7 +c +v8 +d +w1 +w7 +c +2 +w8 +d +2 +w9 +e +2 +w10 +f +2 +w6 +b +2 +Curve +Figure 7.3. The model (T ′ +1, t′ +1) of T1 +20 + +Γ′ +1 +T1 +ϕ1 +v1 +v3 +v4 +v6 +b +v9 +e +v10 +f +v7 +c +v8 +d +w1 +w7 +c +2 +w8 +d +2 +w9 +e +2 +w10 +f +2 +v′ +9 +e +2 +v′ +10 +f +2 +w6 +b +2 +v′ +6 +b +2 +v′ +8 +d +2 +v′ +7 +c +2 +Curve +Figure 8. The tropical morphism ϕ1 : Γ′ +1 → T1 +Case 1.4. Consider the metric graph Γ2 with essential model in Figure +9, where a, b, c, d, and e are real positive numbers such that b > a. Note that +if a = b, then Γ2 is a hyperelliptic metric graph. The model (G1, l1) that is +obtained by subdividing (G, l) is shown in Figure 9.1. +v3 +v1 +v2 +d +b +a +e +c +Curve +Figure 9. The essential model (G, l) of Γ2 +21 + +Γ′ +2 +T ′ +2 +ϕ2 +w1 +w2 +a +w6 +b−a +2 +w9 +e +2 +w8 +d +2 +w7 +c +2 +v4 +v5 +a +v1 +v2 +a +v8 +d +v7 +c +v6 +b − a +v9 +e +Curve +Figure 9.1. The model (G1, l1) of Γ2 +The tropical modification Γ′ +2, the metric tree T2 with models (G′ +2, l′ +2), +(T ′ +2, t′ +2) is given in Figure 9.2, 9.3 respectively. +Γ′ +2 +T ′ +2 +ϕ2 +w1 +w2 +a +w6 +b−a +2 +w9 +e +2 +w8 +d +2 +w7 +c +2 +v4 +v5 +a +v1 +v2 +a +v8 +d +v7 +c +v6 +b − a +v9 +v11 +a +v′ +7 +c +2 +v′ +8 +d +2 +v′ +9 +e +2 +v′ +6 +b−a +2 +e +Curve +Figure 9.2. The model (G′ +2, l′ +2) of Γ′ +2 +Γ′ +2 +T ′ +2 +ϕ2 +w1 +w2 +a +w6 +b−a +2 +w9 +e +2 +w8 +d +2 +w7 +c +2 +v4 +v5 +a +v1 +v2 +a +v8 +d +v7 +c +v6 +b − a +v9 +v11 +a +v′ +7 +c +2 +v′ +8 +d +2 +v′ +9 +e +2 +v′ +6 +b−a +2 +e +Curve +Figure 9.3. The model (T ′ +2, t′ +2) of T2 +The construction of the tropical morphism ϕ2 : Γ′ +2 → T2 of degree 3 is +depicted in Figure 10. +22 + +Γ′ +2 +T2 +ϕ2 +a +b−a +2 +e +2 +d +2 +c +2 +v4 +a +v1 +v2 +a +d +c +b − a +a +c +2 +d +2 +e +2 +b−a +2 +e +Curve +Figure 10. The tropical morphism ϕ2 : Γ′ +2 → T2 +Case 1.5. Consider the metric graph Γ3 with essential model (G, l) in +Figure 11, where a, b, c, and e are real positive numbers such that b > a. Note +that if a = b, then Γ2 is a hyperelliptic metric graph. The model (G1, l1) which is +obtained by subdividing (G, l) is shown in Figure 11.1. The tropical modification +Γ′ +3, the metric tree T3 with models (G′ +3, l′ +3), (T ′ +3, t′ +3) is given in Figure 11.2, 11.3, +respectively. The construction of the tropical morphism ϕ3 : Γ′ +3 → T3 of degree +3 is depicted in Figure 12. +v1 +c +v2 +e +b +a +Figure 11. The essential model (G, l) of Γ3 +23 + +Γ′ +3 +T ′ +3 +ϕ3 +a +b−a +2 +e +2 +c +2 +v1 +v5 +a +v2 +v7 +v6 +b − a +v9 +v11 +a +v′ +7 +c +2 +v′ +9 +e +2 +v′ +6 +b−a +2 +e +Curve +a +c +Figure 11.1. The model (G′ +3, l′ +3) of Γ′ +3 +Γ′ +3 +T ′ +3 +ϕ3 +w1 +w2 +a +w6 +b−a +2 +w9 +e +2 +w7 +c +2 +v1 +v5 +a +v2 +v7 +v6 +b − a +v9 +v11 +a +v′ +7 +c +2 +v′ +9 +e +2 +v′ +6 +b−a +2 +e +Curve +a +c +Figure 11.2. The model (T ′ +3, t′ +3) of T3 +Γ′ +3 +T3 +ϕ3 +a +b−a +2 +e +2 +c +2 +v1 +a +v2 +b − a +a +c +2 +e +2 +b−a +2 +e +Curve +a +c +Figure 12. The tropical morphism ϕ3 : Γ′ +3 → T3 +24 + +Case 1.6. Consider the metric graph Γ4 with essential model (G, l) in +Figure 13, where b, c, d, and e are real positive numbers. The model (G1, l1) +that is obtained by subdividing (G, l) is shown in Figure 13.1. The tropical +modification Γ′ +4, the metric tree T4 with models (G′ +4, l′ +4), (T ′ +4, t′ +4) is given in +Figure 13.2, 13.3, respectively. The construction of the tropical morphism ϕ4 : +Γ′ +4 → T4 of degree 3 is depicted in Figure 14. +v3 +v1 +d +e +c +b +Curve +Figure 13. The essential model (G, l) of Γ4 +Γ′ +4 +T4 +v1 +v′ +3 +d +v′′ +3 +c +v′ +2 +b +v′ +4 +w′ +2 +b +2 +w′ +4 +w′ +3 +d +2 +w′′ +3 +c +2 +v3 +Curve +e +2 +e +Figure 13.1. The model (G1, l1) of Γ4 +Γ′ +4 +T4 +v1 +v′ +3 +d +v′′ +3 +c +v′ +2 +b +x′ +4 +v′ +4 +x′ +2 +w′ +2 +b +2 +w′ +4 +x′ +3 +d +2 +x′′ +3 +c +2 +w′ +3 +d +2 +w′′ +3 +c +2 +v3 +Curve +b +2 +e +2 +e +2 +e +Figure 13.2. The model (G′ +4, l′ +4) of Γ′ +4 +25 + +Γ′ +4 +T4 +v1 +d +c +b +w′ +2 +b +2 +w′ +4 +d +2 +c +2 +w′ +3 +d +2 +w′′ +3 +c +2 +v3 +Curve +ϕ4 +b +2 +e +2 +e +2 +e +Figure 13.3. The model (T ′ +4, t′ +4) of T4 +Γ′ +4 +T4 +v1 +d +c +b +b +2 +d +2 +c +2 +d +2 +c +2 +v3 +Curve +ϕ4 +b +2 +e +2 +e +2 +e +Figure 14. The tropical morphism ϕ4 : Γ′ +4 → T4 +Case 2. If the metric graph Γ has 1 bridge, then Γ is one of the metric +graphs given in Figure 15, 17, or 19. +Solution of Case 2. +Case 2.1. +Consider the metric graph Γ with essential model (G, l) in +Figure 15, where a, b, c, d, e, and f are real positive numbers such that b > a. +Note that if a = b, then Γ is a hyperelliptic metric graph. The model (G1, l1) +which is obtained by subdividing (G, l) is shown in Figure 15.1. The tropical +modification Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure +15.2, 15.3, respectively. The construction of the tropical morphism ϕ : Γ′ → T +of degree 3 is depicted in Figure 16. +26 + +v3 +v1 +v2 +v4 +e +d +c +f +Curve +b +a +Figure 15. The essential model (G, l) of Γ +Γ′ +T +v1 +v2 +a +v3 +v′′ +2 +a +v′ +3 +d +v′′ +3 +c +v′ +2 +b − a +v4 +f +v′ +4 +e +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +w′′ +3 +c +2 +ϕ +Curve +Figure 15.1. The model (G1, l1) of Γ +Γ′ +T +x1 +v1 +v2 +a +v3 +v′′ +2 +a +x2 +a +v′ +3 +d +v′′ +3 +c +v′ +2 +b − a +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +x′ +2 +b−a +2 +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +x′ +3 +d +2 +x′′ +3 +c +2 +w′′ +3 +c +2 +ϕ +Curve +Figure 15.2. The model (G′, l′) of Γ′ +27 + +Γ′ +T +x1 +v1 +v2 +a +v3 +v′′ +2 +a +x2 +a +v′ +3 +d +v′′ +3 +c +v′ +2 +b − a +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +x′ +3 +d +2 +c +2 +w′ +3 +d +2 +w′′ +3 +c +2 +ϕ +Curve +Figure 15.3. The model (T ′, t′) of T +Γ′ +T +v1 +v2 +a +v3 +a +a +d +c +b − a +f +v4 +f +f +e +2 +e +b−a +2 +b−a +2 +f +e +2 +a +d +2 +c +2 +d +2 +c +2 +ϕ +Curve +Figure 16. The tropical morphism ϕ : Γ′ → T +Case 2.2. Consider the metric graph Γ1 with essential model (G, l) in +Figure 17, where a, b, c, d, and e are real positive numbers such that b > a. Note +that if a = b, then Γ2 is a hyperelliptic metric graph. The model (G1, l1) that +obtained by subdividing (G, l) is shown in Figure 17.1. The tropical modification +Γ′ +1, the metric tree T1 with model (G′ +1, l′ +1), (T ′ +1, t′ +1) is given in Figure 17.2, 17.3, +respectively. The construction of the tropical morphism ϕ : Γ′ +1 → T1 of degree +3 is depicted in Figure 18. +v1 +v2 +v4 +d +e +Curve +c +a +b +Figure 17. The essential model (G, l) of Γ1 +28 + +Γ′ +1 +T1 +v1 +v2 +a +v′′ +2 +a +v′′ +3 +v′ +2 +b − a +v4 +f +v′ +4 +e +x′ +2 +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +x′′ +3 +w′′ +3 +c +2 +Curve +c +Figure 17.1. The model (G1, l1) of Γ1 +Γ′ +1 +T1 +x1 +v1 +v2 +a +v′′ +2 +a +x2 +a +v′′ +3 +v′ +2 +b − a +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +x′ +2 +b−a +2 +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +x′′ +3 +c +2 +w′′ +3 +c +2 +Curve +c +Figure 17.2. The model (G′ +1, l′ +1) of Γ′ +1 +Γ′ +1 +T1 +x1 +v1 +v2 +a +v′′ +2 +a +x2 +a +v′′ +3 +v′ +2 +b − a +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +x′ +2 +b−a +2 +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +x′′ +3 +c +2 +w′′ +3 +c +2 +Curve +c +Figure 17.3. The model (T ′ +1, t′ +1) of T1 +29 + +Γ′ +1 +T1 +ϕ1 +x1 +v1 +v2 +a +v′′ +2 +a +x2 +a +v′′ +3 +v′ +2 +b − a +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +x′ +2 +b−a +2 +w2 +w′ +2 +b−a +2 +w4 +f +w′ +4 +e +2 +w1 +a +x′′ +3 +c +2 +w′′ +3 +c +2 +Curve +c +Figure 18. The tropical morphism ϕ1 : Γ1 → T1 +Case 2.3. Consider the metric graph Γ2 with essential model in Figure +19, where a, b, c, d, e, and f are real positive numbers. The model (G1, l1) which +obtained by subdividing (G, l) is shown in Figure 19.1. The tropical modification +Γ′ +2, the metric tree T2 with model (G′ +2, l′ +2), (T ′ +2, t′ +2) 7is given in Figure 19.2, 19.3, +respectively. The construction of the tropical morphism ϕ2 : Γ′ +2 → T2 of degree +3 is depicted in Figure 20. +v1 +v3 +b +d +c +v4 +e +f +Curve +Figure 19. The essential model (G, l) of Γ2 +30 + +Γ′ +2 +T2 +v1 +v′ +3 +d +v′′ +3 +c +v′ +2 +b +v4 +f +v′ +4 +e +x′ +2 +b +2 +w1 +w′ +2 +b +2 +w4 +f +w′ +4 +e +2 +x′′ +3 +w′ +3 +d +2 +w′′ +3 +c +2 +v3 +Curve +ϕ2 +Figure 19.1. The model (G1, l1) of Γ2 +Γ′ +2 +T2 +x1 +v1 +v′ +3 +d +v′′ +3 +c +v′ +2 +b +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +x′ +2 +b +2 +w1 +w′ +2 +b +2 +w4 +f +w′ +4 +e +2 +x′ +3 +d +2 +x′′ +3 +c +2 +w′ +3 +d +2 +w′′ +3 +c +2 +v3 +Curve +ϕ2 +Figure 19.2. The model (G′ +2, l′ +2) of Γ′ +2 +Γ′ +2 +T2 +x1 +v1 +v′ +3 +d +v′′ +3 +c +v′ +2 +b +x4 +f +v4 +f +f +x′ +4 +e +2 +v′ +4 +e +x′ +2 +b +2 +w1 +w′ +2 +b +2 +w4 +f +w′ +4 +e +2 +x′ +3 +d +2 +x′′ +3 +c +2 +w′ +3 +d +2 +w′′ +3 +c +2 +v3 +Curve +ϕ2 +Figure 19.3. The model (T ′ +2, l′ +2) of T2 +31 + +Γ′ +2 +T2 +v1 +d +c +b +f +v4 +f +f +e +2 +e +b +2 +b +2 +f +e +2 +d +2 +c +2 +d +2 +c +2 +v3 +Curve +ϕ2 +Figure 20. The tropical morphism ϕ2 : Γ′ +2 → T2 +Case 3. If the metric graph Γ has 2 bridges, then Γ is one of the metric +graphs given in Figure 21 or 23. +Solution of Case 3. +Case 3.1. +Consider the metric graph Γ with essential model (G, l) in +Figure 21, where a, b, c, d, e, and f are real positive numbers such that b > a. +Note that if b = a, then Γ is a hyperelliptic metric graph. The model (G1, l1) that +obtained by subdividing (G, l) is shown in Figure 21.1. The tropical modification +Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 21.2, 21.3, +respectively. The construction of the tropical morphism ϕ : Γ′ → T of degree 3 +is depicted in Figure 22. +v1 +v2 +v4 +d +e +Curve +v3 +c +a +b +f +Figure 21. The essential model (G, l) of Γ +32 + +Γ′ +T +v3 +v1 +c +v2 +a +v4 +d +v′ +3 +f +v′ +4 +e +v′′ +2 +a +w1 +w2 +a +w′ +2 +b−a +2 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +x′ +2 +b − a +v′ +2 +ϕ +Curve +Figure 21.1. The model (G1, l1) of Γ +Γ′ +T +v3 +v1 +c +v2 +a +v4 +d +v′ +3 +f +v′ +4 +e +v′′ +2 +a +x4 +d +x′ +4 +e +2 +x2 +d +x1 +a +x3 +2c +x′ +3 +f +2 +w1 +w2 +a +w′ +2 +b−a +2 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +x′ +2 +b−a +2 +b − a v′ +2 +ϕ +Curve +Figure 21.2. The model (G′, l′) of Γ′ +Γ′ +T +v3 +v1 +c +v2 +a +v4 +d +v′ +3 +f +v′ +4 +e +v′′ +2 +a +x4 +d +x′ +4 +e +2 +x2 +d +x1 +a +x3 +2c +x′ +3 +f +2 +w1 +w2 +a +w′ +2 +b−a +2 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +x′ +2 +b−a +2 +b − a v′ +2 +ϕ +Curve +Figure 21.3. The model (T ′, t′) of T +33 + +Γ′ +T +v3 +v1 +c +v2 +a +v4 +d +f +e +a +d +e +2 +d +a +2c +f +2 +a +b−a +2 +d +e +2 +2c +f +2 +b−a +2 +b − a +ϕ +Figure 22. The tropical morphism ϕ : Γ′ → T +Case 3.2. Consider the metric graph Γ1 with essential model (G, l) in +Figure 23, where a, b, c, d, e, and f are real positive numbers. The model (G1, l1) +that is obtained by subdividing (G, l) is shown in Figure 23.1. The tropical +modification Γ′ +1, the metric tree T1 with model (G′ +1, l′ +1), (T ′ +1, t′ +1) is given in Figure +23.2, 23.3, respectively. The construction of the tropical morphism ϕ1 : Γ′ +1 → T1 +of degree 3 is depicted in Figure 24. +v3 +v1 +v4 +c +d +e +b +f +Figure 23. The essential model (G, l) of Γ1 +34 + +Γ′ +1 +T1 +ϕ1 +v3 +c +v1 +v4 +d +v′ +3 +f +v′ +4 +e +w1 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +b +v′ +2 +Curve +Figure 23.1. The model (G1, l1) of Γ1 +v3 +c +v1 +v4 +d +v′ +3 +f +v′ +4 +e +x4 +x′ +4 +e +2 +d +x1 +x3 +2c +x′ +3 +f +2 +w1 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +d +b +v′ +2 +x′ +2 +b +2 +w′ +2 +b +2 +Curve +Figure 23.2. The model (G′ +1, l′ +1) of Γ′ +1 +v3 +c +v1 +v4 +d +v′ +3 +f +v′ +4 +e +x4 +x′ +4 +e +2 +d +x1 +x3 +2c +x′ +3 +f +2 +w1 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +d +b +v′ +2 +x′ +2 +b +2 +w′ +2 +b +2 +Curve +Figure 23.3. The model (T ′ +1, t′ +1) of T1 +35 + +Γ′ +1 +T1 +ϕ1 +v3 +c +v1 +v4 +d +v′ +3 +f +v′ +4 +e +x4 +x′ +4 +e +2 +d +x1 +x3 +2c +x′ +3 +f +2 +w1 +w4 +d +w′ +4 +e +2 +w3 +2c +w′ +3 +f +2 +d +b +v′ +2 +x′ +2 +b +2 +w′ +2 +b +2 +Curve +Figure 24. The tropical morphism ϕ1 : Γ′ +1 → T1 +Case 4. If the metric graph Γ has 3 bridges, then Γ is the metric graph +given in Figure 25. +Solution of Case 4. +Consider the metric graph Γ with essential model (G, l) in Figure 25, where +a, b, c, d, e, and f are real positive numbers. Note that the metric graph Γ is +hyperelliptic in the sense of Kawaguchi-Yamaki KY15] i.e., there is a harmonic +morphism from Γ to a metric tree, but it is not hyperelliptic in our sense be- +cause the harmonic map coming from the unique hyperelliptic involution ι on +Γ (see Theorem 3.5, KY15]) is not a tropical morphism in our sense because it +does not satisfy the Riemann-Hurwitz condition. The model (G1, l1) that is ob- +tained by subdividing (G, l) is shown in Figure 25.1. The tropical modification +Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 25.2, 25.3, +respectively. The construction of the tropical morphism ϕ : Γ′ → T of degree 3 +is depicted in Figure 26. This ends our constructive solution of Problem 1. +v1 +v2 +a +v3 +b +v4 +c +f +e +d +Figure 25. The essential model (G, l) of Γ +36 + +Γ′ +T +v′ +2 +d +v2 +v1 +a +w2 +w′ +2 +v3 +b +e +v′ +3 +x3 +x′ +3 +e +2 +w3 +w′ +3 +e +2 +v4 +c +v′ +4 +w′ +4 +ϕ +f +Curve +Figure 25.1. The model (G1, l1) of Γ +Γ′ +T +v′ +2 +d +v2 +v1 +a +x2 +2a +x′ +2 +d +2 +w2 +w′ +2 +v3 +b +e +v′ +3 +x3 +2b +x′ +3 +e +2 +w3 +w′ +3 +e +2 +v4 +c +v′ +4 +x4 +2c +x′ +4 +f +2 +w′ +4 +ϕ +f +Curve +Figure 25.2. The model (G′, l′) of Γ′ +Γ′ +T +d +v2 +v1 +a +2a +d +2 +w1 +w2 +2a +w′ +2 +d +2 +v3 +b +e +2b +w3 +2b +w′ +3 +e +2 +v4 +c +2c +f +2 +w4 +2c +w′ +4 +f +2 +ϕ +f +Curve +Figure 25.3. The model (T ′, t′) of T +37 + +Γ′ +T +d +v2 +v1 +a +2a +d +2 +2a +d +2 +v3 +b +e +2b +e +2 +2b +e +2 +v4 +c +2c +f +2 +2c +f +2 +ϕ +f +Figure 26. The tropical morphism ϕ : Γ′ → T +References +[KY15] Shu Kawaguchi and Kazuhiko Yamaki, Rank of Divisors on Hyperelliptic Curves +and Graphs Under Specialization, Vol. 12, 2015. +[Cha13] Melody Chan, Tropical hyperelliptic curves, Vol. 37, 2013. +[BN07] Matthew Baker and Serguei Norine, Riemann-Roch and Abel - Jacobi theory on a +finite graph, Vol. 215, 2007. +[Cap14] Lucia Caporaso, Gonality of Algebraic Curves and Graphs, Vol. 71, 2014. +[Mik17] Grigory Mikhalkin, Tropical Geometry and Its Applications, 2017. +[BBM11] Benoˆıt Bertrand, Erwan Brugall´e, and Grigory Mikhalkin, Tropical Open Hurwitz +Numbers, Vol. 125, 2011. +[Bak08] Matthew Baker, Specialization of linear systems from curves to graphs, Vol. 2, 2008. +[BN09] Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs, +Vol. 2009, 2009. +[CKK15] Gunther Cornelissen, Fumiharo Kato, and Janne Kool, A combinatorial Li-Yau +inequality and rational points on curves, Vol. 1-2, 2015. +[BN19] Matthew Baker and Serguei Norine, Harmonic morphisms and hyperelliptic graphs, +Vol. 2009, 2019. +[CD18] Filip Cools and Jan Draisma, On Metric Graphs with Prescribed Gonality, Vol. 156, +2018. +[DV19] Jan Draisma and Alejandro Vargas, Catalan-many tropical morphisms to trees; Part +I: Constructions, https: // arxiv. org/ abs/ 1909. 12924 , 2019. +[Cin15] Zubeyir Cinkir, Admissible invariants of genus 3 curves, Manuscripta math 148 +(2015), 317-339. +[Kag18] Yuki Kageyama, Divisorial condition for the stable gonality of tropical curves, +https: // arxiv. org/ abs/ 1801. 07405 , 2018. +38 + diff --git a/JNA0T4oBgHgl3EQfCf8h/content/tmp_files/load_file.txt b/JNA0T4oBgHgl3EQfCf8h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8a05c92453681dd94793d6a7a5669934997ea2f --- /dev/null +++ b/JNA0T4oBgHgl3EQfCf8h/content/tmp_files/load_file.txt @@ -0,0 +1,793 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf,len=792 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='01989v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='AG] 5 Jan 2023 Construction of tropical morphisms from tropical modifications of nonhyperelliptic genus 3 metric graphs with tree gonality 3 to metric trees Hamdi D¨ervodeli Abstract In this article, we look into the tree gonality of genus 3 metric graphs Γ which is defined as the minimum of degrees of all tropical morphisms from any tropical modification of Γ to any metric tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' It is denoted by tgon(Γ) and is at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We define hyperelliptic metric graphs in terms of tropical morphisms and tree gonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a genus 3 metric graph with tgon(Γ) = 3 which is not hyperelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this paper, for such metric graphs Γ, we construct a tropical modification Γ′ of Γ, a metric tree T and a tropical map ϕ : Γ′ → T of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1 Metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2 Harmonic maps and tropical morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3 Tree gonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4 Hyperelliptic metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 8 3 Construction of tropical morphisms 10 1 1 Introduction We look into the tree gonality of metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Its motivation comes from the striking interplay between graphs and algebraic curves discovered over the last two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' For example, there exists a good theory of divisors on graphs (see BN07]) (also including such notions as linear systems, linear equivalences, canonical divisors, degrees, and ranks), and maps between metric graphs with suitable balancing conditions that behave similarly to morphisms between curves (see BN07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Recall that the gonality of an algebraic curve C is the minimum of degrees of all non-constant morphisms from C to the projective line P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' There are two notions of graph gonality in the literature, which are both inspired by the gonality of an algebraic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' They are tree (or geometric) gonality and divisorial gonality e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', studied for ordinary or metric graphs (see Bak08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Yet another variant is stable gonality, which is the infimum of the divisorial gonality over all subdivisions of an ordinary graph (see CKK15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We study a tropical version of gonality, where the roles of algebraic curves and the projective line are played by metric graphs and metric trees, respectively, and the morphisms are replaced by the tropical morphisms (see BN07], Cap14], Mik17], BBM11], Cha13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tree gonality of a metric graph Γ is defined as minimum of degrees of all tropical morphisms from any tropical modification of Γ to any metric tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tree gonality of any metric graph of genus g is at most � g 2 � + 1 (see Theorem 1, DV19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Its proof is entirely combinatorial and provides an explicit method to construct divisors with degree � g 2 � + 1 and rank 1 on genus-g metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this article, we are interested in constructing a degree-( � g 2 � + 1) tropical morphism from a tropical modification of Γ to a metric tree, where Γ is of genus g with tree gonality � g 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Interest for such method dates back to (Bak08], Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this regard, our modest contribution is on the case where g = 3 and Γ is not hyperelliptic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', given a nonhyperelliptic genus 3 metric graph Γ with tree gonality 3, we construct a tropical modification Γ′, a metric tree T , and a degree 3 tropical morphism φ : Γ′ → T (Problem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We emphasize that our constructions are more direct than in DV19] in the sense that we avoid constructing divisors of certain degree and rank, but rather make explicit constructions of tropical morphisms from tropical modifications of metric graphs to metric trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a genus 3 metric graph with tree gonality 3 which is not hyperelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Construct a tropical modification Γ′ of Γ, a metric tree T and a tropical morphism ϕ : Γ′ → T of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 2 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1 Metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A graph G is defined by the following data: a set V called the vertex set, a set E called the edge set and a map ∂ : E → P(V ) such that for any e ∈ E we have |∂(e)| = 1 or |∂(e)| = 2, where P(V ) is the power set of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We write G = (V, E, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The elements of V (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' E) are called vertices (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' edges) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' An edge e ∈ E with |∂(e)| = 1 is called a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Two or more edges e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , en ∈ E are called multiple edges if there exist v1, v2 ∈ V such that ∂(ei) = {vi, vj} for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The graph G is said to be finite if both V and E are finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A length map on G is any function l : E → (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this article, unless stated otherwise, a graph is always assumed to be finite with multiple edges allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let G = (V, E, ∂) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A path in the graph G is a sequence of edges (e1, e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , en−1) for which there exists a sequence of vertices (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , vn) such that ∂(ei) = {vi, vi+1} for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If w = (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , en−1) is a path in G with vertex sequence (v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , vn) then w is said to be a path from v1 to vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A graph G is said to be connected if for any two vertices v1 and v2 there exists a path from v1 to v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let e ∈ E with ∂(e) = {v, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Subdividing the edge e ∈ E with ∂(e) = {v, w} into edges e1, e2 yields the graph G′ = (V ′, E′, ∂′) where V ′ = V ∪ {z}, E′ = (E \\ {e}) ∪ {e1, e2} and ∂′ is given by ∂′|E\\{e} = ∂ and ∂′(e1) = {v, z} , ∂′(e2) = {z, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let G = (V, E, ∂) be a connected graph with no loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' An orientation on G is a map ⃗∂ : E → V × V such that if we write ⃗∂(e) = (v1, v2) then ∂(e) = {v1, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that giving an orientation ⃗∂ on G is equivalent to giving a map (∂0, ∂1) : E → V × V where ∂0, ∂1 : E → V are endpoint maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Fix an orientation (∂0, ∂1) : E → V ×V on G and choose a length map l on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let (X, d) be the disjoint union of the real metric spaces [0, l(e)] for e ∈ E i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', the set X = � e∈E [0, l(e)] := � e∈E [0, l(e)] × {e} together with the metric d : X × X → [0, ∞] given by d((x1, e1), (x2, e2)) = � |x1 − x2|, if e1 = e2 ∞, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the relation ∼1 on X defined by x ∼1 y if there exists a vertex v ∈ V such that x, y ∈ {(0, e) ∈ X | ∂0(e) = v} ∪ {(l(e), e) ∈ X | ∂1(e) = v} and let ∼ be the equivalence relation on X generated by ∼1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', x ∼ y if and only if x = y or there exists a finite subset {z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , zn} ⊂ X such that x = z1, zn = y and zi ∼1 zi+1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let ¯X := X/∼ be the quotient space of X 3 with respect to the equivalence relation ∼ and ¯d : ¯X × ¯X → [0, ∞) be given by ¯d(¯x, ¯y) := inf k � i=1 d(xi, yi) where the infimum is taken over all k ∈ N and sequences (x1, y1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , xk, yk) in X such that x1 ∈ ¯x, xi+1 ∼ yi for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', k − 1 and yk ∈ ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, Γ := ( ¯X, ¯d) is a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this case, we say that the metric space ( ¯X, ¯d) is obtained from (G, l) by gluing intervals [0, l(e)], one for each e ∈ E, along their endpoints in the manner prescribed by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We often regard each edge e ∈ E as a subset of Γ and each vertex v ∈ V as a point in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1 A metric graph is a metric space Γ such that there exists a loopless connected graph G with a length map l such that Γ is isometric to the metric space obtained from (G, l) by gluing intervals [0, l(e)], one for each e ∈ E(G), along their endpoints in the manner prescribed by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The pair (G, l) is called a model of Γ whereas Γ is called a realization of the model (G, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of a metric graph from a graph that may have loops will be given in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let G = (V, E, ∂) be a connected graph with loops and l a length function on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Subdividing all the loops e ∈ E, say into e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , en, yields a graph G′ with no loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map l′ on G′ is given by l′ = l on E \\ {e ∈ E | ∂(e) = 1} and l′(e1) + l′(e2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' + l′(en) = l(e) for edges e1, e2 for which a loop e ∈ E subdivided to e1, e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then Γ does not depend on the choice of the subdivision (G′, l′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Thus, we define Γ to be the realization of (G, l), and we also call (G, l) a model (that may have loops) of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The first Betti number of Γ is equal to g(G) := |E(G)| − |V (G)| + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' It is called the genus of Γ and it is denoted by g(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A metric graph Γ of genus g(Γ) = 0 is called a metric tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A vertex set of Γ is a finite subset S ⊂ Γ such that the subspace Γ \\ S is isometric to a disjoint union of finitely many real open intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Any vertex set S ̸= ∅ of Γ induces a model (GS, lS) of Γ in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The graph GS = (V, E, ∂) is given by its vertex set V := S, its edge set E defined to be the set of closures of finitely many connected components of Γ \\ V and the map ∂ : E → P(V ) given by e �−→ ∂(int(e)), where int(e) = e \\ S and ∂(int(e)) ⊂ V is its boundary in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Each edge e ∈ E is isometric to either a segment or a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map lS : E → (0, ∞) assigns each edge e ∈ E the length of the segment or circle isometric to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We single out a particular model for Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A point x ∈ Γ is called an essential vertex if for any ε > 0, the open ball B(x, ε) := � y ∈ Γ | ¯d(x, y) < ε � is not isometric to (−ε, ε) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If x ∈ Γ is an essential vertex, then for any model (G, l) of Γ and any edge e ∈ E(G) we have x /∈ int (e), and so, the set of essential vertices of Γ is a subset of E(G) for any model (G, l) of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Since G is a finite graph, Γ has only finitely many essential vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 4 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2 Let Γ be a metric graph, E the set of essential vertices of Γ, and S a finite nonempty subset of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, the set S is a vertex set of Γ if and only if E ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Suppose that ∅ ̸= S is a vertex set in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, S induces a model (G, l) of Γ where S = V (G) and, so Γ \\ S = Γ \\ V (G) ≡ � e∈E(G) (0, l(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If E ∩ (Γ \\ S) ̸= ∅ then there exists x ∈ E and an edge e ∈ E(G) such that x ∈ int (e) which contradicts x being an essential vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Therefore, E ∩ (Γ \\ S) = ∅ and E ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Now, assume that E ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If E = ∅ then Γ is isometric to a circle, and so, any non-empty finite subset of Γ is a vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Suppose that E ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let (G, l) be a model of Γ, and V , E be the set of vertices, edges of G respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, the set V = � e∈E ∂(e), where ∂(e) is the boundary set of e ⊂ Γ, is a vertex set of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' As E is the set of essential vertices, and V is a vertex set, it follows, from what we have shown above, that E ⊆ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Now, if E = V , then E is a vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Assume that E ⊊ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We know that the set V \\ E is always finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If this is a one-element set i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', V \\ E = {x1}, then there exist unique edges e1, f1 ∈ E, e1 ̸= f1 such that x1 is a common endpoint of e1 and f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, we obtain that Γ \\ E = (Γ \\ V ) ∪ {x1} ≡ � e∈E (0, l(e)) ∪ {x1} ≡ � e∈E e̸=e1,f1 (0, l(e)) ⊔ (0, l(e1) + l(f1)) which implies that E is a vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If V \\ E = {x1, x2}, then there exist unique edges ei, fi ∈ E with ei ̸= fi such that xi is a common endpoint of ei and fi for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In the case when one of e1 and e2 is equal to one of f1 and f2, say, f1 = e2, we have that Γ \\ E ≡ � e∈E e̸=e1,e2,f2 (0, l(e)) ⊔ (0, l(e1) + l(e2) + l(f2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If both e1 and e2 are different to both f1 and f2, then Γ \\ E ≡ � e∈E e̸=e1,e2,f1,f2 (0, l(e)) ⊔ (0, l(e1) + l(e2)) ⊔ (0, l(f1) + l(f2)) and therefore, E is a vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Similarly we get we get that E is a vertex set if V \\ E = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , xn}, Thus, Γ \\ E is isometric to a disjoint union of finitely many open real intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Since Γ \\ S ⊂ Γ \\ E, we have that Γ \\ S is also is isometric to a disjoint union of finitely many open intervals, and therefore, S is a vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' □ 5 A metric graph is said to be a metric loop if it is isometric to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If Γ is not a metric loop, then E ̸= ∅ is a vertex set of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (GE, lE) induced by the essential vertex set E is called the essential model of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1, the essential model (GE, lE) is minimal in the sense that any other model of Γ can be obtained by a sequence of edge subdivisions of GE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Thus, all models are refinements of the essential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In addition, this implies that the valence of a point x ∈ Γ defined as the valence of x in GS for S a vertex set of Γ and x ∈ S, is well-defined notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The valence of the point x ∈ Γ is denoted by val(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2 Harmonic maps and tropical morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3 Let Γ1 and Γ2 be metric graphs with loopless models (G1, l1) and (G2, l2) respectively, where E(G1) = {e1} and E(G2) = {e2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A map ϕ : Γ1 → Γ2 is said to be linear if there exist isometries ρ1 : Γ1 → [0, l1(e1)] and ρ2 : Γ2 → [0, l2(e2)] such that the map ρ2 ◦ ϕ ◦ ρ−1 1 : [0, l1(e1)] → [0, l2(e2)] is an affine linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4 Let Γ1 and Γ2 be two metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A continuous map ϕ : Γ1 → Γ2 is said to be piecewise linear if there exist loopless models (G1, l1) and (G2, l2) of Γ1 and Γ2 respectively, such that for any edge e1 ∈ E(G1) there exists an edge e2 ∈ E(G2) such that ϕ(e1) ⊆ e2 and ϕ|e1 : e1 → e2 is a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let ϕ : Γ1 → Γ2 be a piecewise linear map of metric graphs, v ∈ Γ1 and w := ϕ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let (G1, l1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', (G2, l2)) be loopless models of Γ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', Γ2) such that for all e1 ∈ E(G1) there exists e2 ∈ E(G2) such that ϕ(e1) ⊆ e2, ϕ|e1 : e1 → e2 is a linear map, and assume that v ∈ V (G1) and w ∈ V (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Fix a direction ⃗w at w (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', a ’unit vector’ starting at w with direction of a path emanating from w), and let e2 ∈ E(G2) such that w is an endpoint of e2 and e2 is in the direction ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let {ev1, ev2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , evr} ⊆ E(G1) be the set of edges emanating from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Without loss of generality, assume that {ev1, ev2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , evs} = {evj | ϕ(evj) ⊆ e2, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , r} for some s such that 0 ⩽ s ⩽ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, ϕ|evj : evj → e is a linear map for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , s because of the choice of models (G1, l1) and (G2, l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Denote by mϕ, ⃗w(v) the sum of slopes of these linear maps ϕ|evj, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', mϕ, ⃗w(v) = s � j=1 slope (ρ ◦ ϕ ◦ ρ−1 vj ) where ρ : e2 → [0, l2(e2)] and ρvj : evj → [0, l1(evj)] are the chosen isometries with unique parametrizations ρ(w) = ρvj(v) = 0 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , s i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', that map initial endpoints of e2, evj, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' This definition of the slope of the linear maps ϕ|evj, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , s, and their sum mϕ, ⃗w(v) is independent of the choice of such models (G1, l1) and (G2, l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 6 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='5 A continuous map ϕ : Γ1 → Γ2 is said to be a harmonic map of metric graphs if it is piecewise linear with integer slopes and satisfies the harmonicity condition: For any point v ∈ Γ and any two directions ⃗w1, ⃗w2 emanating from w := ϕ(v) we have mϕ, ⃗ w1(v) = mϕ, ⃗ w2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let ϕ : Γ1 → Γ2 be a harmonic map and v ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, mϕ(v) := mϕ, ⃗ w1(v) = mϕ, ⃗ w2(v) for any two directions ⃗w1, ⃗w2 emanating from ϕ(v) is said to be the local degree of ϕ at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The degree of a non-constant harmonic map ϕ : Γ1 → Γ2 is defined to be the sum of all local degrees of ϕ at the pre-images under ϕ of any point w ∈ Γ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', deg ϕ := � v∈Γ,ϕ(v)=w mϕ(w) for any w ∈ Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The degree of ϕ is independent of the choice of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4, Kag18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='6 A non-constant harmonic map ϕ : Γ → Γ′ of metric graphs is said to be a tropical morphism between metric graphs if the slopes of ϕ along the edges of linearity are nonzero and the following inequality (k − 2) ⩾ mϕ(v) · (l − 2) holds for all points v ∈ Γ, where k is the valence of v, and l is the valence of w := ϕ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The above inequality is known as the Riemann-Hurwitz condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3 Tree gonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a metric graph, T a metric tree, and let v ∈ Γ, w ∈ T be two points such that val(w) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Denote by Γ′ the quotient space of Γ ⊔ T with respect to the equivalence relation ∼ that identifies v with w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The metric space Γ′ is a metric graph, and we say that Γ′ is obtained by grafting the metric tree T onto the point v ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this article, we allow the inverse operation of grafting a metric tree onto a point of a metric graph, and we call it deleting a metric tree onto a point of the metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='7 A tropical modification of a metric graph Γ is another metric graph Γ′ that is obtained by grafting or deleting a finite number of metric trees onto points of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Given a tropical modification Γ′ of Γ and a tropical morphism ϕ : Γ′ → T of metric graphs, then there exists a tropical modification Γ′′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' T ′) of Γ′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' T ) respectively and a tropical morphism ϕ′ : Γ′′ → T ′ that extends ϕ and has the same degree as ϕ (CD18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The following definition is the key definition in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='8 The tree gonality of a metric graph Γ, denoted by tgon(Γ), is defined as the minimum of degrees of all tropical morphisms from any tropical modification of Γ to any metric tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 7 In order to study tree gonality and tropical morphisms of metric graphs, we consider the equivalence relation on metric graphs under tropical modification called tropical equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Metric graphs under tropical equivalence are said to be tropically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' First, we recall the notions of contracting and deleting an edge of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let G = (V, E, ∂) be a graph and e ∈ E with ∂(e) = {v, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Contracting G at the edge e ∈ E yields the graph G1 = (V1, E1, ∂1) where V1 := V/ ∼ where ∼ identifies v with w, E1 := E \\ {e} and ∂1 : E1 → P(V1) given as follows: for e′ ∈ E1 such that ∂(e′) = {v′, w′} we define ∂1(e′) = {p(v′), p(w′)}, where p : V → V1 is the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Deleting the edge e ∈ E yields the graph G′ := (V, E \\ {e} , ∂|E\\{e}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Next, we work with the notion of dangling edges which is due to DV19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that we regard a singleton graph (a graph without an edge) as a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='9 Let G be a connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' An edge e ∈ E(G) is said to be dangling if deleting e gives a graph with two connected components and one of them is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a metric graph with model (G, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Assume that g(Γ) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Denote by ˜G the graph obtained by successively contracting the dangling edges of G, and let ˜l be a length map on ˜G given as the restriction of l on E( ˜G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let ˜Γ be metric graph which is the realization of ( ˜G, ˜l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, the metric graph Γ is a tropical modification of ˜Γ, and note that by construction, ˜Γ satisfies the following property: ˜Γ is the unique metric graph tropically equivalent to Γ whose essential model (E, lE) has valency at least 3 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', every vertex point has valence at least three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4 Hyperelliptic metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We first recall the basic theory of divisors on metric graphs (Cha13], BN07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' An element of the free abelian group Div(Γ) generated by points of Γ is called a divisor on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If D = � v∈Γ D(v) · v is a divisor in Γ, then define the degree of D to be deg(D) := � x∈Γ D(v) ∈ Z Denote by Div0(Γ) the subgroup of divisors of degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A function f : Γ → R is called rational function on Γ if it is continuous, piecewise-linear with integer slopes along its domains of linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We denote by Rat(Γ) the set of rational functions on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' For f ∈ Rat(Γ) and a point v in Γ, the sum of the outgoing 8 slopes of f at v is denoted by ordv(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' This sum is 0 except for all but finitely many points of Γ, and therefore, div(f) := � v∈Γ ordv(f) is a divisor on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The set of principal divisors on Γ is defined to be Prin(Γ) := {div(f) | f ∈ Rat(Γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that Prin(Γ) is a subgroup of Div0(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Two divisors D and D′ are said to be linearly equivalent, and we write D ∼ D′, if D − D′ ∈ Prin(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' A divisor D = � v∈Γ D(v) · v ∈ Div(Γ) is said to be effective, and we write D ⩾ 0, if D(v) ⩾ 0 for all v ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Denote by Divk +(Γ) the set of all effective divisors with degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' For a divisor D ∈ Div(Γ) a complete linear system |D| is defined to be |D| := {D′ ∈ Div(Γ) | D′ ⩾ 0, D′ ∼ D} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The rank of a divisor D is defined to be −1 if |D| = ∅, and max � k ∈ Z | ∀D′ ∈ Divk +(Γ) we have |D − D′| ̸= ∅ � if |D| ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The rank of the divisor D is simply denoted by r(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In the literature, there exists a notion of a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' For example in Cha13], a metric graph Γ is said to be hyperelliptic if there exists a divisor D ∈ Div(Γ) such that deg(D) = 2 and r(D) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In this article, we give a definiton of hyperelliptic metric graphs in terms of tropical morphisms and their tree gonality and which is different to the one given in Cha13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='10 A metric graph Γ is said to be hyperelliptic if there exists a tropical morphism from Γ to a metric tree with degree tgon(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' One of our goals in this article is to investigate genus 3 nonhyperelliptic metric graphs Γ with tree gonality 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if Γ is hyperelliptic in the sense of Kawaguchi-Yamaki (KY15]) that does not imply that Γ is hyperelliptic in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' For example, the metric graph Γ in Figure 25 is hyperelliptic in the sense of Kawaguchi-Yamaki but is not hyperelliptic in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' This is because the harmonic map coming from the unique hyperelliptic involution ι (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='5, KY15]) does not satisfy the Riemann-Hurwitz condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 9 3 Construction of tropical morphisms The main result in this article is the constructive solution given to the Problem 1 stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Before we do that, we give the following lemma, which will be useful to construct tropical morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1 Let Γ = (G, l), T = (H, m) be two metric graphs where H does not have multiple edges and ψ : V (G) → V (H) a map on the set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Suppose that for any v, w ∈ V (G) that are the endpoints of some non-loop edge e ∈ E(G), we have ψ(v) = ψ(w), or ψ(v) and ψ(w) are endpoints of some edge e′ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, there exists a unique continuous map ϕ : Γ → T such that ϕ|V (G) = ψ and ϕ is linear over each edge e in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If e ∈ E(G) is an edge with endpoints v, w such that ψ(v) = ψ(w), then take ϕe : e → T to be the constant map on e with image ψ(v) = ψ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In the case when e ∈ E(G) is an edge with endpoints v, w such that ψ(v) and ψ(w) are endpoints of some edge e′ ∈ H, then choose ϕe : e → e′ to be the linear map with slope m(e′)/l(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Now, we take ϕ : Γ → T to be the unique continuous map such that ϕ|e = ϕe for all edges e ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' □ Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ be a genus 3 metric graph with tree gonality 3 which is not hyperelliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Construct a tropical modification Γ′ of Γ, a metric tree T and a tropical morphism ϕ : Γ′ → T of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Solution of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider genus 3 nonhyperelliptic metric graphs with tree gonality 3 up to tropical equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' There is a complete list (up to tropical equivalence) of genus 3 metric graphs (Figure 4, Cin15]), and also a complete list of genus 3 hyperelliptic metric graphs (the tropical hyperelliptic curves of genus 3 with unmarked vertices in Figure 2, Cha13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that there is a hyperelliptic metric graph in the latter list, namely the one in Figure 25, which is not hyperelliptic in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Based on this, now it is enough to make the constructions for the tropically equivalent metric graphs Γ whose essential model (G, l) has valency at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' They are depicted in Figures 1,5,7,9,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' ,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We divide the constructions into four cases depending on the number bridges (edges of a connected graph whose deletion increases its number of connected components) that the essential model (G, l) possesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If the metric graph Γ has no bridges, then Γ is one of the metric graphs given in Figure 1, 5, 7, 9, 11, or 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Solution of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ whose essential model (G, l) is given in Figure 1, where the graph G = (V, E, ∂) is given by V = {v1, v2, v3, v4}, E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , e6}, and ∂(e1) = {v1, v2}, ∂(e2) = {v1, v3}, ∂(e3) = {v4, v1}, ∂(e5) = {v2, v3}, and ∂(e6) = {v2, v4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map l on E is defined by assigning e1 �→ a, e2 �→ b, e3 �→ c, e4 �→ d, e5 �→ e and e6 �→ f, where a, b, c, d, e, and f are real positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 10 v1 v2 v3 v4 a e d c f b Curve Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ Choose any vertex, say v1 ∈ V (G), and without loss of generality, assume that c ⩽ b ⩽ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' It is enough to consider the following three subcases: (A) c < b ⩽ a, (B) c = b < a, and (C) c = b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We give the constructions for each subcase separately as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let (G1, l1) be another model of Γ given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' where the graph G1 = (V1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' ∂1) is obtained by subdividing the edges ei ∈ E into e′ i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e′′ i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e′′′ i (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' and ej ∈ E into e′ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e′′ j (j = 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 6) with orientation given by: ∂1(e′ 1) = {v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v6} ∂1(e′′ 1) = {v6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v4} ∂1(e′′′ 1 ) = {v5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v2} ∂1(e′′ 2) = {v8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v7} ∂1(e′′′ 2 ) = {v7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v3} ∂1(e′ 4) = {v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v9} ∂1(e′′ 4) = {v9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v3} ∂1(e′ 5) = {v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v10} ∂1(e′′ 5) = {v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v10} ∂1(e′ 2) = {v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v8} ∂1(e′′ 6) = {v4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v11} ∂1(e′ 6) = {v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v11} and length map l1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' which is equal to l on E \\ {e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e6},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' whereas on {e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e6} it is equal to l1(e′ 1) = l1(e′′ 1) = (a − c)/2 l1(e′ 4) = l1(e′′ 4) = d/2 l1(e′ 2) = l1(e′′ 2) = (b − c)/2 l1(e′ 5) = l1(e′′ 5) = e/2 l1(e′ 6) = l1(e′′′ 1 ) = l1(e′′′ 2 ) = c l1(e′′ 6) = f/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' T = (T ′, t′) Γ′ = (G′, l′) v5 v2 c v3 c v1 v4 c w1 w0 w6 w8 w10 w11 w9 d e f v9 v11 v10 ϕ Curve v7 b − c a − c v8 v6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ 11 Let Γ′ be the the tropical modification of Γ with model (G′, l′) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, where the graph G′ is given by its vertex set V (G′) = V1 ∪ {v′ 6, v′ 8, v′ 9, v′ 10, v′ 11}, and edge set E(G′) = E1 ∪{v2v′ 9, v3v′ 11, v4v′ 10, v5v′ 8, v7v′ 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map l′ on G′ is defined by l′ = l1 on E1, and l′(v2v′ 9) = d 2 l′(v3v′ 11) = f 2 l′(v4v′ 10) = e 2 l′(v5v′ 8) = b − c 2 l′(v7v′ 6) = a − c 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' T = (T ′, t′) Γ′ = (G′, l′) v5 v2 v3 v1 v4 w1 w0 w6 w8 w10 w11 w9 v9 v11 v10 v′ 11 v′ 10 v′ 9 ϕ Curve v7 v8 v6 v′ 8 v′ 6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′, l′) of Γ′ Let T be the metric tree with model (T ′, t′) in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, where the tree T ′ is given with its vertex set V (T ′) = {w0, w1, w6, w8, w9, w10, w11}, and edge set E(T ′) = {w0w1, w0w9, w0w10, w0w11, w1w6, w1w8}, whereas the length map t′ on T is defined by t′(w0w9) = d 2 t′(w0w10) = e 2 t′(w0w11) = f 2 t′(w0w1) = c t′(w1w8) = b − c 2 t′(w1w6) = a − c 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 12 T = (T ′, t′) Γ′ = (G′, l′) v5 v2 c v3 c v1 v4 c w1 w0 w6 w8 w10 w11 w9 d e f v′ 9 v11 v10 ϕ Curve v7 b − c a − c v8 v6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′, t′) of T Let ψ : V (G′) → V (T ) the map on the set of vertices given by v2, v3, v4 �→ w0, v1, v7, v5 �→ w1, and vi, v′ i �→ wi for i = 6, 8, 9, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, the map ψ satisfies the condition in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1, and so, there exist a unique continuous map ϕ : Γ′ → T such that ϕ|V (G′) = ψ, and ϕ is linear on each edge e′ ∈ E(G′) with slope t′(e)/l′(e′), where e = ϕ(e′) ∈ E(T ) with endpoints ψ(v) and ψ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The map ϕ given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' By construction, the models (G′, l′) and (T ′, t′) satisfy the condition in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4, and therefore, ϕ is a piecewise linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' From our choice of length maps t′, l′, the slope of ϕ|e′ is equal to 1 for all edges e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Thus, ϕ has non-zero integer slopes along its edges of linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' It is remaining to show that the map ϕ satisfies (i) the harmonicity condition and (ii) the Riemann-Hurtswitz condition on every point v ∈ Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' (i) Assume that v ∈ Γ ′ is a vertex point, say v = v1 ∈ V (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, for all the directions ⃗w at ϕ(v) = w1, we have that mϕ, ⃗w(v1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' We check that the harmonicity condition holds on every other vertex point in a similar fashion, and this checking process terminates because the vertex set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Whenever v is not a vertex point, say v ∈ int(e) for some edge e ∈ E(G′), we have that ϕ(v) ∈ int(e′) where e′ = ϕ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the new vertex sets on Γ′ and T by adding v and w respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' There are only two directions ⃗w1 and ⃗w2 at ϕ(v) because val(ϕ(v)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The slopes of ϕ at v with directions ⃗w1 and ⃗w2 at ϕ(v) are equal to the slope of the same linear map ϕ|e i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', mϕ, ⃗w1(v) = mϕ, ⃗w2(v), and so, we get that ϕ is a harmonic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Its degree is 3 because for a fixed w ∈ T , say w1, the degree of ϕ is given by deg(ϕ) = � v∈Γ′,ϕ(v)=w1 mϕ(v) = mϕ(v1) + mϕ(v7) + mϕ(v5) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' (ii) Assume that v ∈ Γ′ is a vertex point, say v = v8 ∈ V (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, mϕ(v8) = 2, val(v8) = 2, and val(ϕ(v8)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Therefore, (val(v8) − 2) − mϕ(v8) · � val (ϕ(v8)) − 2 � = 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Similarly, we check that the Riemann-Hurwitz condition holds on every other vertex point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Now, assume that v is not a vertex point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider 13 the new vertex sets on Γ′ and T (just like in part (i)) by adding v and w respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Then, we have that val(v) = val(ϕ(v)) = 2, and so the Riemann-Hurwitz condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' From (i) and (ii), we obtain that the map ϕ : Γ′ → T is a tropical morphism of metric graphs of degree 3, and so, the solution for the Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='A is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' T Γ′ v2 c v3 c v1 v4 c c a−c 2 b−c 2 e 2 f 2 d 2 d e f f 2 e 2 d 2 ϕ Curve b − c a − c b−c 2 a−c 2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ : Γ′ → T Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1 Let ϕ : Γ′ → T be non-constant piecewise linear map with nonzero integer slopes (as in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='A), where the models (G′, l′), (T ′, t′) of Γ′, T respectively, are taken so that the condition in the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In order to show that ϕ satisfies the harmonicity and the Riemann-Hurwitz condition on Γ′, it is enough to check those conditions on vertex points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' This is due to the parts (i) and (ii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ′ 1 be the tropical modification of Γ with model (G′ 1, l′ 1), where the graph G′ 1 is obtained by contracting the edges v1v8, v8v7, and v5v′ 8 of G′ in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let T1 be the metric tree with model (T ′ 1, t′ 1), where the tree T ′ 1 is obtained by contracting the edge w1w8 of T ′ in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Next, let ψ1 : V (G′ 1) → V (T1) the map on the set of vertices given by v2, v3, v4 �→ w0, v1, v5 �→ w1, and vi, v′ i �→ wi for i = 6, 9, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' This map ψ satisfies the condition in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1, and so, there exist a unique continuous map ϕ1 : Γ′ 1 → T1, given in Figure 3, such that ϕ1|V (G′ 1) = ψ1 and ϕ1 is linear on each edge e′ ∈ E(G′ 1) with slope t′ 1(e)/l′ 1(e′), where e = ϕ1(e′) ∈ E(T1) with endpoints ψ1(v) and ψ1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Following the reasoning in (i) and (ii), we get that ϕ1 is a tropical map of degree 3, and thus, the solution of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='B is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 14 T1 Γ′ 1 v2 c v3 v1 v4 c c a−c 2 e 2 f 2 d 2 d e f f 2 e 2 d 2 ϕ1 Curve a − c a−c 2 c Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical map ϕ1 : Γ′ 1 → T1 T ′ 2 Γ′ 2 v2 v3 v1 v4 c c e 2 f 2 d 2 d e f f 2 e 2 d 2 ϕ2 Curve c c Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical map ϕ2 : Γ′ 2 → T2 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let Γ′ 2 be the tropical modification of Γ with model (G′ 2, l′ 2), where G′ 2 is obtained by contracting the edges v1v6, v6v5, v7v′ 6, v1v8, v8v7 and v5v′ 8 of the graph G′ as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Let T2 be the metric tree with model (T ′ 2, t′ 2), where the tree T ′ 2 is obtained by contracting the edges w1w6, w1w8 of the tree T ′ as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Next, let ψ2 : V (G′ 2) → V (T2) the map on the set of vertices given by v2, v3, v4 �→ w0, v1 �→ w1 and vi, v′ i �→ wi for i = 9, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The function ψ2 satisfies the condition in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1 and so, there exist a unique continuous map ϕ2 : Γ′ 2 → T2, given in Figure 4, such that ϕ2|V (G′ 2) = ψ2 and ϕ2 is linear on each edge e′ ∈ E(G′ 2) with slope t′ 2(e)/l′ 2(e′), 15 where e = ϕ2(e′) ∈ E(T2) with endpoints ψ2(v) and ψ2(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Following the reasoning in (i) and (ii), we conclude that ϕ2 is a tropical map of degree 3, and therefore, the solution of Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='C is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ with essential model (G, l) in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The graph G is given by its vertex set V (G) = {v1, v2, v3, v4}, and edge set E(G) = {v1v2, v3v4, e1, e2, e3, e4}, where e1, e2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', e3, e4) are two edges with endpoints v1, v4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', v2, v3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map l : E(G) → (0, ∞) is defined by assigning v1v2 �→ a, v3v4 �→ b, e1 �→ c, e2 �→ d, e3 �→ e and e4 �→ f where a, b, c, d, e, and f are real positive numbers such that a < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if a = b, then Γ is a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v1 v2 v4 v3 a e b d c f Curve Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ Let (G1, l1) be another model of Γ as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The graph G1 is obtained from G by subdividing the following edges: v3v4 ∈ E(G) into v3v6, v6v5, v5v4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e1 ∈ E(G) into v1v7, v7v4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e2 ∈ E(G) into v1v8, v8v4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' e3 ∈ E(G) into v2v9, v9v3, and e4 ∈ E(G) into v2v10, v10v3, such that l1(v3v6) = l1(v6v5) = b − a 2 l1(v1v7) = l1(v7v4) = c 2 l1(v1v8) = l1(v8v4) = d 2 l1(v4v5) = a l1(v2v9) = l1(v9v3) = e 2 l1(v2v10) = l1(v10v3) = f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 16 Γ′ T v1 v2 a v3 v4 v5 a v9 e v10 f a v6 b − a v8 d v7 c d 2 c 2 f 2 e 2 b−a 2 ϕ Curve Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ Let Γ′ be the tropical modification of Γ with model (G′, l′) in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, where the graph G′ is given with its vertex set V (G′) = V (G1)∪{v′ 6, v′ 7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , v′ 11}, and edge set E(G′) = {v2v′ 6, v3v′ 11, v′ 11v′ 7, v′ 11v′ 8, v5v′ 9, v5v′ 10}∪E(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map l′ on G′ is given by l′ = l1 on E(G1), and l′(v1v7) = l′(v7v4) = l′(v11v′ 7) = c 2 l′(v5v′ 10) = l′(v2v10) = l′(v10, v3) = f 2 l′(v11v′ 8) = l′(v1v8) = l′(v8v4) = d 2 l′(v2v′ 6) = l′(v3v6) = l′(v6v5) = b − a 2 l′(v5v′ 9) = l′(v2v9) = l′(v9, v3) = e 2 l′(v1v2) = l′(v4v5) = l′(v3v11) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Γ′ T v1 v2 a v3 a v4 v5 a v9 e v10 f a v6 b − a v8 d v7 c d 2 c 2 v′ 8 v′ 7 f 2 e 2 b−a 2 v′ 6 b−a 2 v′ 10 f 2 v′ 9 e 2 ϕ Curve Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′, l′) of Γ′ Choose T to be the metric tree with model (T ′, t′) in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, where the tree T ′ is given by its vertex set V (T ′) = {w1, w2, w6, w7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' , w10}, and edge set E(T ′) = {w1w2, w1w7, w1w8, w2w6, w2w9, w2w10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The length map t′ on T ′ 17 is given by t′(w2w1) = a t′(w2w6) = b − a 2 t′(w1w7) = c 2 t′(w1w8) = d 2 t′(w2w9) = e 2 t′(w2w10) = f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Γ′ T v1 v2 a v3 v4 v5 a v9 e v10 f w1 w2 a v6 b − a v8 d v7 c w8 d 2 w7 c 2 w10 f 2 w9 e 2 w6 b−a 2 ϕ Curve Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′, t′) of T Let ψ : V (G′) → V (T ) the map on the set of vertices given by v1, v4, v′ 11 �→ w1, v2, v3, v5 �→ w2, and vi, v′ i �→ wi for i = 6, 8, 9, 10, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The function ψ satisfies the condition in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1, and so, there exist a unique continuous map ϕ : Γ′ → T , shown in Figure 6, such that ϕ|V (G′) = ψ, and ϕ is linear on each edge e′ ∈ E(G′) with slope t′(e)/l′(e′), where e = ϕ(e′) ∈ E(T ) with endpoints ψ(v) and ψ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ : Γ′ → T is of degree 3 essentially because of the reasoning in (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2 The constructions of tropical morphisms of the remaining metric graphs are done similarly as for the metric graph in the Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' In order to avoid tedious writing, we give the construction of a model, a tropical modifica- tion, a metric tree, and a tropical morphism, using only figures from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The vertices labeled with a small × are the ’midpoints’ of the edges i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', when subdividing an edge e into e1 and e2 then both lengths of e1 and e2 are equal to the half of the length of edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 18 Γ′ T v1 v2 a v3 a v4 a e f a b − a d c d 2 c 2 f 2 e 2 b−a 2 b−a 2 f 2 e 2 ϕ Curve Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ : Γ′ → T v1 v4 v3 d e c f b Curve Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ1 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ1 with essential model (G, l) in Figure 7, where b, c, d, e, and f are real positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) which is obtained by subdividing (G, l) is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′ 1, the metric tree T1 with models (G′ 1, l′ 1), (T ′ 1, t′ 1) is given in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ1 : Γ′ 1 → T1 of degree 3 is depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 19 v1 v3 v4 v6 b v9 e v10 f v7 c v8 d w1 w7 c 2 w8 d 2 w9 e 2 w10 f 2 w6 b 2 Curve Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ1 Γ′ 1 T1 ϕ1 v1 v3 v4 v6 b v9 e v10 f v7 c v8 d w1 w7 c 2 w8 d 2 w9 e 2 w10 f 2 v′ 9 e 2 v′ 10 f 2 w3 b 2 v′ 6 b 2 v′ 8 d 2 v′ 7 c 2 Curve Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 1, l′ 1) of Γ′ 1 v1 v3 v4 v6 b v9 e v10 f v7 c v8 d w1 w7 c 2 w8 d 2 w9 e 2 w10 f 2 w6 b 2 Curve Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 1, t′ 1) of T1 20 Γ′ 1 T1 ϕ1 v1 v3 v4 v6 b v9 e v10 f v7 c v8 d w1 w7 c 2 w8 d 2 w9 e 2 w10 f 2 v′ 9 e 2 v′ 10 f 2 w6 b 2 v′ 6 b 2 v′ 8 d 2 v′ 7 c 2 Curve Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ1 : Γ′ 1 → T1 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ2 with essential model in Figure 9, where a, b, c, d, and e are real positive numbers such that b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if a = b, then Γ2 is a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) that is obtained by subdividing (G, l) is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v3 v1 v2 d b a e c Curve Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ2 21 Γ′ 2 T ′ 2 ϕ2 w1 w2 a w6 b−a 2 w9 e 2 w8 d 2 w7 c 2 v4 v5 a v1 v2 a v8 d v7 c v6 b − a v9 e Curve Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ2 The tropical modification Γ′ 2, the metric tree T2 with models (G′ 2, l′ 2), (T ′ 2, t′ 2) is given in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Γ′ 2 T ′ 2 ϕ2 w1 w2 a w6 b−a 2 w9 e 2 w8 d 2 w7 c 2 v4 v5 a v1 v2 a v8 d v7 c v6 b − a v9 v11 a v′ 7 c 2 v′ 8 d 2 v′ 9 e 2 v′ 6 b−a 2 e Curve Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 2, l′ 2) of Γ′ 2 Γ′ 2 T ′ 2 ϕ2 w1 w2 a w6 b−a 2 w9 e 2 w8 d 2 w7 c 2 v4 v5 a v1 v2 a v8 d v7 c v6 b − a v9 v11 a v′ 7 c 2 v′ 8 d 2 v′ 9 e 2 v′ 6 b−a 2 e Curve Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 2, t′ 2) of T2 The construction of the tropical morphism ϕ2 : Γ′ 2 → T2 of degree 3 is depicted in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 22 Γ′ 2 T2 ϕ2 a b−a 2 e 2 d 2 c 2 v4 a v1 v2 a d c b − a a c 2 d 2 e 2 b−a 2 e Curve Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ2 : Γ′ 2 → T2 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ3 with essential model (G, l) in Figure 11, where a, b, c, and e are real positive numbers such that b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if a = b, then Γ2 is a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) which is obtained by subdividing (G, l) is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′ 3, the metric tree T3 with models (G′ 3, l′ 3), (T ′ 3, t′ 3) is given in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ3 : Γ′ 3 → T3 of degree 3 is depicted in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v1 c v2 e b a Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ3 23 Γ′ 3 T ′ 3 ϕ3 a b−a 2 e 2 c 2 v1 v5 a v2 v7 v6 b − a v9 v11 a v′ 7 c 2 v′ 9 e 2 v′ 6 b−a 2 e Curve a c Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 3, l′ 3) of Γ′ 3 Γ′ 3 T ′ 3 ϕ3 w1 w2 a w6 b−a 2 w9 e 2 w7 c 2 v1 v5 a v2 v7 v6 b − a v9 v11 a v′ 7 c 2 v′ 9 e 2 v′ 6 b−a 2 e Curve a c Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 3, t′ 3) of T3 Γ′ 3 T3 ϕ3 a b−a 2 e 2 c 2 v1 a v2 b − a a c 2 e 2 b−a 2 e Curve a c Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ3 : Γ′ 3 → T3 24 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ4 with essential model (G, l) in Figure 13, where b, c, d, and e are real positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) that is obtained by subdividing (G, l) is shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′ 4, the metric tree T4 with models (G′ 4, l′ 4), (T ′ 4, t′ 4) is given in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ4 : Γ′ 4 → T4 of degree 3 is depicted in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v3 v1 d e c b Curve Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ4 Γ′ 4 T4 v1 v′ 3 d v′′ 3 c v′ 2 b v′ 4 w′ 2 b 2 w′ 4 w′ 3 d 2 w′′ 3 c 2 v3 Curve e 2 e Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ4 Γ′ 4 T4 v1 v′ 3 d v′′ 3 c v′ 2 b x′ 4 v′ 4 x′ 2 w′ 2 b 2 w′ 4 x′ 3 d 2 x′′ 3 c 2 w′ 3 d 2 w′′ 3 c 2 v3 Curve b 2 e 2 e 2 e Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 4, l′ 4) of Γ′ 4 25 Γ′ 4 T4 v1 d c b w′ 2 b 2 w′ 4 d 2 c 2 w′ 3 d 2 w′′ 3 c 2 v3 Curve ϕ4 b 2 e 2 e 2 e Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 4, t′ 4) of T4 Γ′ 4 T4 v1 d c b b 2 d 2 c 2 d 2 c 2 v3 Curve ϕ4 b 2 e 2 e 2 e Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ4 : Γ′ 4 → T4 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If the metric graph Γ has 1 bridge, then Γ is one of the metric graphs given in Figure 15, 17, or 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Solution of Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ with essential model (G, l) in Figure 15, where a, b, c, d, e, and f are real positive numbers such that b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if a = b, then Γ is a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) which is obtained by subdividing (G, l) is shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ : Γ′ → T of degree 3 is depicted in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 26 v3 v1 v2 v4 e d c f Curve b a Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ Γ′ T v1 v2 a v3 v′′ 2 a v′ 3 d v′′ 3 c v′ 2 b − a v4 f v′ 4 e w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a w′′ 3 c 2 ϕ Curve Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ Γ′ T x1 v1 v2 a v3 v′′ 2 a x2 a v′ 3 d v′′ 3 c v′ 2 b − a x4 f v4 f f x′ 4 e 2 v′ 4 e x′ 2 b−a 2 w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a x′ 3 d 2 x′′ 3 c 2 w′′ 3 c 2 ϕ Curve Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′, l′) of Γ′ 27 Γ′ T x1 v1 v2 a v3 v′′ 2 a x2 a v′ 3 d v′′ 3 c v′ 2 b − a x4 f v4 f f x′ 4 e 2 v′ 4 e w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a x′ 3 d 2 c 2 w′ 3 d 2 w′′ 3 c 2 ϕ Curve Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′, t′) of T Γ′ T v1 v2 a v3 a a d c b − a f v4 f f e 2 e b−a 2 b−a 2 f e 2 a d 2 c 2 d 2 c 2 ϕ Curve Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ : Γ′ → T Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ1 with essential model (G, l) in Figure 17, where a, b, c, d, and e are real positive numbers such that b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if a = b, then Γ2 is a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) that obtained by subdividing (G, l) is shown in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′ 1, the metric tree T1 with model (G′ 1, l′ 1), (T ′ 1, t′ 1) is given in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ : Γ′ 1 → T1 of degree 3 is depicted in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v1 v2 v4 d e Curve c a b Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ1 28 Γ′ 1 T1 v1 v2 a v′′ 2 a v′′ 3 v′ 2 b − a v4 f v′ 4 e x′ 2 w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a x′′ 3 w′′ 3 c 2 Curve c Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ1 Γ′ 1 T1 x1 v1 v2 a v′′ 2 a x2 a v′′ 3 v′ 2 b − a x4 f v4 f f x′ 4 e 2 v′ 4 e x′ 2 b−a 2 w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a x′′ 3 c 2 w′′ 3 c 2 Curve c Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 1, l′ 1) of Γ′ 1 Γ′ 1 T1 x1 v1 v2 a v′′ 2 a x2 a v′′ 3 v′ 2 b − a x4 f v4 f f x′ 4 e 2 v′ 4 e x′ 2 b−a 2 w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a x′′ 3 c 2 w′′ 3 c 2 Curve c Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 1, t′ 1) of T1 29 Γ′ 1 T1 ϕ1 x1 v1 v2 a v′′ 2 a x2 a v′′ 3 v′ 2 b − a x4 f v4 f f x′ 4 e 2 v′ 4 e x′ 2 b−a 2 w2 w′ 2 b−a 2 w4 f w′ 4 e 2 w1 a x′′ 3 c 2 w′′ 3 c 2 Curve c Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ1 : Γ1 → T1 Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ2 with essential model in Figure 19, where a, b, c, d, e, and f are real positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) which obtained by subdividing (G, l) is shown in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′ 2, the metric tree T2 with model (G′ 2, l′ 2), (T ′ 2, t′ 2) 7is given in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ2 : Γ′ 2 → T2 of degree 3 is depicted in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v1 v3 b d c v4 e f Curve Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ2 30 Γ′ 2 T2 v1 v′ 3 d v′′ 3 c v′ 2 b v4 f v′ 4 e x′ 2 b 2 w1 w′ 2 b 2 w4 f w′ 4 e 2 x′′ 3 w′ 3 d 2 w′′ 3 c 2 v3 Curve ϕ2 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ2 Γ′ 2 T2 x1 v1 v′ 3 d v′′ 3 c v′ 2 b x4 f v4 f f x′ 4 e 2 v′ 4 e x′ 2 b 2 w1 w′ 2 b 2 w4 f w′ 4 e 2 x′ 3 d 2 x′′ 3 c 2 w′ 3 d 2 w′′ 3 c 2 v3 Curve ϕ2 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 2, l′ 2) of Γ′ 2 Γ′ 2 T2 x1 v1 v′ 3 d v′′ 3 c v′ 2 b x4 f v4 f f x′ 4 e 2 v′ 4 e x′ 2 b 2 w1 w′ 2 b 2 w4 f w′ 4 e 2 x′ 3 d 2 x′′ 3 c 2 w′ 3 d 2 w′′ 3 c 2 v3 Curve ϕ2 Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 2, l′ 2) of T2 31 Γ′ 2 T2 v1 d c b f v4 f f e 2 e b 2 b 2 f e 2 d 2 c 2 d 2 c 2 v3 Curve ϕ2 Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ2 : Γ′ 2 → T2 Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If the metric graph Γ has 2 bridges, then Γ is one of the metric graphs given in Figure 21 or 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Solution of Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ with essential model (G, l) in Figure 21, where a, b, c, d, e, and f are real positive numbers such that b > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that if b = a, then Γ is a hyperelliptic metric graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) that obtained by subdividing (G, l) is shown in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ : Γ′ → T of degree 3 is depicted in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v1 v2 v4 d e Curve v3 c a b f Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ 32 Γ′ T v3 v1 c v2 a v4 d v′ 3 f v′ 4 e v′′ 2 a w1 w2 a w′ 2 b−a 2 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 x′ 2 b − a v′ 2 ϕ Curve Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ Γ′ T v3 v1 c v2 a v4 d v′ 3 f v′ 4 e v′′ 2 a x4 d x′ 4 e 2 x2 d x1 a x3 2c x′ 3 f 2 w1 w2 a w′ 2 b−a 2 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 x′ 2 b−a 2 b − a v′ 2 ϕ Curve Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′, l′) of Γ′ Γ′ T v3 v1 c v2 a v4 d v′ 3 f v′ 4 e v′′ 2 a x4 d x′ 4 e 2 x2 d x1 a x3 2c x′ 3 f 2 w1 w2 a w′ 2 b−a 2 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 x′ 2 b−a 2 b − a v′ 2 ϕ Curve Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′, t′) of T 33 Γ′ T v3 v1 c v2 a v4 d f e a d e 2 d a 2c f 2 a b−a 2 d e 2 2c f 2 b−a 2 b − a ϕ Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ : Γ′ → T Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ1 with essential model (G, l) in Figure 23, where a, b, c, d, e, and f are real positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) that is obtained by subdividing (G, l) is shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′ 1, the metric tree T1 with model (G′ 1, l′ 1), (T ′ 1, t′ 1) is given in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ1 : Γ′ 1 → T1 of degree 3 is depicted in Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v3 v1 v4 c d e b f Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ1 34 Γ′ 1 T1 ϕ1 v3 c v1 v4 d v′ 3 f v′ 4 e w1 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 b v′ 2 Curve Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ1 v3 c v1 v4 d v′ 3 f v′ 4 e x4 x′ 4 e 2 d x1 x3 2c x′ 3 f 2 w1 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 d b v′ 2 x′ 2 b 2 w′ 2 b 2 Curve Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′ 1, l′ 1) of Γ′ 1 v3 c v1 v4 d v′ 3 f v′ 4 e x4 x′ 4 e 2 d x1 x3 2c x′ 3 f 2 w1 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 d b v′ 2 x′ 2 b 2 w′ 2 b 2 Curve Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′ 1, t′ 1) of T1 35 Γ′ 1 T1 ϕ1 v3 c v1 v4 d v′ 3 f v′ 4 e x4 x′ 4 e 2 d x1 x3 2c x′ 3 f 2 w1 w4 d w′ 4 e 2 w3 2c w′ 3 f 2 d b v′ 2 x′ 2 b 2 w′ 2 b 2 Curve Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ1 : Γ′ 1 → T1 Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' If the metric graph Γ has 3 bridges, then Γ is the metric graph given in Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Solution of Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Consider the metric graph Γ with essential model (G, l) in Figure 25, where a, b, c, d, e, and f are real positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' Note that the metric graph Γ is hyperelliptic in the sense of Kawaguchi-Yamaki KY15] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=', there is a harmonic morphism from Γ to a metric tree, but it is not hyperelliptic in our sense be- cause the harmonic map coming from the unique hyperelliptic involution ι on Γ (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='5, KY15]) is not a tropical morphism in our sense because it does not satisfy the Riemann-Hurwitz condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) that is ob- tained by subdividing (G, l) is shown in Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical modification Γ′, the metric tree T with model (G′, l′), (T ′, t′) is given in Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The construction of the tropical morphism ϕ : Γ′ → T of degree 3 is depicted in Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' This ends our constructive solution of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' v1 v2 a v3 b v4 c f e d Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The essential model (G, l) of Γ 36 Γ′ T v′ 2 d v2 v1 a w2 w′ 2 v3 b e v′ 3 x3 x′ 3 e 2 w3 w′ 3 e 2 v4 c v′ 4 w′ 4 ϕ f Curve Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G1, l1) of Γ Γ′ T v′ 2 d v2 v1 a x2 2a x′ 2 d 2 w2 w′ 2 v3 b e v′ 3 x3 2b x′ 3 e 2 w3 w′ 3 e 2 v4 c v′ 4 x4 2c x′ 4 f 2 w′ 4 ϕ f Curve Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (G′, l′) of Γ′ Γ′ T d v2 v1 a 2a d 2 w1 w2 2a w′ 2 d 2 v3 b e 2b w3 2b w′ 3 e 2 v4 c 2c f 2 w4 2c w′ 4 f 2 ϕ f Curve Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The model (T ′, t′) of T 37 Γ′ T d v2 v1 a 2a d 2 2a d 2 v3 b e 2b e 2 2b e 2 v4 c 2c f 2 2c f 2 ϕ f Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' The tropical morphism ϕ : Γ′ → T References [KY15] Shu Kawaguchi and Kazuhiko Yamaki, Rank of Divisors on Hyperelliptic Curves and Graphs Under Specialization, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' 12, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNA0T4oBgHgl3EQfCf8h/content/2301.01989v1.pdf'} +page_content=' [Cha13] Melody Chan, Tropical hyperelliptic curves, Vol.' 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+1,2623 @@ +1 +Marine IoT Systems with Space-Air-Sea Integrated +Networks: Hybrid LEO and UAV Edge Computing +Sooyeob Jung, Seongah Jeong, Jinkyu Kang, and Joonhyuk Kang +Abstract—Marine Internet of Things (IoT) systems have grown +substantially with the development of non-terrestrial networks +(NTN) via aerial and space vehicles in the upcoming sixth- +generation (6G), thereby assisting environment protection, mili- +tary reconnaissance, and sea transportation. Due to unpredictable +climate changes and the extreme channel conditions of maritime +networks, however, it is challenging to efficiently and reliably +collect and compute a huge amount of maritime data. In +this paper, we propose a hybrid low-Earth orbit (LEO) and +unmanned aerial vehicle (UAV) edge computing method in space- +air-sea integrated networks for marine IoT systems. Specifically, +two types of edge servers mounted on UAVs and LEO satellites +are endowed with computational capabilities for the real-time +utilization of a sizable data collected from ocean IoT sensors. +Our system aims at minimizing the total energy consumption +of the battery-constrained UAV by jointly optimizing the bit +allocation of communication and computation along with the +UAV path planning under latency, energy budget and opera- +tional constraints. For availability and practicality, the proposed +methods were developed for three different cases according to +the accessibility of the LEO satellite, “Always On,” “Always +Off” and “Intermediate Disconnected”, by leveraging successive +convex approximation (SCA) strategies. Via numerical results, +we verify that significant energy savings can be accrued for +all cases of LEO accessibility by means of joint optimization +of bit allocation and UAV path planning compared to partial +optimization schemes that design for only the bit allocation or +trajectory of the UAV. +Index terms — Marine networks, Internet of Things (IoT), edge +computing, low-Earth orbit (LEO) satellite, unmanned aerial +vehicles (UAVs), successive convex approximation (SCA). +I. INTRODUCTION +M +ARINE Internet of Things (IoT) systems have evolved +significantly with the rapid development of non- +terrestrial network (NTN) technologies composed of space +and airborne platforms to collect and process a variety of +ocean data. The vast amount of ocean data plays an important +This work was supported by the Institute of Information & communica- +tions Technology Planning & Evaluation (IITP) grant funded by the Korea +government (MSIT) (No.2021-0-00847, Development of 3D Spatial Satellite +Communications Technology). +This research was supported by the Ministry of Science and ICT (MSIT), +Korea, under the Information Technology Research Center (ITRC) support +program (IITP-2020-0-01787) supervised by the IITP. +Sooyeob Jung is with the Department of Electrical Engineering, Korea +Advanced Institute of Science and Technology (KAIST), and with the Satellite +Wide-Area Infra Research Section, Electronics and Telecommunications Re- +search Institute (ETRI), Daejeon, South Korea (Email: jung2816@kaist.ac.kr). +Seongah Jeong is with the School of Electronics Engineering, Kyungpook +National University, Daegu 14566, South Korea (Email: seongah@knu.ac.kr). +Jinkyu Kang is with the Department of Information and Communications +Engineering, Myongji University, Gyeonggi-do 17058, South Korea (Email: +jkkang@mju.ac.kr). +Joonhyuk Kang is with the Department of Electrical Engineering, Korea +Advanced Institute of Science and Technology (KAIST), Daejeon, South +Korea (Email: jhkang@ee.kaist.ac.kr). +role in marine monitoring, which contributes to environ- +mental protection, natural disaster prevention, oceanographic +research, mineral exploration, military surveillance, etc. [1]- +[3]. In particular, continuous monitoring of various physical +phenomena of marine networks, such as sounds, vibrations and +images, requires high-precision and wide-range measurements. +Currently, three types of marine monitoring platforms are +being investigated according to the relay node: shore-based +radar, survey vessels and satellites [1], most of which have the +following procedures. By using existing information communi- +cation technologies, the marine data collected from ocean IoT +sensors is transferred to a ground cloud server with sufficient +computation storage capacity. The ground cloud server stores +and analyzes the collected data, thereby managing various ap- +plications based on ocean utilization and exploration. In shore- +based radar systems installed on offshore buoys and automatic +weather stations located on the coast or islands, there are +difficulties in installation and maintenance due to their spatial +constraints. Meanwhile, survey vessel-based platforms have +temporal constraints, which limit the time for data collection. +In addition, unexpected loss and defects of collected data may +occur in point measurements attained by platforms with shore- +based radar or survey vessel platforms due to extreme channel +environments and unpredictable climate changes in the ocean +[2]. +To address these spatial and temporal limitations, satellite- +based monitoring can be an alternative that provides full +coverage of the area of interest with one or multiple satellites. +With the participation of global companies in the satellite +business such as SpaceX, Amazon, and Telesat [4], low- +Earth orbit (LEO) satellites are gaining more attention than +ever before, and cost-effective easy-to-deploy large-scale satel- +lite networks are being established. In addition, conventional +satellite operators such as Spire, Kepler, Fleet, Lacuna space +and Eutelsat, are preparing to provide satellite IoT services +with global coverage [5], [6]. Until recently, satellites have +mostly been adopted as a relay with terrestrial networks; +however, for future 6G IoT services, they can operate as +functional network components, e.g., computing servers [7]- +[12]. Traditionally, the critical drawback of satellite-assisted +networks is the latency resulting from round-trip delays due +to the IoT sensor-satellite-terrestrial station link as well as +the rapidly increasing volume of transmitted data. Therefore, +it is beneficial to bring computing functions in the satellite +to handle processing capabilities of the collected data, rather +than sending it to the ground cloud server. In the following +section, we briefly summarize the recent research activities +that focus on hierarchical integrated networks using satellites +as computing servers. +arXiv:2301.03815v1 [eess.SY] 10 Jan 2023 + +2 +A. Related Works +Satellite-assisted edge computing systems have been ac- +tively studied in space-ground integrated networks [7]-[13], +space-air-ground integrated networks (SAGIN) [14]-[21] and +space-air-sea-based non-terrestrial networks (SAS-NTN) [22], +[23]. In particular, the authors in [7] propose a three-tier +computation architecture consisting of ground users, LEO +satellites and ground servers to minimize the total energy +consumption of the system. In [8], network slice scheduling +for satellite-assisted computing architecture is studied, where +satellite servers and ground servers are considered for IoT +applications. Although satellite-assisted edge computing can +provide real-time offloading services to large areas, such as +the ocean, it still faces several practical problems. For long- +distance communication with a satellite, more transmit power +and larger antenna size are preferred at ground user terminals, +which is costly and spatially-limited in real applications. +Moreover, the transceiver for satellite communications must +be robustly designed against severe fading due to atmospheric +turbulence. +Unmanned aerial vehicles (UAVs) can be adopted to provide +enhanced coverage for overcoming path loss and fading issues +of satellite-assisted edge computing. UAVs can receive and +compute data in close proximity to ocean IoT sensors, or can +relay the data to the cloud server for computing. Recently, +UAV-assisted satellite IoT networks have been suggested in +several studies [14], [15]. Cheng et al. [14] propose offloading +systems of remote IoT applications in the space-air-ground +scenario, where UAVs provide computational capability to +nearby users as edge servers, while satellites relay the of- +floaded data to the ground cloud server. In [15], LEO satellite- +assisted UAV data collection for IoT sensors is proposed, +where the delay-tolerant data and delay-sensitive data are +transferred to the ground cloud server via UAV and LEO +satellite, respectively. +As briefly reviewed above, most of existing works on +hierarchical offloading systems in the integrated space and +air networks assume terrestrial infrastructures, which may +result in latency caused by the extreme channel variation of +marine IoT systems. Furthermore, even though space or aerial +computing platforms are considered, most studies assume full +accessibility of the LEO satellite during mission time, which +may not be guaranteed according to the orbit of revolution of +the LEO satellite under insufficient deployments. To perform +real-time data mining and analysis of ocean data in marine +IoT systems, the use of aerial/space moving cloudlets play an +important role considering their availability. +B. Main Contributions +In this paper, we focus on a marine IoT system with +space-air-sea integrated networks, as illustrated in Fig. 1, +where both UAV and LEO satellite-mounted cloudlets are +deployed to offer computing opportunities. In the proposed +system, a number of ocean IoT sensors are distributed only to +collect abundant marine information with limited battery, and +transmit the collected data to a designated computing server +among UAV or LEO-mounted cloudlets so as to satisfy the +LEO satellite +(Cloud server) +UAV +(Edge server) +IoT 1 +End user +Frame 1 +Frame n +Frame N +IoT k +IoT K +( +) +, +,0 +I +I +I +k +k +k +x +y += +p +( +) +1 +, +, +E +E +E +n +n +n +x +y +h += +p +( +) +1 +2 +, +, +C +C +C +x +y +h +h += ++ +p +1 +I +K + +p +1 +E +N + +p +IoT → UAV (for UAV computing) +UAV → LEO +LEO → UAV +UAV → User +IoT → UAV → LEO → UAV (for LEO computing) +IoT → UAV (for edge computing) +IoT → UAV (for cloud computing) +UAV → LEO (offloading) +LEO → UAV +UAV → User +IoT → UAV (for edge computing) +IoT → UAV (for cloud computing) +UAV → LEO (offloading) +LEO → UAV +UAV → User +LEO satellite-mounted cloudlet +End user +Ocean +IoT sensor k +( +) +, +,0 +I +I +I +k +k +k +x +y += +p +( +) +, +, +U +U +U +n +n +n +U +x +y +h += +p +UAV-mounted cloudlet +LEO → Ground +Orbit +Ocean +IoT sensor 1 +Ocean +IoT sensor K +IoT → UAV (for UAV computing) +IoT → UAV → LEO → UAV (for LEO computing) +Space +Air +Sea +( +) +, +, +L +L +L +n +n +n +U +L +x +y +h +h += ++ +p +LEO satellite-mounted cloudlet +End user +Ocean +IoT sensor k +UAV-mounted +cloudlet +Orbit +Space +Air +Sea +( +) +, +, +L +L +L +n +n +n +U +L +x +y +h +h += ++ +p +( +) +, +, +U +U +U +n +n +n +U +x +y +h += +p +( +) +, +,0 +I +I +I +k +k +k +x +y += +p +1 +U +p +U +N +p +1 +Ip +I +K +p +UAV computing: Sensor → UAV → +LEO computing: Sensor → UAV → +UAV computing +LEO computing +LEO satellite-mounted cloudlet +End user +Ocean +IoT sensor k +UAV-mounted cloudlet +Orbit +Space +Air +Sea +( +) +, +, +L +L +L +n +n +n +U +L +x +y +h +h += ++ +p +( +) +, +, +U +U +U +n +n +n +U +x +y +h += +p +( +) +, +,0 +I +I +I +k +k +k +x +y += +p +1 +U +p +U +N +p +1 +I +p +I +K +p +End user +: +: Sensor → UAV → +LEO → UAV → End user +UAV computing +LEO computing +UAV computing: Sensor → UAV → End user +LEO computing: Sensor → UAV →LEO → UAV → End user +Fig. 1: Marine IoT system model with a space-air-sea inte- +grated network using hybrid LEO and UAV edge computing +for real-time data utilization. +system design criterion. Here, the LEO satellite is assumed +to have a higher computational capability to process the task +than that of the UAV. When the IoT data size exceeds the +computation capacity of the UAV, the computational task is +totally offloaded to the LEO satellite. The computation results +executed at LEO are retransmitted to the UAV, are stored +until it arrives over the end user, and is finally sent to the +end user. To this end, we tackle the key design problem of +jointly optimizing the bit allocation for communication and +computing and the trajectory of the UAV, with the aim of +minimizing its energy consumption. The main contributions +of this paper are summarized as follows: +• For marine IoT systems with extreme channel environ- +ments and unpredictable climate changes, we propose +a hybrid LEO and UAV edge computing method. The +scheduling between UAV and LEO satellite-mounted +cloudlets depends on the size of the offloaded ocean data +and the LEO connection status. +• For practicality and usability, we consider three different +scenarios according to LEO availability such as “Always +On,” “Always Off” and “Intermediate Disconnected”. +For each case, we develop the joint optimization of bit +allocation required for offloading and UAV path planning. +• The non-convex optimization problems formulated for +three different cases depending on the availability of the +LEO satellite are tackled by means of a successive convex +approximation (SCA) algorithm [24], [25], which can +guarantee the local minimum of the original non-convex +problems by using an efficient iterative algorithm. +The rest of this paper is organized as follows. The system +model is presented in Section II. Section III, IV and V provide +problem formulations and proposed methods for the LEO +access status of “Always On,” “Always Off” and “Intermediate +Disconnected”, respectively. Simulation results are given in +Section VI, and conclusions are summarized in Section VII. + +C3 +Frame n-1 +Frame n +IoT sensor 1 +〮〮〮 +IoT sensor k +〮〮〮 +IoT sensor K + +… +… +K + +1 +U +n− +p +U +np +1 +U +n+ +p +2 +U +n+ +p +Frame n+1 +Fig. 2: Frame structure of orthogonal access for multiple ocean +IoT sensors. +II. SYSTEM MODEL +A. Set-up +Fig. 1 illustrates a marine IoT system with a space-air- +sea integrated network using hybrid LEO and UAV edge +computing, where 𝐾 ocean IoT sensors collect marine data +to be entirely transferred to available cloudlets for computing. +The computed results are then designated to an end user. For +real-time data utilization, two types of cloudlets mounted on +the UAV and LEO satellite are considered, between which +the scheduling depends on the UAV computing capability and +LEO accessibility. Specifically, when the collected data size +exceeds the computation capacity of the UAV, the data should +be entirely offloaded to the LEO. The computing capability +of the LEO satellite is assumed to be higher than that of +the UAV. Another major factor for scheduling is whether +the LEO satellite is available or not since its beam coverage +varies according to the orbit of revolution. Here, we consider +three different cases according to the availability of the LEO +satellite during mission time: “Always On,” “Always Off” and +“Intermediate Disconnected”. For each scenario, we developed +the joint optimization of the bit allocation for communication +and computation and the trajectory of the UAV. Depending on +the types of cloudlets, we refer to UAV computing and LEO +computing, where computing of the IoT sensor task is executed +at the UAV and LEO, respectively. In UAV computing, the task +of the IoT sensor 𝑘 is offloaded to the UAV-mounted cloudlet +until the UAV arrives over the end user and the output results +are conveyed to them. In LEO computing, the UAV receives +and relays the offloaded data of the IoT sensor to LEO for the +LEO execution. The computed results at LEO are then sent to +the end user via the UAV when the UAV arrives above them. +For communication links between IoT sensors and the UAV, +and between the UAV and LEO satellite, a frequency division +duplex (FDD) scheme is assumed with equal bandwidth 𝐵 for +the uplink and downlink. Each IoT sensor 𝑘 has the number +𝐼𝑘 of input information bits to be processed. The results for +LEO computing and UAV computing are characterized as the +number 𝑂𝐿 +𝑘 and 𝑂𝑈 +𝑘 of bits produced per input bit of the IoT +sensor 𝑘, and the number 𝐶𝐿 +𝑘 and 𝐶𝑈 +𝑘 of CPU cycles per input +bit for computing, respectively. We assume that all tasks must +be computed within the total mission time 𝑇. Here, a three- +dimensional Cartesian coordinate system is adopted based on +the metric unit. We assume that the IoT sensor 𝑘 is deployed +at position 𝒑𝐼 +𝑘 = (𝑥𝐼 +𝑘, 𝑦𝐼 +𝑘, 𝑎𝑘), for 𝑘 ∈ {1, · · ·, 𝐾 + 1}, with +𝑎𝑘 being the average sea surface level, where the position +TABLE I: List of Symbols +Symbol +Definition +𝐾 +Number of ocean IoT sensors +𝑇 +Total mission time +Δ +Frame duration +𝑁 +Number of frames within 𝑇 +ℎ𝑈 , ℎ𝐿 +Altitudes of UAV and LEO satellite with respect to +average sea surface level and UAV, respectively +𝑔𝑘,𝑛, ℎ𝑛 +Path loss between the IoT sensor 𝑘 and UAV and +between the UAV and LEO at the 𝑛th frame +𝑔0 +Channel gain at reference distance 1 m +𝑇𝑣 +Visible time of an LEO satellite +𝑣𝑠 +Speed of an LEO satellite +ℎ +Height of an LEO satellite orbit +𝜃, 𝜑 +Elevation angle and beamwidth of the LEO satellite +𝑀 +the gross mass of the UAV +𝒗𝑈 +𝑛 +velocity vector of the UAV at the 𝑛th frame +𝜀 +Energy budget of the IoT sensor 𝑘 at each frame +𝐼𝑘 +Number of input bits of the IoT sensor 𝑘 +𝐸𝐼,𝑈 +𝑘,𝑛 +Energy consumption for uplink communication at the +IoT sensor 𝑘 at the 𝑛th frame +𝐸𝑈 +𝑘,𝑛, 𝐸𝑈,𝐿 +𝑘,𝑛 +Energy consumption for computing and uplink com- +munication at the UAV-mounted cloudlet for the IoT +sensor 𝑘 at the 𝑛th frame +𝐸𝑈,𝐸 +Energy consumption for downlink communication at +the UAV-mounted cloudlet +𝐸 𝐿 +𝑘,𝑛, 𝐸 𝐿,𝑈 +𝑘,𝑛 +Energy consumption for computing and downlink com- +munication at the LEO-mounted cloudlet for the IoT +sensor 𝑘 at the 𝑛th frame +𝐸𝐹 +𝑛 +Energy consumption for a UAV flying at the 𝑛th frame +𝐿𝐼,𝑈 +𝑘,𝑛 +Number of bits for uplink communication at the IoT +sensor 𝑘 at the 𝑛th frame +𝑙𝑈 +𝑘,𝑛, 𝐿𝑈,𝐿 +𝑘,𝑛 +Number of bits for computing and uplink communica- +tion at a UAV-mounted cloudlet for the IoT sensor 𝑘 at +the 𝑛th frame +𝐿𝑈,𝐸 +Number of bits for downlink communication at the +UAV-mounted cloudlet +𝑙𝐿 +𝑘,𝑛, 𝐿𝐿,𝑈 +𝑘,𝑛 +Number of bits for computing and downlink communi- +cation at the LEO-mounted cloudlet for the IoT sensor +𝑘 at the 𝑛th frame +𝑂𝐿 +𝑘 , 𝑂𝑈 +𝑘 +Number of output bits produced per input bit of the IoT +sensor 𝑘 +𝑓 𝐿 +𝑛 , 𝑓 𝑈 +𝑛 +CPU frequency at the LEO and UAV-mounted cloudlets +for the 𝑛th frame +𝐶𝐿 +𝑘 , 𝐶𝑈 +𝑘 +CPU cycles per input bit at the LEO and UAV-mounted +cloudlets for the task of the IoT sensor 𝑘 +𝛾𝐿, 𝛾𝑈 +Effective switched capacitances of the LEO and UAV, +respectively +𝒑𝐼 +𝑘, 𝒑𝑈 +𝑛 , 𝒑𝐿𝑛 +Positions of the IoT sensor 𝑘, UAV and LEO for the +𝑛th frame +𝛼𝑘,𝑛, 𝛽𝑘,𝑛 +Variables to indicate LEO connection and offloading +scheduling of the IoT sensor 𝑘 at the 𝑛th frame +𝑁𝑡 +Frame number during LEO disconnection +of the end user is considered with an index of 𝐾 + 1. The +UAV flies along a trajectory 𝒑𝑈 (𝑡) = (𝑥𝑈 (𝑡), 𝑦𝑈 (𝑡), ℎ𝑈) +with a fixed altitude ℎ𝑈 assumed for system stability, for +0 ≤ 𝑡 ≤ 𝑇, and the position of the LEO satellite is defined as +𝒑𝐿(𝑡) = (𝑥𝐿(𝑡), 𝑦𝐿(𝑡), ℎ𝑈 + ℎ𝐿) with a fixed altitude ℎ𝑈 + ℎ𝐿, +for 0 ≤ 𝑡 ≤ 𝑇, all the altitudes are measured with respect to +the average sea surface level 𝑎𝑘. For the multiple access of 𝐾 +ocean IoT sensors, orthogonal access is assumed, as shown in +Fig. 2. For tractability, in this paper, the total time duration 𝑇 +is divided into 𝑁 frames of duration Δ seconds, each of which +is equally divided as Δ/𝐾 seconds, and is preallocated to +IoT sensors for uplink and downlink communication required + +4 +Er +h +L +LEO +Satellite +s + + +Earth +Orbit +IoT Sensor + +sv +Er +h +L +LEO +Satellite +s + + +Earth +Orbit +IoT Sensor + +sv +Fig. 3: Geometric relationship between the ground user and +the LEO satellite. +for offloading. Accordingly, the IoT sensors do not interfere +with each other in the offloading procedure. Moreover, the +information data collected from the IoT sensor 𝑘 at the 𝑛 th +frame is assumed to be entirely computed and transferred to +the designated node within the corresponding frame during +Δ/𝐾 seconds, for 𝑛 ∈ {1, · · ·, 𝑁}, so that the computational +task cannot be partitioned. According to the discretized time +unit, the trajectory of the UAV 𝒑𝑈 (𝑡) and the position of the +LEO satellite 𝒑𝐿(𝑡) is expressed as 𝒑𝑈 +𝑛 = (𝑥𝑈 +𝑛 , 𝑦𝑈 +𝑛 , ℎ𝑈) and +𝒑𝐿 +𝑛 = (𝑥𝐿 +𝑛 , 𝑦𝐿 +𝑛, ℎ𝑈 + ℎ𝐿), for 𝑛 ∈ N, respectively. The LEO +satellite generally flies at a constant speed along its orbit and +the relative positional coordinates of the LEO and UAV should +vary constantly. For the task mission of marine IoT systems, +the initial location 𝒑𝑈 +𝐼 and the final location 𝒑𝑈 +𝐹 of the UAV +are assigned to 𝒑𝑈 +1 and 𝒑𝑈 +𝑁 +1, respectively, and its maximum +speed constraint is given as +��𝒗𝑈 +𝑛 +�� = +�� 𝒑𝑈 +𝑛+1 − 𝒑𝑈 +𝑛 +�� +Δ +≤ 𝑣max, +(1) +where the velocity vector 𝒗𝑈 +𝑛 +of the UAV is defined as +( 𝒑𝑈 +𝑛+1 − 𝒑𝑈 +𝑛 )/Δ, and 𝑣max is its maximum velocity. The overall +system variables and parameters are summarized in Table I. +We assume that communication channels between the IoT +sensors and UAV [16], [26], and between the UAV and LEO +satellite [15], [16] are dominated by line-of-sight (LoS) links. +At the 𝑛th frame, the channel gains for the IoT sensor 𝑘-UAV +link and UAV-LEO link are written as +𝑔𝑘,𝑛( 𝒑𝑈 +𝑛 ) = +𝑔0 +(𝑥𝑈𝑛 − 𝑥𝐼 +𝑘)2 + (𝑦𝑈𝑛 − 𝑦𝐼 +𝑘)2 + ℎ𝑈 2 +(2) +and +ℎ𝑛( 𝒑𝑈 +𝑛 ) = +𝑔0𝐺 +(𝑥𝐿𝑛 − 𝑥𝑈𝑛 )2 + (𝑦𝐿𝑛 − 𝑦𝑈𝑛 )2 + ℎ𝐿2 , +(3) +respectively, where 𝑔0 represents the channel gain at the +reference distance 1 m, and 𝐺 is an antenna gain for the long- +distance satellite communication consisting of the transmission +antenna gain of the UAV and the receiver antenna gain of +the LEO satellite [15], [27]. In real applications, note that +ℎ𝑛( 𝒑𝑈 +𝑛 ) ≫ 𝑔𝑘,𝑛( 𝒑𝑈 +𝑛 ) is guaranteed. For communication links, +an additive white Gaussian noise is considered with zero mean +and power spectral density 𝑁0 [dBm/Hz]. +B. Coverage Model of the LEO Satellite +In this section, we explore the beam coverage model [7], +[28] of an LEO satellite that accounts for the effect of the +orbit of revolution. As shown in Fig. 3, when the LEO satellite +makes an orbit round, the available communication time with +the UAV can be limited, which is referred to as the LEO visible +time window. The length of the visible time window is defined +as +𝑇𝑣 = 𝐿 +𝑣𝑠 += 2 (𝑟𝐸 + ℎ) 𝛾 +𝑣𝑠 +, +(4) +where 𝑣𝑠 is the speed of the LEO satellite. 𝐿 is the arc length to +define the coverage where IoT sensors can communicate with +the LEO satellite, and is calculated by 𝐿 = 2 (𝑟𝐸 + ℎ) 𝛾 with +𝑟𝐸 being the radius of Earth, ℎ being the height of the LEO +satellite orbit, and 𝛾 being the angle of the satellite coverage. +In general, due to the very low altitude of a UAV in comparison +to the orbit height, the same visible time window is applied to +the UAV and IoT sensors. The maximum length of the LEO +visible time window can be achieved when 𝛾 = 𝜋. The angle +𝛾 of the satellite coverage is calculated by +𝛾 = cos−1 +� +𝑟𝐸 +𝑟𝐸 + ℎ · cos 𝜃 +� +− 𝜃, +(5) +where 𝜃 and 𝜑 are the elevation angle and the beamwidth +of the satellite, respectively, and are derived as 𝜃 += +cos−1 � +𝑟𝐸+ℎ +𝑠 +· cos (𝜃 + 𝜑) +� +and 𝜑 = 𝜋/2 − (𝜃 + 𝛾) with 𝑠 +indicating the distance between the IoT sensor and LEO +satellite. We assume that the UAV can fully access the LEO +satellite within the visible time window of 𝑇𝑣. According to +the availability of LEO communication based on the coverage +model, three different cases can be considered: “Always On,” +“Always Off” and “Intermediate Disconnected”, the details for +which are described below. +1) “Always On” scenario (𝑇 ≤ 𝑇𝑣): The first scenario is +when the UAV can communicate with the LEO satellite during +the entire mission time since the total mission time is within +the LEO visible time, i.e., 𝑇 ≤ 𝑇𝑣. In this scenario, we have +𝛼𝑘,𝑛 = 1 for all 𝑛 ∈ N; therefore, the computation capability +of the UAV determines whether the UAV or LEO will be used +for computing. +2) “Always Off” scenario (𝑇𝑣 = 0): The second scenario +is when LEO communication is not available during the entire +mission time since the UAV flies outside the beam coverage +of the LEO satellite, i.e., 𝑇𝑣 = 0. In this scenario, we have +𝛼𝑘,𝑛 = 0 for all 𝑛 ∈ N, and only the UAV computing can be +performed. Furthermore, when the offloaded data size exceeds +the UAV computation capability, it is transferred to the end +user via the UAV without computing. +3) “Intermediate Disconnected” scenario (𝑇 > 𝑇𝑣): The +final scenario is when LEO connection is lost during the +mission time, since the total mission time is larger than the +LEO visible time, i.e., 𝑇 > 𝑇𝑣. In this scenario, when 𝑡 ≤ 𝑇𝑣, +we have 𝛼𝑘,𝑛 = 1 for 𝑛 ∈ {1, · · ·, 𝑁𝑡}, with 𝑁𝑡 being the +last frame within 𝑇𝑣, where both LEO computing and UAV +computing can be performed: that is, 𝛽𝑘,𝑛 ∈ {0, 1}. When +𝑡 > 𝑇𝑣, 𝛼𝑘,𝑛 = 0 for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁}, where only UAV +computing is available: that is, 𝛽𝑘,𝑛 = 0. For example, if the + +5 +TABLE II: Three different scenarios according to LEO availability. +Scenario +𝛼𝑘,𝑛 +𝛽𝑘,𝑛 +Available types of computing +“Always On” (𝑇 ≤ 𝑇𝑣) +1, for all 𝑛 ∈ N +0, for all 𝑛 ∈ N +UAV Computing +1, for all 𝑛 ∈ N +LEO Computing +“Always Off” (𝑇𝑣 = 0) +0, for all 𝑛 ∈ N +0, for all 𝑛 ∈ N +UAV Computing +“Intermediate Disconnected” (𝑇 > 𝑇𝑣) +1, for 𝑛 ∈ {1, · · ·, 𝑁𝑡 }, +0, for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁 } +0, for 𝑛 ∈ {1, · · ·, 𝑁𝑡 }, +0, for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁 } +UAV Computing +1, for 𝑛 ∈ {1, · · ·, 𝑁𝑡 }, +0, for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁 } +LEO Computing → +UAV Computing +LEO connection is lost at 𝑇𝑣 = 𝑇/2, 𝑁𝑡 is defined as 𝑁/2. The +frame data of 𝑛 ∈ {1, · · ·, 𝑁𝑡} is computed by the LEO or UAV, +while the frame data of 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁} is computed by +the UAV. The details for these three scenarios are summarized +in Table II. +C. Energy Consumption Model for Offloading +In the proposed hierarchical architecture, IoT sensors and +the UAV are battery-limited, while the available energy of the +LEO satellite is much more sufficient due to its larger size +and mass, which is therefore negligible for the system design. +With the aim of minimizing the total energy consumption of +the UAV, we cover the energy consumption model for compu- +tation, communication and flying required for offloading. Here, +the LEO satellite is assumed to have sufficient battery capacity +compared to the UAV and IoT sensors [7], [13], which is not +reflected in the system design. +1) Computation energy model: First, we define the +amount of computation energy consumption at the LEO and +UAV-mounted cloudlets at the 𝑛th frame for IoT sensor 𝑘 as +[29], [30] +𝐸𝑑 +𝑘,𝑛(𝑙𝑑 +𝑘,𝑛) = +𝛾𝑑𝐶𝑑 +𝑘 𝑙𝑑 +𝑘,𝑛 +Δ2 +� 𝐾 +∑︁ +𝑘′=1 +𝐶𝑑 +𝑘′𝑙𝑑 +𝑘′,𝑛 +�2 +, +(6) +where 𝑑 ∈ {𝐿,𝑈} with 𝐿 indicating the LEO satellite and 𝑈 +indicating the UAV; 𝑙𝑑 +𝑘,𝑛 is the number of bits to be computed +at the cloudlet and 𝛾𝑑 is the effective switched capacitance of +the cloudlet. +2) Communication energy model: In the proposed system, +the transmit energy consumption from the UAV to LEO at the +𝑛th frame for offloading the task of the IoT sensor 𝑘 is defined +as [26], [31] +𝐸𝑈,𝐿 +𝑘,𝑛 (𝐿𝑈,𝐿 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) = 𝑁0𝐵Δ/𝐾 +ℎ𝑛( 𝒑𝑈𝑛 ) +� +2 +𝐿𝑈,𝐿 +𝑘,𝑛 +𝐵Δ/𝐾 − 1 +� +, +(7) +where 𝐿𝑈,𝐿 +𝑘,𝑛 +is the number of uplink bits. At the final +destination of the UAV above the end user, the downlink +communication energy consumption is required so that the +UAV can transmit the computing results accumulated during +flying, which is given as +𝐸𝑈,𝐸 (𝐿𝑈,𝐸, 𝒑𝑈 +𝑁 +1) = +𝑁0𝐵Δ/𝐾 +𝑔𝐾+1,𝑁 +1( 𝒑𝑈 +𝑁 +1) +� +2 +𝐿𝑈,𝐸 +𝐵Δ/𝐾 − 1 +� +, +(8) +where 𝐿𝑈,𝐸 is the number of downlink bits and is the same as +the sum of output bits of the UAV and LEO-mounted cloudlets +as follows: +𝐿𝑈,𝐸 = 𝑂𝑈 +𝑘 +𝑁 −2 +∑︁ +𝑛=1 +𝑙𝑈 +𝑘,𝑛+1 + 𝑂𝐿 +𝑘 +𝑁 −4 +∑︁ +𝑛=1 +𝑙𝐿 +𝑘,𝑛+2. +(9) +In addition, the transmit energy consumption from the LEO +and IoT sensor 𝑘 to the UAV at the 𝑛th frame is defined as +𝐸 𝐿,𝑈 +𝑘,𝑛 (𝐿𝐿,𝑈 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) = 𝑁0𝐵Δ/𝐾 +ℎ𝑛( 𝒑𝑈𝑛 ) +� +2 +𝐿𝐿,𝑈 +𝑘,𝑛 +𝐵Δ/𝐾 − 1 +� +(10) +and +𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝐿𝐼 ,𝑈 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) = 𝑁0𝐵Δ/𝐾 +𝑔𝑘,𝑛( 𝒑𝑈𝑛 ) +� +2 +𝐿𝐼,𝑈 +𝑘,𝑛 +𝐵Δ/𝐾 − 1 +� +, +(11) +where 𝐿𝐿,𝑈 +𝑘,𝑛 +is the number of downlink bits transmitted at +the LEO and 𝐿𝐼 ,𝑈 +𝑘,𝑛 is the number of uplink bits transmitted +at the IoT sensor 𝑘. The energy consumption for reception is +excluded since it is much smaller than the transmission energy +consumption. +3) Flying energy model: Following [32], [33], the flying +energy consumption of the UAV at the 𝑛th frame is written as +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) = 𝜅∥𝒗𝑈 +𝑛 ∥2, +(12) +where 𝜅 = 0.5𝑀Δ and 𝑀 is the mass of the UAV. The flying +energy consumption depends only on the velocity vector 𝒗𝑈 +𝑛 +of the UAV, and the level flight entails no change in the +gravitational potential energy. +Our purpose is to minimize the total energy consumption of +the UAV, which must be calculated as the sum of the energy +consumption of computation, communication and flying: +𝐸𝑡𝑜𝑡𝑎𝑙 +𝑘,𝑛 += 𝛼𝑘,𝑛 +� +𝛽𝑘,𝑛𝐸𝑈,𝐿 +𝑘,𝑛 (𝐿𝑈,𝐿 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) + (1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛(𝑙𝑈 +𝑘,𝑛) +� ++ (1 − 𝛼𝑘,𝑛)(1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛(𝑙𝑈 +𝑘,𝑛) + 𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ), +(13) +where 𝛼𝑘,𝑛 and 𝛽𝑘,𝑛 are variables for the LEO availability +and scheduling between LEO computing and UAV computing, +respectively, which are given as +𝛼𝑘,𝑛 = +� +1, +if LEO communication is available, +0, +otherwise, +(14) +𝛽𝑘,𝑛 = +� +1, +if LEO computing is performed, +0, +if UAV computing is performed. +(15) +Note that the energy consumption 𝐸𝑈,𝐸 for downlink com- +munication with the end user in (8) is excluded from (13) + +6 +since it is constant regardless of optimization. In addition, +LEO computing is considered by 𝛽𝑘,𝑛 = 1 when the input +bits of the IoT sensor 𝑘 exceeds the computation capability of +the UAV: that is, +𝑁 +∑︁ +𝑛=1 +𝐿𝐼 ,𝑈 +𝑘,𝑛 > +𝑁 +∑︁ +𝑛=1 +� +𝑓 𝑈 +𝑛 · Δ +𝐾 +� +1 +𝐶𝑈 +𝑘 +, +(16) +where 𝑓 𝑈 +𝑛 +[CPU cycles/s] is the CPU frequency at the UAV +edge server. +III. OPTIMAL ENERGY CONSUMPTION FOR THE +“ALWAYS ON” SCENARIO +In this section, we formulate an optimization problem and +the proposed algorithm to obtain a solution for the “Always +On” scenario. Depending on the size of the offloaded data, +either LEO computing or UAV computing is selected. As +mentioned above, the total UAV energy consumption 𝐸𝑡𝑜𝑡𝑎𝑙 +𝑘,𝑛 +in (13) is rewritten with 𝛼𝑘,𝑛 = 1, for all 𝑛 ∈ N, as +𝐸𝑡𝑜𝑡𝑎𝑙 +𝑘,𝑛 += 𝛽𝑘,𝑛𝐸𝑈,𝐿 +𝑘,𝑛 (𝐿𝑈,𝐿 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) + (1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛(𝑙𝑈 +𝑘,𝑛) ++ 𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ). +(17) +When LEO computing is considered, i.e., 𝛽𝑘,𝑛 += +1, +we +need +to +jointly +optimize +the +bit +allocation +of +{𝐿𝐼 ,𝑈 +𝑘,𝑛 }𝑛∈{1,···,𝑁 −4},𝑘 ∈K, +{𝐿𝑈,𝐿 +𝑘,𝑛 }𝑛∈{2,···,𝑁 −3},𝑘 ∈K, +{𝑙𝐿 +𝑘,𝑛}𝑛∈{3,···,𝑁 −2},𝑘 ∈K +and +{𝐿𝐿,𝑈 +𝑘,𝑛 }𝑛∈{4,···,𝑁 −1},𝑘 ∈K +along +with +the +UAV +trajectory +{ 𝒑𝑈 +𝑛 }𝑛∈{2,···,𝑁 }. +When +UAV +computing is performed, that is, 𝛽𝑘,𝑛 = 0, we must jointly +optimize +the +bit +allocation +of +{𝐿𝐼 ,𝑈 +𝑘,𝑛 }𝑛∈{1,···,𝑁 −2},𝑘 ∈K +and {𝑙𝑈 +𝑘,𝑛}𝑛∈{2,···,𝑁 −1},𝑘 ∈K along with the UAV trajectory +{ 𝒑𝑈 +𝑛 }𝑛∈{2,···,𝑁 }. This problem is formulated with (17) as +follows: +min +𝐿𝐼,𝑈 +𝑘,𝑛 ,𝐿𝑈,𝐿 +𝑘,𝑛 ,𝐿𝐿,𝑈 +𝑘,𝑛 +𝑙𝑈 +𝑘,𝑛,𝑙𝐿 +𝑘,𝑛,𝒑𝑈 +𝑛 +𝐾 +∑︁ +𝑘=1 +�𝑁 −4 +∑︁ +𝑛=1 +𝛽𝑘,𝑛𝐸𝑈,𝐿 +𝑘,𝑛+1(𝐿𝑈,𝐿 +𝑘,𝑛+1, 𝒑𝑈 +𝑛+1) ++ +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛+1(𝑙𝑈 +𝑘,𝑛+1) +� ++ +𝑁 +∑︁ +𝑛=1 +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) +(18a) +s.t. 𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝐿𝐼 ,𝑈 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) ≤ 𝜀, ∀𝑘 ∈ K, 𝑛 ∈ N +(18b) +𝑛 +∑︁ +𝑖=1 +𝑙𝑈 +𝑘,𝑖+1 ≤ +𝑛 +∑︁ +𝑖=1 +𝐿𝐼 ,𝑈 +𝑘,𝑖 , ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 2 +(18c) +𝑛 +∑︁ +𝑖=1 +𝐿𝑈,𝐿 +𝑘,𝑖+1 ≤ +𝑛 +∑︁ +𝑖=1 +𝐿𝐼 ,𝑈 +𝑘,𝑖 , ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 4 +(18d) +𝑛 +∑︁ +𝑖=1 +𝑙𝐿 +𝑘,𝑖+2 ≤ +𝑛 +∑︁ +𝑖=1 +𝐿𝑈,𝐿 +𝑘,𝑖+1, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 4 +(18e) +𝑛 +∑︁ +𝑖=1 +𝐿𝐿,𝑈 +𝑘,𝑖+3 ≤ 𝑂𝐿 +𝑘 +𝑛 +∑︁ +𝑖=1 +𝑙𝐿 +𝑘,𝑖+2, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁 − 4 (18f) +𝑁 −4 +∑︁ +𝑛=1 +𝛽𝑘,𝑛𝐿𝐼 ,𝑈 +𝑘,𝑛 + +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝐿𝐼 ,𝑈 +𝑘,𝑛 = 𝐼𝑘, ∀𝑘 ∈ K +(18g) +𝑁 −4 +∑︁ +𝑛=1 +𝛽𝑘,𝑛𝑙𝐿 +𝑘,𝑛+2 + +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝑙𝑈 +𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K +(18h) +𝑁 −4 +∑︁ +𝑛=1 +𝑙𝐿 +𝑘,𝑛+2 = +𝑁 −4 +∑︁ +𝑛=1 +𝐿𝑈,𝐿 +𝑘,𝑛+1, ∀𝑘 ∈ K +(18i) +𝑁 −4 +∑︁ +𝑛=1 +𝐿𝐿,𝑈 +𝑘,𝑛+3 = 𝑂𝐿 +𝑘 +𝑁 −4 +∑︁ +𝑛=1 +𝐿𝑈,𝐿 +𝑘,𝑛+1, ∀𝑘 ∈ K +(18j) +𝐿𝐼 ,𝑈 +𝑘,𝑛 , 𝐿𝑈,𝐿 +𝑘,𝑛 , 𝐿𝐿,𝑈 +𝑘,𝑛 , 𝑙𝑈 +𝑘,𝑛, 𝑙𝐿 +𝑘,𝑛 ≥ 0, ∀𝑘 ∈ K, 𝑛 ∈ N +(18k) +𝒑𝑈 +1 = 𝒑𝑈 +𝐼 , 𝒑𝑈 +𝑁 +1 = 𝒑𝑈 +𝐹 , +(18l) +��𝒗𝑈 +𝑛 +�� ≤ 𝑣max, ∀𝑛 ∈ N, +(18m) +where 𝜀 in (18b) represents the energy budget constraint per +frame for the IoT sensors. The inequality constraint (18c) +and (18e) ensures that the number of bits computed at the +UAV and LEO-mounted cloudlet is less than or equal to the +number of uplink bits transmitted from the IoT sensor and +UAV, respectively. The inequality constraints (18d) and (18f) +ensure that the number of uplink bits from the UAV is less than +or equal to the number of uplink bits from the IoT sensor, and +the number of downlink bits from the LEO is limited by the +number of output bits from the LEO. The equality constraints +(18g) and (18h) enforce that the sum of the uplink bits of +the IoT sensor and the sum of the computation bits for the +LEO and UAV computing are equal to the input bits of the +IoT sensor. The equality constraints (18i) and (18j) enforce the +completion of LEO computing, while (18k) is imposed for the +non-negative bit allocations. The constraints (18l) and (18m) +represent the flying UAV’s initial and final position constraint +and the maximum speed constraint, respectively. +Problem (18) is non-convex because the objective function +and the energy budget constraint are non-convex. To address +this non-convexity, we apply the SCA-based strategy [24], [25] +which builds on the inner convex approximation framework. +In particular, we develop proposed algorithm 1 by using the +following lemmas. +Lemma 1: Given that a non-convex objective function +𝑈(𝒙) = 𝑓1(𝒙) 𝑓2(𝒙) is the product of 𝑓1 and 𝑓2 convex and +non-negative for any 𝒚 in the domain of 𝑈(𝒙), a convex +approximation that satisfies the conditions required by the +SCA algorithm is given as +¯𝑈 (𝒙; 𝒚) = 𝑓1(𝒙) 𝑓2(𝒚) + 𝑓1(𝒚) 𝑓2(𝒙) ++ 𝜏𝑖 +2 (𝒙 − 𝒚)T𝑯(𝒚)(𝒙 − 𝒚), +(19) +where 𝜏𝑖 > 0 is a positive constant, 𝑯(𝒚) is a positive definite +matrix, and (·)T indicates the transpose. +Lemma 2: Given a non-convex constraint 𝑔(𝒙1, 𝒙2) ≤ 0, +where 𝑔(𝒙1, 𝒙2) = ℎ1(𝒙1)ℎ2(𝒙2) is the product of the ℎ1 and +ℎ2 convex and non-negative, for any (𝒚1, 𝒚2) in the domain of +𝑔(𝒙1, 𝒙2), a convex approximation that satisfies the conditions +required by the SCA algorithm is given as +¯𝑔 (𝒙1, 𝒙2; 𝒚1, 𝒚2) +Δ= 1 +2 (ℎ1(𝒙1) + ℎ2(𝒙2))2 − 1 +2 (ℎ12(𝒚1) + ℎ22(𝒚2)) +− ℎ1(𝒚1)ℎ1 +′(𝒚1)(𝒙1 − 𝒚1) − ℎ2(𝒚2)ℎ2 +′(𝒚2)(𝒙2 − 𝒚2), +(20) +where the partial derivative of 𝑓 (·) is 𝑓 +′ (·). + +7 +We set the primal variables for the formulated Problem +(18) as 𝒛 = {𝒛𝑛}𝑛∈N with 𝒛𝑛 = ({𝐿𝐼 ,𝑈 +𝑘,𝑛 }𝑘 ∈K, {𝐿𝑈,𝐿 +𝑘,𝑛 }𝑘 ∈K, +{𝐿𝐿,𝑈 +𝑘,𝑛 }𝑘 ∈K, {𝑙𝑈 +𝑘,𝑛}𝑘 ∈K, {𝑙𝐿 +𝑘,𝑛}𝑘 ∈K, 𝒑𝑈 +𝑛 ). We observe that the +function 𝐸𝑈,𝐿 +𝑘,𝑛 (𝒛𝑛) +Δ= 𝐸𝑈,𝐿 +𝑘,𝑛 (𝐿𝑈,𝐿 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) in (18a) is the product +of two convex and non-negative functions, namely +𝑓1(𝐿𝑈,𝐿 +𝑘,𝑛 ) = 𝑁0𝐵Δ/𝐾 +𝑔0𝐺 +� +2 +𝐿𝑈,𝐿 +𝑘,𝑛 +𝐵Δ/𝐾 − 1 +� +(21) +and +𝑓2( 𝒑𝑈 +𝑛 ) = (𝑥𝐿 +𝑛 − 𝑥𝑈 +𝑛 )2 + (𝑦𝐿 +𝑛 − 𝑦𝑈 +𝑛 )2 + ℎ𝐿2. +(22) +Then, +by +using +Lemma +1 +and +defining +𝒛𝑛(𝑣) += +({𝐿𝐼 ,𝑈 +𝑘,𝑛 (𝑣)}𝑘 ∈K, +{𝐿𝑈,𝐿 +𝑘,𝑛 (𝑣)}𝑘 ∈K, +{𝐿𝐿,𝑈 +𝑘,𝑛 (𝑣)}𝑘 ∈K, +{𝑙𝑈 +𝑘,𝑛(𝑣)}𝑘 ∈K, {𝑙𝐿 +𝑘,𝑛(𝑣)}𝑘 ∈K, 𝒑𝑈 +𝑛 (𝑣))∈ X for the 𝑣th iterate +within the feasible set X of (18), we obtain a strongly convex +surrogate function ¯𝐸𝑈,𝐿 +𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) of 𝐸𝑈,𝐿 +𝑘,𝑛 (𝒛𝑛) as +¯𝐸𝑈,𝐿 +𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) +Δ= ¯𝐸𝑈,𝐿 +𝑘,𝑛 (𝐿𝑈,𝐿 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ; 𝐿𝑈,𝐿 +𝑘,𝑛 (𝑣), 𝒑𝑈 +𝑛 (𝑣)) += 𝑓1(𝐿𝑈,𝐿 +𝑘,𝑛 ) 𝑓2( 𝒑𝑈 +𝑛 (𝑣)) + 𝑓1(𝐿𝑈,𝐿 +𝑘,𝑛 (𝑣)) 𝑓2( 𝒑𝑈 +𝑛 ) ++ +𝜏𝐿𝑈,𝐿 +𝑘,𝑛 +2 +(𝐿𝑈,𝐿 +𝑘,𝑛 − 𝐿𝑈,𝐿 +𝑘,𝑛 (𝑣))2 + +𝜏𝑥𝑈 +𝑛 +2 (𝑥𝑈 +𝑛 − 𝑥𝑈 +𝑛 (𝑣))2 ++ +𝜏𝑦𝑈 +𝑛 +2 (𝑦𝑈 +𝑛 − 𝑦𝑈 +𝑛 (𝑣))2, +(23) +where 𝜏𝐿𝑈,𝐿 +𝑘,𝑛 , 𝜏𝑥𝑈 +𝑛 , 𝜏𝑦𝑈 +𝑛 +> 0. Also, the function 𝐸𝑈 +𝑘,𝑛(𝒛𝑛) +Δ= +𝐸𝑈 +𝑘,𝑛(𝑙𝑈 +𝑘,𝑛) in (18a) is the product of two convex and non- +negative functions, namely +𝑓1(𝑙𝑈 +𝑘,𝑛) = +𝛾𝑈𝐶𝑈 +𝑘 𝑙𝑈 +𝑘,𝑛 +Δ2 +(24) +and +𝑓2(𝑙𝑈 +𝑘′,𝑛) = +� 𝐾 +∑︁ +𝑘′=1 +𝐶𝑈 +𝑘′𝑙𝑈 +𝑘′,𝑛 +�2 +. +(25) +As in (23), we obtain a strongly convex surrogate function +¯𝐸𝑈 +𝑘,𝑛(𝒛𝑛; 𝒛𝑛(𝑣)) of 𝐸𝑈 +𝑘,𝑛(𝒛𝑛) as +¯𝐸𝑈 +𝑘,𝑛(𝒛𝑛; 𝒛𝑛(𝑣)) +Δ= ¯𝐸𝑈 +𝑘,𝑛(𝑙𝑈 +𝑘,𝑛, 𝑙𝑈 +𝑘′,𝑛; 𝑙𝑈 +𝑘,𝑛(𝑣), 𝑙𝑈 +𝑘′,𝑛(𝑣)) += 𝑓1(𝑙𝑈 +𝑘,𝑛) 𝑓2(𝑙𝑈 +𝑘′,𝑛(𝑣)) + 𝑓1(𝑙𝑈 +𝑘,𝑛(𝑣)) 𝑓2(𝑙𝑈 +𝑘′,𝑛) ++ +𝜏𝑙𝑈 +𝑘,𝑛 +2 (𝑙𝑈 +𝑘,𝑛 − 𝑙𝑈 +𝑘,𝑛(𝑣))2 + +𝜏𝑙𝑈 +𝑘′,𝑛 +2 +(𝑙𝑈 +𝑘′,𝑛 − 𝑙𝑈 +𝑘′,𝑛(𝑣))2, +(26) +where 𝜏𝑙𝑈 +𝑘,𝑛, 𝜏𝑙𝑈 +𝑘′,𝑛 > 0. +For the non-convex energy budget constraint (18b), we +derive a convex upper bound by using Lemma 2. The function +𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝒛𝑛) +Δ= 𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝐿𝐼 ,𝑈 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ) is the product of two convex and +non-negative functions, namely +ℎ1(𝐿𝐼 ,𝑈 +𝑘,𝑛 ) = 2 +𝐿𝐼,𝑈 +𝑘,𝑛 +𝐵Δ/𝐾 − 1 +(27) +and +ℎ2( 𝒑𝑈 +𝑛 ) = (𝑥𝑈 +𝑛 − 𝑥𝐼 +𝑘)2 + (𝑦𝑈 +𝑛 − 𝑦𝐼 +𝑘)2 + ℎ𝑈 2. +(28) +Algorithm 1 Proposed algorithm for the “Always On” scenario +Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X; Set 𝑣 = 0. +Output: {𝐿𝐼 ,𝑈 +𝑘,𝑛 }, {𝐿𝑈,𝐿 +𝑘,𝑛 }, {𝐿𝐿,𝑈 +𝑘,𝑛 }, {𝑙𝑈 +𝑘,𝑛}, {𝑙𝐿 +𝑘,𝑛}, { 𝒑𝑈 +𝑛 }. +1: If 𝒛(𝑣) is a stationary solution of (18): STOP. +2: Compute ˆ𝒛 (𝒛(𝑣)) of (30) using dual decomposition or +CVX. +3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)). +4: 𝑣 ← 𝑣 + 1 and go to step 1. +Then, by using Lemma 2 and defining 𝒛𝑛(𝑣) for the 𝑣th +iterate, we obtain a strongly convex surrogate function +¯𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) of 𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝒛𝑛) as +¯𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) +Δ= 𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝐿𝐼 ,𝑈 +𝑘,𝑛 , 𝒑𝑈 +𝑛 ; 𝐿𝐼 ,𝑈 +𝑘,𝑛 (𝑣), 𝒑𝑈 +𝑛 (𝑣)) += 𝑁0𝐵Δ/𝐾 +2𝑔0 +������ +� +2 +𝐿𝐼,𝑈 +𝑘,𝑛 +𝐵Δ/𝐾 − 1 + (𝑥𝑈 +𝑛 − 𝑥𝐼 +𝑘) +2 + (𝑦𝑈 +𝑛 − 𝑦𝐼 +𝑘) +2 + ℎ𝑈 2 +�2 +− +� +2 +𝐿𝐼,𝑈 +𝑘,𝑛 (𝑣) +𝐵Δ/𝐾 +− 1 +�2 +− +� +(𝑥𝑈 +𝑛 (𝑣) − 𝑥𝐼 +𝑘) +2 + (𝑦𝑈 +𝑛 (𝑣) − 𝑦𝐼 +𝑘) +2 + ℎ𝑈 2�2������ +− 𝑁0 ln 2 +𝑔0 +2 +𝐿𝐼,𝑈 +𝑘,𝑛 (𝑣) +𝐵Δ/𝐾 +� +2 +𝐿𝐼,𝑈 +𝑘,𝑛 (𝑣) +𝐵Δ/𝐾 +− 1 +� � +𝐿𝐼 ,𝑈 +𝑘,𝑛 − 𝐿𝐼 ,𝑈 +𝑘,𝑛 (𝑣) +� +− 2𝑁0𝐵Δ/𝐾 +𝑔0 +� +(𝑥𝑈 +𝑛 (𝑣) − 𝑥𝐼 +𝑘) +2 + (𝑦𝑈 +𝑛 (𝑣) − 𝑦𝐼 +𝑘) +2 + ℎ𝑈 2� +� +(𝑥𝑈 +𝑛 (𝑣) − 𝑥𝐼 +𝑘)(𝑥𝑈 +𝑛 − 𝑥𝑈 +𝑛 (𝑣)) + (𝑦𝑈 +𝑛 (𝑣) − 𝑦𝐼 +𝑘)(𝑦𝑈 +𝑛 − 𝑦𝑈 +𝑛 (𝑣)) +� +. +(29) +Finally, the problem in Equation (18) can be transformed +into the strongly convex inner approximation for a given +feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as +min +𝒛 +𝐾 +∑︁ +𝑘=1 +�𝑁 −4 +∑︁ +𝑛=1 +𝛽𝑘,𝑛 ¯𝐸𝑈,𝐿 +𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) ++ +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛) ¯𝐸𝑈 +𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) +� ++ +𝑁 +∑︁ +𝑛=1 +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) +(30a) +s.t. ¯𝐸 𝐼 ,𝑈 +𝑘,𝑛 (𝒛𝑛; 𝒛𝑛(𝑣)) ≤ 𝜀, ∀𝑘 ∈ K, 𝑛 ∈ N +(30b) +(18c) − (18m), +(30c) +which has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)). Since Prob- +lem (30) is convex, we can obtain the closed-form solutions +via dual decomposition [34] or a standard convex optimization +solver such as CVX [35]. The proposed algorithm based +on the SCA method is summarized as Algorithm 1. The +sequence {𝒛(𝑣)} generated by Algorithm 1 converges if the +step size 𝛾(𝑣) is chosen so that 𝛾(𝑣) ∈ (0, 1], 𝛾(𝑣) → 0, and +� +𝑣 𝛾(𝑣) = ∞. Also, {𝒛(𝑣)} is bounded and every limit point +of {𝒛(𝑣)} is stationary. Furthermore, if Algorithm 1 does not +stop after a finite number of steps, none of the stationary points +are a local minimum of Problem (18). + +8 +Algorithm 2 Proposed algorithm for the “Always Off” sce- +nario +Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X; Set 𝑣 = 0. +Output: {𝐿𝐼 ,𝑈 +𝑘,𝑛 }, {𝑙𝑈 +𝑘,𝑛}, { 𝒑𝑈 +𝑛 }. +1: If 𝒛(𝑣) is a stationary solution of (32): STOP. +2: Compute ˆ𝒛 (𝒛(𝑣)) of (33) using dual decomposition or +CVX. +3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)). +4: 𝑣 ← 𝑣 + 1 and go to step 1. +IV. OPTIMAL ENERGY CONSUMPTION FOR THE +“ALWAYS OFF” SCENARIO +In this section, we find the optimal bit allocation and UAV +path planning when the LEO communication is not available +during the entire mission time. Therefore, the total UAV +energy consumption 𝐸𝑡𝑜𝑡𝑎𝑙 +𝑘,𝑛 +in (13) is rewritten with 𝛼𝑘,𝑛 = 0 +for all 𝑛 ∈ N, as +𝐸𝑡𝑜𝑡𝑎𝑙 +𝑘,𝑛 += (1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛(𝑙𝑈 +𝑘,𝑛) + 𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ). +(31) +For UAV computing with 𝛽𝑘,𝑛 = 0, the problem is given with +(31) by +min +𝐿𝐼,𝑈 +𝑘,𝑛 ,𝑙𝑈 +𝑘,𝑛,𝒑𝑈 +𝑛 +𝐾 +∑︁ +𝑘=1 +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛+1(𝑙𝑈 +𝑘,𝑛+1) + +𝑁 +∑︁ +𝑛=1 +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) +(32a) +s.t. +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝐿𝐼 ,𝑈 +𝑘,𝑛 = 𝐼𝑘, ∀𝑘 ∈ K +(32b) +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝑙𝑈 +𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K +(32c) +(18b), (18c), (18k) − (18m), +(32d) +where the equality constraints (32b) and (32c) guarantee that +the total number of uplink bits from the IoT sensor and the +total number of computation bits at the UAV must be equal to +the input bits of the IoT sensor for complete offloading. +In the “Always Off” case, the primal variables are defined as +𝒛 = {𝒛𝑛}𝑛∈N with 𝒛𝑛 = ({𝐿𝐼 ,𝑈 +𝑘,𝑛 }𝑘 ∈K, {𝑙𝑈 +𝑘,𝑛}𝑘 ∈K, 𝒑𝑈 +𝑛 ). Since +Problem (32) is non-convex, it can be transformed into the +strongly convex inner approximation, for a given a feasible +𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as +min +𝒛 +𝐾 +∑︁ +𝑘=1 +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛) ¯𝐸𝑈 +𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) + +𝑁 +∑︁ +𝑛=1 +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) +(33a) +s.t. (32b), (32c), (30b), (18c), (18k) − (18m), +(33b) +where ¯𝐸𝑈 +𝑘,𝑛 of the objective function is defined equally in (26). +Problem (33) has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)) due to +its convexity. As in Problem (30), the locally optimal solution +can be obtained by dual decomposition or a standard convex +optimization solver. The proposed SCA-based algorithm is +summarized in Algorithm 2. +Algorithm 3 Proposed algorithm for the “Intermediate Dis- +connected” scenario +Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X; Set 𝑣 = 0. +Output: {𝐿𝐼 ,𝑈 +𝑘,𝑛 }, {𝐿𝑈,𝐿 +𝑘,𝑛 }, {𝐿𝐿,𝑈 +𝑘,𝑛 }, {𝑙𝑈 +𝑘,𝑛}, {𝑙𝐿 +𝑘,𝑛}, { 𝒑𝑈 +𝑛 }. +1: If 𝒛(𝑣) is a stationary solution of (34): STOP. +2: Compute ˆ𝒛 (𝒛(𝑣)) of (35) using dual decomposition or +CVX. +3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)). +4: 𝑣 ← 𝑣 + 1 and go to step 1. +V. OPTIMAL ENERGY CONSUMPTION FOR THE +“INTERMEDIATE DISCONNECTED” SCENARIO +For the “Intermediate Disconnected” case, we provide joint +path planning and resource allocation when the LEO commu- +nication is intermediately disconnected. The total UAV energy +consumption in this case follows (13). +During the LEO computing for 𝑛 +∈ +{1, · · ·, 𝑁𝑡} with +𝛼𝑘,𝑛 += +1 +and +𝛽𝑘,𝑛 += +1, +we +jointly +optimize +the +bit allocation {𝐿𝐼 ,𝑈 +𝑘,𝑛 }𝑛∈{1,···,𝑁𝑡 },𝑘 ∈K, {𝐿𝑈,𝐿 +𝑘,𝑛 }𝑛∈{2,···,𝑁𝑡+1},𝑘 ∈K, +{𝑙𝐿 +𝑘,𝑛}𝑛∈{3,···,𝑁𝑡+2},𝑘 ∈K +and +{𝐿𝐿,𝑈 +𝑘,𝑛 }𝑛∈{4,···,𝑁𝑡+3},𝑘 ∈K +along +with the UAV trajectory { 𝒑𝑈 +𝑛 }𝑛∈{2,···,𝑁𝑡+4}. During UAV com- +puting for 𝑛 ∈ {1, · · ·, 𝑁𝑡} with 𝛼𝑘,𝑛 = 1 and 𝛽𝑘,𝑛 = 0 and +𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁} with 𝛼𝑘,𝑛 = 0 and 𝛽𝑘,𝑛 = 0, the bit +allocation and the UAV path planning are jointly designed +as in the UAV computing process of the “Always On” case. +Accordingly, we can formulate the problem as +min +𝐿𝐼,𝑈 +𝑘,𝑛 ,𝐿𝑈,𝐿 +𝑘,𝑛 ,𝐿𝐿,𝑈 +𝑘,𝑛 +𝑙𝑈 +𝑘,𝑛,𝑙𝐿 +𝑘,𝑛,𝒑𝑈 +𝑛 +𝐾 +∑︁ +𝑘=1 +𝑁𝑡 +∑︁ +𝑛=1 +𝛼𝑘,𝑛 +� +𝛽𝑘,𝑛𝐸𝑈,𝐿 +𝑘,𝑛+1(𝐿𝑈,𝐿 +𝑘,𝑛+1, 𝒑𝑈 +𝑛+1) ++ �1 − 𝛽𝑘,𝑛 +� 𝐸𝑈 +𝑘,𝑛+1(𝑙𝑈 +𝑘,𝑛+1) +� ++ +𝐾 +∑︁ +𝑘=1 +𝑁 −2 +∑︁ +𝑛=𝑁𝑡+1 +�1 − 𝛼𝑘,𝑛 +� (1 − 𝛽𝑘,𝑛)𝐸𝑈 +𝑘,𝑛+1(𝑙𝑈 +𝑘,𝑛+1) ++ +𝑁 +∑︁ +𝑛=1 +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) +(34a) +s.t. +𝑛 +∑︁ +𝑖=1 +𝐿𝑈,𝐿 +𝑘,𝑖+1 ≤ +𝑛 +∑︁ +𝑖=1 +𝐿𝐼 ,𝑈 +𝑘,𝑖 , ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁𝑡 +(34b) +𝑛 +∑︁ +𝑖=1 +𝑙𝐿 +𝑘,𝑖+2 ≤ +𝑛 +∑︁ +𝑖=1 +𝐿𝑈,𝐿 +𝑘,𝑖+1, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁𝑡 +(34c) +𝑛 +∑︁ +𝑖=1 +𝐿𝐿,𝑈 +𝑘,𝑖+3 ≤ 𝑂𝐿 +𝑘 +𝑛 +∑︁ +𝑖=1 +𝑙𝐿 +𝑘,𝑖+2, ∀𝑘 ∈ K, 𝑛 = 1, · · ·, 𝑁𝑡 +(34d) +𝑁𝑡 +∑︁ +𝑛=1 +𝛽𝑘,𝑛𝑙𝐿 +𝑘,𝑛+2 + +𝑁 −2 +∑︁ +𝑛=1 +(1 − 𝛽𝑘,𝑛)𝑙𝑈 +𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K +(34e) +𝑁𝑡 +∑︁ +𝑛=1 +𝑙𝐿 +𝑘,𝑛+2 = +𝑁𝑡 +∑︁ +𝑛=1 +𝐿𝑈,𝐿 +𝑘,𝑛+1, ∀𝑘 ∈ K +(34f) +𝑁𝑡 +∑︁ +𝑛=1 +𝐿𝐿,𝑈 +𝑘,𝑛+3 = 𝑂𝐿 +𝑘 +𝑁𝑡 +∑︁ +𝑛=1 +𝐿𝑈,𝐿 +𝑘,𝑛+1, ∀𝑘 ∈ K +(34g) +(18b), (18c), (32b), (18k) − (18m), +(34h) + +9 +TABLE III: Simulation Parameters +Parameter +Value +Parameter +Value +𝑣𝑠 +7.5 km/s +𝑟𝐸 +6371 km +𝜃 +10 ◦ +𝑇𝑣 +830 s +ℎ𝑈 +1 km +ℎ𝐿 +600 km +𝐾 +10 +𝑣max +50 m/s +𝑀 +9.65 kg +𝑂𝐿 +𝑘 , 𝑂𝑈 +𝑘 +0.5 +𝑓 𝑈 +𝑛 +19.5 × 109 cycles/s [7] +𝐺 +10 dB +𝛾𝐿, 𝛾𝑈 +10−28 [29], [30] +𝐶𝐿 +𝑘 , 𝐶𝑈 +𝑘 +1550.7 [29], [30] +𝐵 +40 MHz +𝑁0 +-174 dBm/Hz +𝜀 +0.11 J +ref. SNR +80 dB +where the inequality constraints (34b)-(34d) and equality con- +straints (34e)-(34g) limit the number of frames to 𝑛 = 1, ···, 𝑁𝑡 +instead of 𝑛 = 1, ···, 𝑁 −4 in constraints (18d)-(18f) and (18h)- +(18j), respectively. +In the“Intermediate Disconnected” case, the primal vari- +ables are defined the same as in the“Always On” case. By +applying the SCA method,the non-convex Problem (34) can +be transformed into the strongly convex inner approximation +for a given a feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as +min +𝒛 +𝐾 +∑︁ +𝑘=1 +𝑁𝑡 +∑︁ +𝑛=1 +𝛼𝑘,𝑛 +� +𝛽𝑘,𝑛 ¯𝐸𝑈,𝐿 +𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) ++ �1 − 𝛽𝑘,𝑛 +� ¯𝐸𝑈 +𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) +� ++ +𝐾 +∑︁ +𝑘=1 +𝑁 −2 +∑︁ +𝑛=𝑁𝑡+1 +�1 − 𝛼𝑘,𝑛 +� (1 − 𝛽𝑘,𝑛) ¯𝐸𝑈 +𝑘,𝑛+1(𝒛𝑛+1; 𝒛𝑛+1(𝑣)) ++ +𝑁 +∑︁ +𝑛=1 +𝐸𝐹 +𝑛 (𝒗𝑈 +𝑛 ) +(35a) +s.t. (34b) − (34g), (30b), (18c), (32b), (18k) − (18m), (35b) +which has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)) to be obtained +by dual decomposition or a standard convex optimization +solver. Algorithm 3 describes the proposed method for the +“Intermediate Disconnected” scenario. +VI. SIMULATION RESULTS +In this section, we evaluate the performance of the proposed +algorithms to jointly optimize the bit allocation and the UAV +trajectory for marine IoT systems in various LEO accessible +statuses. For reference, we consider the following schemes: +(i) No optimization: The equal bit allocation is considered for +communication and computation per frame, while the UAV +flies at constant velocity between the initial and final positions +as 𝒑𝑈 +𝑛 = 𝒑𝑈 +𝐼 + (𝑛 − 1) � 𝒑𝑈 +𝐹 − 𝒑𝑈 +𝐼 +��𝑁, for 𝑛 ∈ N; (ii) Opti- +mized bit allocation: The communication and computation bits +are optimized by the proposed algorithms while considering +the UAV trajectory with the constant-velocity as in (i); (iii) +Optimized UAV trajectory: The path planning of the UAV +is obtained by the proposed algorithms with fixed equal bit +allocation per frame. The simulation parameters are provided +in Table III. Particularly, the space segment considers Iridium- +like LEO satellite networks that provide global coverage with +66 satellites distributed in 6 polar orbits [15], where the orbit +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +x [km] +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +y [km] +IoT2 +IoT5 +IoT3 +IoT4 +IoT7 +IoT8 +IoT9 +IoT10 +IoT6 +IoT1 +LEO +UAV trajectory +("Always On") +UAV trajectory +("Always Off") +UAV trajectory +("Intermediate Disconnected") +Fig. 4: Optimal UAV trajectories according to the different +LEO access scenarios. +height is ℎ = 601 km with the elevation angle 𝜃 = 10 ◦, and +satellites in the orbit travel at a speed of around 𝑣𝑠 = 7.5 km/s. +To better understand the proposed algorithms, Figs. 4 and 5 +consider the partial optimization of UAV path planning or bit +allocation. As shown in Fig. 4, there are 𝐾 = 10 IoT sensors +distributed randomly in a 10 km × 10 km area within the +beam coverage of the central LEO satellite, i.e., 𝛼𝑘,𝑛 = 1, +for all 𝑛 ∈ N and 𝑘 ∈ K. With the LEO visible time of +𝑇𝑣 = 830 s obtained from (4) and a bandwidth of 𝐵 = 40 +MHz [15], the data size collected from each IoT sensor is +randomly determined based on the computation capability of +the UAV in (16). In our simulation, the scheduling variable +𝛽𝑘,𝑛 is defined from (16), i.e., 𝛽𝑘,𝑛 = [0 0 0 1 1 1 0 1 0 0], +for 𝑘 ∈ K and 𝑛 ∈ N, as shown in Fig. 4. The IoT sensors +with 𝛽𝑘,𝑛 = 0 for UAV computing and with 𝛽𝑘,𝑛 = 1 for LEO +computing are indicated by black-colored circles and green- +colored circles, respectively, while the LEO satellite, indicated +by a red-colored hexagram, travels along the red dotted line. +The initial and final positions of the UAV are 𝒑𝑈 +𝐼 = (5, 0, 0) +to 𝒑𝑈 +𝐹 = (10, 5, 0). +Fig. 4 shows the optimized UAV trajectories with the fixed +equal bit allocation according to the different LEO satellite +access scenarios. For this experiment, the latency constraint is +𝑇 = 360 s with 𝑁 = 60 and Δ = 6 s. In the “Always On” case, +the optimized UAV trajectory, represented by a blue asterisk +line, is designed to fly close to the IoT sensors with LEO +computing until its final destination. This can significantly +reduce the large amount of uplink communication energy +consumption induced by the long distance between the LEO +satellite and UAV. In the “Always Off” case, where only UAV +computing is considered, the UAV flies along a straight path +to a destination, which is represented by a yellow crossed +line. In this case, the flying energy consumption must be +reduced to to minimize the total UAV energy due to the fixed +computation bit allocation. In the “Intermediate Disconnected” +case, where the LEO communication is lost at 𝑁𝑡 = 𝑁/2, the + +10 +10 +20 +30 +40 +50 +60 +Frame number +0 +1 +2 +IoT6's number of bits +107 + LI,U +6,n + lU +6,n + LU,L +6,n + lL +6,n + LL,U +6,n +(a) “Always On” scenario +10 +20 +30 +40 +50 +60 +Frame number +0 +5 +10 +IoT6's number of bits +106 +(b) “Always Off” scenario +10 +20 +30 +40 +50 +60 +Frame number +0 +0.5 +1 +1.5 +2 +IoT6's number of bits +107 +(c) “Intermediate Disconnected” scenario +Fig. 5: Optimal bit allocations for IoT sensor 6 in Fig. 4 +according to the different LEO access scenarios. +optimized UAV trajectory, represented by a purple square line, +tends to fly close to the IoT sensors with LEO computing for +𝑛 = 1, · · ·, 𝑁𝑡. Then, in the frame period of 𝑛 = 𝑁𝑡 + 1, · · ·, 𝑁 +where LEO communication is disconnected, the UAV flies +straight to the final destination because it performs only UAV +computing. +Fig. 5 illustrates the optimized bit allocations for IoT sensor +6 shown in Fig. 4 with the fixed constant-velocity UAV +trajectory according to different LEO access scenarios. Except +for the UAV trajectory, the simulation environment is the same +as in Fig. 4. In Fig. 5(a), the optimal bit allocations 𝐿𝐼 ,𝑈 +𝑘,𝑛 , +𝐿𝑈,𝐿 +𝑘,𝑛 , 𝑙𝐿 +𝑘,𝑛, 𝐿𝐿,𝑈 +𝑘,𝑛 +by proposed Algorithm 1 are shown for +LEO computing in the “Always On” case. First, most of the +uplink bits 𝐿𝐼 ,𝑈 +𝑘,𝑛 are allocated between frames 20 to 35, which +corresponds to the period where the UAV flies closest to IoT +sensor 6. The offloading bits 𝐿𝑈,𝐿 +𝑘,𝑛 +are allocated equally in +the entire frame because the equal bit allocation can achieve +the minimal communication energy from (7). Finally, the LEO +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +x [km] +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +y [km] +IoT3 +IoT4 +IoT10 +IoT6 +IoT2 +IoT8 +IoT1 +IoT7 +IoT9 +IoT5 +LEO +2nd +orbit +1st +orbit +3rd +orbit +UAV trajectory +(1st orbit) +UAV trajectory +(2nd orbit) +UAV trajectory +(3rd orbit) +Fig. 6: Optimal UAV trajectories according to different LEO +satellite orbits, where the IoT sensors with LEO computing +are deployed at the corner. +computing bits 𝑙𝐿 +𝑘,𝑛 and LEO downlink bits 𝐿𝐿,𝑈 +𝑘,𝑛 are mostly +allocated in the latter parts between frames 50 to 60 to satisfy +the inequality constraints of (18e) and (18f). In Fig. 5(b), the +optimized bit allocations 𝐿𝐼 ,𝑈 +𝑘,𝑛 and 𝑙𝑈 +𝑘,𝑛 obtained by proposed +Algorithm 2 are shown for UAV computing of the“Always +Off” case. Since the UAV cannot communicate with the LEO +satellite, the computing process is entirely at the UAV-mounted +cloudlet. The uplink bits 𝐿𝐼 ,𝑈 +𝑘,𝑛 and the computing bits 𝑙𝑈 +𝑘,𝑛 are +assigned the same as 𝐿𝐼 ,𝑈 +𝑘,𝑛 and 𝐿𝑈,𝐿 +𝑘,𝑛 in Fig. 5(a), respectively. +However, 𝑙𝑈 +𝑘,𝑛 is dramatically reduced to 8 × 106 per frame +compared to 10 × 106, as illustrated in Fig. 5(a). This is +because the amount of data exceeding the UAV computation +capability is excluded from the UAV computing. Fig. 5(c) +shows the optimization result of bit allocation attained by +proposed Algorithm 3 in the “Intermediate Disconnected” +case. LEO computing is performed during the first half of +frames, i.e., 𝑛 = 1, ···, 𝑁𝑡, while UAV computing is performed +during the second half of frames, i.e., 𝑛 = 𝑁𝑡 + 1, · · ·, 𝑁. The +computing bits 𝑙𝐿 +𝑘,𝑛 at LEO and the downlink bits 𝐿𝐿,𝑈 +𝑘,𝑛 are +reduced in proportion to the reduced frame duration of LEO +computing compared to those shown in Fig. 5(a). For UAV +computing, there are more computing bits 𝑙𝑈 +𝑘,𝑛 allocated at +the UAV than those from the case in Fig. 5(b). This means +that less data exceeds the computational capability of the UAV +thanks to the LEO computing. +Fig. 6 shows the optimal UAV trajectories according to the +different LEO satellite orbits in the “Always On” scenario, +where the IoT sensors that need LEO computing are clustered +at the corner, i.e., 𝛽𝑘,𝑛 = [0 1 1 0 1 1 0 0 0 0], for 𝑘 ∈ K and +𝑛 ∈ N. In this deployment, the three different movements of +the LEO satellite in different orbital directions are considered. +In the first orbit moving from the upper right corner to the +lower left corner, the UAV flies near the corner area with IoT +sensors with LEO computing to its final destination. In the + +11 +400 +600 +800 +1000 +1200 +1400 +1600 +Total time T (s) +0 +1 +2 +3 +4 +5 +6 +7 +Total UAV energy consumption (J) +106 +No opt. - "Always On" +No opt. - "Always Off" +No opt. - "Intermediate Disconnected" +Opt. bit allocation - "Always On" +Opt. UAV trajectory - "Always On" +Joint opt. - "Always On" +Joint opt. - "Always Off" +Joint opt. - "Intermediate Disconnected" +Fig. 7: Comparison of the total UAV energy consumption +for different optimization schemes in the three LEO satellite +access scenarios. +second orbit moving from the upper left corner to the lower +right corner, the UAV flies in a diagonally downward direction +along its own orbit rather than the optimized UAV trajectory +for the first orbit. In the third orbit moving upwards from +below the midpoint, the UAV flies in an upward direction along +its own orbit rather than the optimal UAV trajectory for the first +orbit. From these results, we can see that the LEO movements +resulting from the orbit influences the optimal UAV path so +as to reduce the communication energy consumption between +the UAV and the LEO satellite. +Fig. 7 compares the total UAV energy consumption of the +joint optimization scheme with reference schemes in three +LEO satellite access scenarios. For this experiment, the latency +constraint is 𝑇 = [360:90:1620] s with 𝑁 = [60:15:270] and +Δ = 6 s, while the remaining simulation parameters are +the same as in Figs. 4 and 5. First, the no optimization +scheme consumes the highest energy in the three scenarios, +among which the largest energy consumption takes place in +the “Always Off” case, where only the UAV computing is +performed. This is natural since the UAV-mounted cloudlet has +a slightly larger burden in terms of the energy consumption +with no support of the LEO. In the “Always On” case, for +𝑇 = 360 s, the total UAV energy consumption for the joint +optimization scheme is the lowest at 4.6 × 106 J, whereas +the optimized UAV trajectory scheme with fixed equal bit +allocation requires 5.5×106 J, and the optimized bit allocation +with the constant-velocity UAV and no optimization schemes +requires 6.1 × 106 J. This implies that the UAV path planning +is more effective in terms of UAV energy consumption than +bit allocation. Moreover, the total energy consumption in all +schemes decreases as the total time increases. This is because +the same amount of data is processed over a longer period +of time. Compared to the total UAV energy consumption of +the joint optimization scheme in the “Always Off” scenario, +those of the joint optimization scheme in other scenarios +0 +1/8 +2/8 +4/8 +6/8 +7/8 +1 +LEO satellite access time rate +2.3 +2.4 +2.5 +2.6 +2.7 +2.8 +2.9 +3 +3.1 +3.2 +3.3 +3.4 +Total UAV energy consumption (J) +106 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Collected data usage rate (%) +Total energy - "Always On" +Total energy - "Always Off" +Total energy - "Intermediate Disconnected" +Data usage rate - "Always On" +Data usage rate - "Always Off" +Data usage rate - "Intermediate Disconnected" +Fig. 8: Relationship between the total UAV energy consump- +tion and the collected data usage rate in three LEO satellite +access scenarios according to the LEO satellite access time +rate. +are much higher since the UAV flies straight to its final +destination when the LEO satellite connection is lost, as in Fig. +4. However, there is a trade-off between the total UAV energy +consumption and the collected data usage rate for computing, +which determines the amount of data executed at cloudlet, +which is analyzed in the following figure. +Fig. 8 shows the relationship between the total UAV energy +consumption and the collected data usage rate for computing in +the different LEO accessibility scenarios. Any amount of data +exceeding the UAV computation capability is excluded from +UAV computing. For this experiment, the scheduling variables +are defined as 𝛽𝑘,𝑛 = [0 0 1 1 1 1 0 1 0 0], for 𝑘 ∈ K +and 𝑛 ∈ N. The UAV computation capability is applied to +226 Mbits by using the CPU frequency at the UAV server +𝑓 𝑈 +𝑛 += 9.75 × 109 cycles/s. In the “Always On” scenario, the +LEO satellite access time rate is 1. At this time, the total +UAV energy consumption is 3.3×106 J and the collected data +usage rate is 100%. In the “Always Off” case, where the LEO +satellite access time rate is 0, the total UAV energy consump- +tion is 2.24 × 106 J and the collected data usage rate is 54%. +Although the energy consumption in the “Always Off” case is +dramatically reduced, the utilization rate of the collected data +is also cut in half. In the “Intermediate Disconnected” case, +as the LEO satellite access time rate increases, the total UAV +energy consumption and the collected data usage rate increase +differently. When the LEO satellite access time rate is above +6/8, the total UAV energy consumption is saturated with the +total UAV energy consumption of the “Always On” case. This +is because the straight flight segment of the UAV to the final +destination after disconnecting with the LEO satellite matches +that of the “Always On” case. Also, when the LEO satellite +access time rate is more than 7/8, the collected data usage +rate is more than about 95%. In this simulation environment, +adequate data usage and energy consumption is achieved with +more than a 7/8 LEO satellite access time rate. + +12 +VII. CONCLUSIONS +In this paper, a marine IoT system using hybrid LEO +and UAV computing for real-time utilization of marine data +has been analyzed according to the different LEO satellite +access scenarios: “Always On,” “Always Off” and “Inter- +mediate Disconnected”. For each scenario, we proposed the +joint optimization problem of bit allocation for computing +and communication in offloading and UAV path planning to +minimize the total UAV energy consumption under latency, +energy budget, and UAV operational constraints. To solve the +optimization problem, we developed an SCA-based algorithm +whose performance in terms of energy efficiency was validated +via numerical results compared to conventional approaches +with partial optimization that design only the bit allocation or +UAV trajectory. 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Commun., vol. 64, no. +12, pp. 4983-4996, Dec. 2016. +[32] N. Xue, “Design and optimization of lithium-ion batteries for electric- +vehicle applications,” Doctoral dissertation, Univ. Michigan, Ann Arbor, +MI, USA, 2014. +[33] A. Chakrabarty and J. Langelaan, “Energy-based long-range path plan- +ning for soaring-capable unmanned aerial vehicles,” J. Guid., Control, +Dyn., vol. 34, no. 41, pp. 1002-1015, Jul. 2011. +[34] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, UK: +Cambridge University Press, 2004. +[35] M. Grant and S. Boyd, “Cvx: Matlab software for disciplined +convex programming, version 2.1, Mar. 2014,” Available on-line at +http://cvxr.com/cvx. + diff --git a/JtE2T4oBgHgl3EQfUwcR/content/tmp_files/load_file.txt b/JtE2T4oBgHgl3EQfUwcR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b867b1615578e77ba5f8015ab2f6fb6acc937f85 --- /dev/null +++ b/JtE2T4oBgHgl3EQfUwcR/content/tmp_files/load_file.txt @@ -0,0 +1,1043 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf,len=1042 +page_content='1 Marine IoT Systems with Space-Air-Sea Integrated Networks: Hybrid LEO and UAV Edge Computing Sooyeob Jung, Seongah Jeong, Jinkyu Kang, and Joonhyuk Kang Abstract—Marine Internet of Things (IoT) systems have grown substantially with the development of non-terrestrial networks (NTN) via aerial and space vehicles in the upcoming sixth- generation (6G), thereby assisting environment protection, mili- tary reconnaissance, and sea transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Due to unpredictable climate changes and the extreme channel conditions of maritime networks, however, it is challenging to efficiently and reliably collect and compute a huge amount of maritime data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this paper, we propose a hybrid low-Earth orbit (LEO) and unmanned aerial vehicle (UAV) edge computing method in space- air-sea integrated networks for marine IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Specifically, two types of edge servers mounted on UAVs and LEO satellites are endowed with computational capabilities for the real-time utilization of a sizable data collected from ocean IoT sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Our system aims at minimizing the total energy consumption of the battery-constrained UAV by jointly optimizing the bit allocation of communication and computation along with the UAV path planning under latency, energy budget and opera- tional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For availability and practicality, the proposed methods were developed for three different cases according to the accessibility of the LEO satellite, “Always On,” “Always Off” and “Intermediate Disconnected”, by leveraging successive convex approximation (SCA) strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Via numerical results, we verify that significant energy savings can be accrued for all cases of LEO accessibility by means of joint optimization of bit allocation and UAV path planning compared to partial optimization schemes that design for only the bit allocation or trajectory of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Index terms — Marine networks, Internet of Things (IoT), edge computing, low-Earth orbit (LEO) satellite, unmanned aerial vehicles (UAVs), successive convex approximation (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' INTRODUCTION M ARINE Internet of Things (IoT) systems have evolved significantly with the rapid development of non- terrestrial network (NTN) technologies composed of space and airborne platforms to collect and process a variety of ocean data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The vast amount of ocean data plays an important This work was supported by the Institute of Information & communica- tions Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='2021-0-00847, Development of 3D Spatial Satellite Communications Technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This research was supported by the Ministry of Science and ICT (MSIT), Korea, under the Information Technology Research Center (ITRC) support program (IITP-2020-0-01787) supervised by the IITP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Sooyeob Jung is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), and with the Satellite Wide-Area Infra Research Section, Electronics and Telecommunications Re- search Institute (ETRI), Daejeon, South Korea (Email: jung2816@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Seongah Jeong is with the School of Electronics Engineering, Kyungpook National University, Daegu 14566, South Korea (Email: seongah@knu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Jinkyu Kang is with the Department of Information and Communications Engineering, Myongji University, Gyeonggi-do 17058, South Korea (Email: jkkang@mju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Joonhyuk Kang is with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea (Email: jhkang@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' role in marine monitoring, which contributes to environ- mental protection, natural disaster prevention, oceanographic research, mineral exploration, military surveillance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' [1]- [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In particular, continuous monitoring of various physical phenomena of marine networks, such as sounds, vibrations and images, requires high-precision and wide-range measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Currently, three types of marine monitoring platforms are being investigated according to the relay node: shore-based radar, survey vessels and satellites [1], most of which have the following procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' By using existing information communi- cation technologies, the marine data collected from ocean IoT sensors is transferred to a ground cloud server with sufficient computation storage capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The ground cloud server stores and analyzes the collected data, thereby managing various ap- plications based on ocean utilization and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In shore- based radar systems installed on offshore buoys and automatic weather stations located on the coast or islands, there are difficulties in installation and maintenance due to their spatial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Meanwhile, survey vessel-based platforms have temporal constraints, which limit the time for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In addition, unexpected loss and defects of collected data may occur in point measurements attained by platforms with shore- based radar or survey vessel platforms due to extreme channel environments and unpredictable climate changes in the ocean [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' To address these spatial and temporal limitations, satellite- based monitoring can be an alternative that provides full coverage of the area of interest with one or multiple satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' With the participation of global companies in the satellite business such as SpaceX, Amazon, and Telesat [4], low- Earth orbit (LEO) satellites are gaining more attention than ever before, and cost-effective easy-to-deploy large-scale satel- lite networks are being established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In addition, conventional satellite operators such as Spire, Kepler, Fleet, Lacuna space and Eutelsat, are preparing to provide satellite IoT services with global coverage [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Until recently, satellites have mostly been adopted as a relay with terrestrial networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' however, for future 6G IoT services, they can operate as functional network components, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', computing servers [7]- [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Traditionally, the critical drawback of satellite-assisted networks is the latency resulting from round-trip delays due to the IoT sensor-satellite-terrestrial station link as well as the rapidly increasing volume of transmitted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Therefore, it is beneficial to bring computing functions in the satellite to handle processing capabilities of the collected data, rather than sending it to the ground cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the following section, we briefly summarize the recent research activities that focus on hierarchical integrated networks using satellites as computing servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='03815v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='SY] 10 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Related Works Satellite-assisted edge computing systems have been ac- tively studied in space-ground integrated networks [7]-[13], space-air-ground integrated networks (SAGIN) [14]-[21] and space-air-sea-based non-terrestrial networks (SAS-NTN) [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In particular, the authors in [7] propose a three-tier computation architecture consisting of ground users, LEO satellites and ground servers to minimize the total energy consumption of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In [8], network slice scheduling for satellite-assisted computing architecture is studied, where satellite servers and ground servers are considered for IoT applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Although satellite-assisted edge computing can provide real-time offloading services to large areas, such as the ocean, it still faces several practical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For long- distance communication with a satellite, more transmit power and larger antenna size are preferred at ground user terminals, which is costly and spatially-limited in real applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Moreover, the transceiver for satellite communications must be robustly designed against severe fading due to atmospheric turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Unmanned aerial vehicles (UAVs) can be adopted to provide enhanced coverage for overcoming path loss and fading issues of satellite-assisted edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' UAVs can receive and compute data in close proximity to ocean IoT sensors, or can relay the data to the cloud server for computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Recently, UAV-assisted satellite IoT networks have been suggested in several studies [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' [14] propose offloading systems of remote IoT applications in the space-air-ground scenario, where UAVs provide computational capability to nearby users as edge servers, while satellites relay the of- floaded data to the ground cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In [15], LEO satellite- assisted UAV data collection for IoT sensors is proposed, where the delay-tolerant data and delay-sensitive data are transferred to the ground cloud server via UAV and LEO satellite, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' As briefly reviewed above, most of existing works on hierarchical offloading systems in the integrated space and air networks assume terrestrial infrastructures, which may result in latency caused by the extreme channel variation of marine IoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Furthermore, even though space or aerial computing platforms are considered, most studies assume full accessibility of the LEO satellite during mission time, which may not be guaranteed according to the orbit of revolution of the LEO satellite under insufficient deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' To perform real-time data mining and analysis of ocean data in marine IoT systems, the use of aerial/space moving cloudlets play an important role considering their availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Main Contributions In this paper, we focus on a marine IoT system with space-air-sea integrated networks, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1, where both UAV and LEO satellite-mounted cloudlets are deployed to offer computing opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the proposed system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' a number of ocean IoT sensors are distributed only to collect abundant marine information with limited battery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' and transmit the collected data to a designated computing server among UAV or LEO-mounted cloudlets so as to satisfy the LEO satellite (Cloud server) UAV (Edge server) IoT 1 End user Frame 1 Frame n Frame N IoT k IoT K ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='0 I I I k k k x y = p ( ) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' E E E n n n x y h = p ( ) 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' C C C x y h h = + p 1 I K + p 1 E N + p IoT → UAV (for UAV computing) UAV → LEO LEO → UAV UAV → User IoT → UAV → LEO → UAV (for LEO computing) IoT → UAV (for edge computing) IoT → UAV (for cloud computing) UAV → LEO (offloading) LEO → UAV UAV → User IoT → UAV (for edge computing) IoT → UAV (for cloud computing) UAV → LEO (offloading) LEO → UAV UAV → User LEO satellite-mounted cloudlet End user Ocean IoT sensor k ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='0 I I I k k k x y = p ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' U U U n n n U x y h = p UAV-mounted cloudlet LEO → Ground Orbit Ocean IoT sensor 1 Ocean IoT sensor K IoT → UAV (for UAV computing) IoT → UAV → LEO → UAV (for LEO computing) Space Air Sea ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' L L L n n n U L x y h h = + p LEO satellite-mounted cloudlet End user Ocean IoT sensor k UAV-mounted cloudlet Orbit Space Air Sea ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' L L L n n n U L x y h h = + p ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' U U U n n n U x y h = p ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='0 I I I k k k x y = p 1 U p U N p 1 Ip I K p UAV computing: Sensor → UAV → LEO computing: Sensor → UAV → UAV computing LEO computing LEO satellite-mounted cloudlet End user Ocean IoT sensor k UAV-mounted cloudlet Orbit Space Air Sea ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' L L L n n n U L x y h h = + p ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' U U U n n n U x y h = p ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='0 I I I k k k x y = p 1 U p U N p 1 I p I K p End user : : Sensor → UAV → LEO → UAV → End user UAV computing LEO computing UAV computing: Sensor → UAV → End user LEO computing: Sensor → UAV →LEO → UAV → End user Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1: Marine IoT system model with a space-air-sea inte- grated network using hybrid LEO and UAV edge computing for real-time data utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' system design criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Here, the LEO satellite is assumed to have a higher computational capability to process the task than that of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' When the IoT data size exceeds the computation capacity of the UAV, the computational task is totally offloaded to the LEO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The computation results executed at LEO are retransmitted to the UAV, are stored until it arrives over the end user, and is finally sent to the end user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' To this end, we tackle the key design problem of jointly optimizing the bit allocation for communication and computing and the trajectory of the UAV, with the aim of minimizing its energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows: For marine IoT systems with extreme channel environ- ments and unpredictable climate changes, we propose a hybrid LEO and UAV edge computing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The scheduling between UAV and LEO satellite-mounted cloudlets depends on the size of the offloaded ocean data and the LEO connection status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For practicality and usability, we consider three different scenarios according to LEO availability such as “Always On,” “Always Off” and “Intermediate Disconnected”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For each case, we develop the joint optimization of bit allocation required for offloading and UAV path planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The non-convex optimization problems formulated for three different cases depending on the availability of the LEO satellite are tackled by means of a successive convex approximation (SCA) algorithm [24], [25], which can guarantee the local minimum of the original non-convex problems by using an efficient iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The system model is presented in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Section III, IV and V provide problem formulations and proposed methods for the LEO access status of “Always On,” “Always Off” and “Intermediate Disconnected”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Simulation results are given in Section VI, and conclusions are summarized in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' C3 Frame n-1 Frame n IoT sensor 1 〮〮〮 IoT sensor k 〮〮〮 IoT sensor K \uf044 … … K \uf044 1 U n− p U np 1 U n+ p 2 U n+ p Frame n+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2: Frame structure of orthogonal access for multiple ocean IoT sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' SYSTEM MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Set-up Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1 illustrates a marine IoT system with a space-air- sea integrated network using hybrid LEO and UAV edge computing, where 𝐾 ocean IoT sensors collect marine data to be entirely transferred to available cloudlets for computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The computed results are then designated to an end user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For real-time data utilization, two types of cloudlets mounted on the UAV and LEO satellite are considered, between which the scheduling depends on the UAV computing capability and LEO accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Specifically, when the collected data size exceeds the computation capacity of the UAV, the data should be entirely offloaded to the LEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The computing capability of the LEO satellite is assumed to be higher than that of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Another major factor for scheduling is whether the LEO satellite is available or not since its beam coverage varies according to the orbit of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Here, we consider three different cases according to the availability of the LEO satellite during mission time: “Always On,” “Always Off” and “Intermediate Disconnected”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For each scenario, we developed the joint optimization of the bit allocation for communication and computation and the trajectory of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Depending on the types of cloudlets, we refer to UAV computing and LEO computing, where computing of the IoT sensor task is executed at the UAV and LEO, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In UAV computing, the task of the IoT sensor 𝑘 is offloaded to the UAV-mounted cloudlet until the UAV arrives over the end user and the output results are conveyed to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In LEO computing, the UAV receives and relays the offloaded data of the IoT sensor to LEO for the LEO execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The computed results at LEO are then sent to the end user via the UAV when the UAV arrives above them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For communication links between IoT sensors and the UAV, and between the UAV and LEO satellite, a frequency division duplex (FDD) scheme is assumed with equal bandwidth 𝐵 for the uplink and downlink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Each IoT sensor 𝑘 has the number 𝐼𝑘 of input information bits to be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The results for LEO computing and UAV computing are characterized as the number 𝑂𝐿 𝑘 and 𝑂𝑈 𝑘 of bits produced per input bit of the IoT sensor 𝑘, and the number 𝐶𝐿 𝑘 and 𝐶𝑈 𝑘 of CPU cycles per input bit for computing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' We assume that all tasks must be computed within the total mission time 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Here, a three- dimensional Cartesian coordinate system is adopted based on the metric unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' We assume that the IoT sensor 𝑘 is deployed at position 𝒑𝐼 𝑘 = (𝑥𝐼 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑦𝐼 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑎𝑘),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑘 ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐾 + 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' with 𝑎𝑘 being the average sea surface level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' where the position TABLE I: List of Symbols Symbol Definition 𝐾 Number of ocean IoT sensors 𝑇 Total mission time Δ Frame duration 𝑁 Number of frames within 𝑇 ℎ𝑈 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ℎ𝐿 Altitudes of UAV and LEO satellite with respect to average sea surface level and UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' respectively 𝑔𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ℎ𝑛 Path loss between the IoT sensor 𝑘 and UAV and between the UAV and LEO at the 𝑛th frame 𝑔0 Channel gain at reference distance 1 m 𝑇𝑣 Visible time of an LEO satellite 𝑣𝑠 Speed of an LEO satellite ℎ Height of an LEO satellite orbit 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝜑 Elevation angle and beamwidth of the LEO satellite 𝑀 the gross mass of the UAV 𝒗𝑈 𝑛 velocity vector of the UAV at the 𝑛th frame 𝜀 Energy budget of the IoT sensor 𝑘 at each frame 𝐼𝑘 Number of input bits of the IoT sensor 𝑘 𝐸𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Energy consumption for uplink communication at the IoT sensor 𝑘 at the 𝑛th frame 𝐸𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐸𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Energy consumption for computing and uplink com- munication at the UAV-mounted cloudlet for the IoT sensor 𝑘 at the 𝑛th frame 𝐸𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸 Energy consumption for downlink communication at the UAV-mounted cloudlet 𝐸 𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐸 𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Energy consumption for computing and downlink com- munication at the LEO-mounted cloudlet for the IoT sensor 𝑘 at the 𝑛th frame 𝐸𝐹 𝑛 Energy consumption for a UAV flying at the 𝑛th frame 𝐿𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Number of bits for uplink communication at the IoT sensor 𝑘 at the 𝑛th frame 𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Number of bits for computing and uplink communica- tion at a UAV-mounted cloudlet for the IoT sensor 𝑘 at the 𝑛th frame 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸 Number of bits for downlink communication at the UAV-mounted cloudlet 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Number of bits for computing and downlink communi- cation at the LEO-mounted cloudlet for the IoT sensor 𝑘 at the 𝑛th frame 𝑂𝐿 𝑘 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑂𝑈 𝑘 Number of output bits produced per input bit of the IoT sensor 𝑘 𝑓 𝐿 𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑓 𝑈 𝑛 CPU frequency at the LEO and UAV-mounted cloudlets for the 𝑛th frame 𝐶𝐿 𝑘 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐶𝑈 𝑘 CPU cycles per input bit at the LEO and UAV-mounted cloudlets for the task of the IoT sensor 𝑘 𝛾𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝛾𝑈 Effective switched capacitances of the LEO and UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' respectively 𝒑𝐼 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝑈 𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝐿𝑛 Positions of the IoT sensor 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' UAV and LEO for the 𝑛th frame 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Variables to indicate LEO connection and offloading scheduling of the IoT sensor 𝑘 at the 𝑛th frame 𝑁𝑡 Frame number during LEO disconnection of the end user is considered with an index of 𝐾 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The UAV flies along a trajectory 𝒑𝑈 (𝑡) = (𝑥𝑈 (𝑡), 𝑦𝑈 (𝑡), ℎ𝑈) with a fixed altitude ℎ𝑈 assumed for system stability, for 0 ≤ 𝑡 ≤ 𝑇, and the position of the LEO satellite is defined as 𝒑𝐿(𝑡) = (𝑥𝐿(𝑡), 𝑦𝐿(𝑡), ℎ𝑈 + ℎ𝐿) with a fixed altitude ℎ𝑈 + ℎ𝐿, for 0 ≤ 𝑡 ≤ 𝑇, all the altitudes are measured with respect to the average sea surface level 𝑎𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For the multiple access of 𝐾 ocean IoT sensors, orthogonal access is assumed, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For tractability, in this paper, the total time duration 𝑇 is divided into 𝑁 frames of duration Δ seconds, each of which is equally divided as Δ/𝐾 seconds, and is preallocated to IoT sensors for uplink and downlink communication required 4 Er h L LEO Satellite s \uf067 \uf071 Earth Orbit IoT Sensor \uf06a sv Er h L LEO Satellite s \uf067 \uf071 Earth Orbit IoT Sensor \uf06a sv Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3: Geometric relationship between the ground user and the LEO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Accordingly, the IoT sensors do not interfere with each other in the offloading procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Moreover, the information data collected from the IoT sensor 𝑘 at the 𝑛 th frame is assumed to be entirely computed and transferred to the designated node within the corresponding frame during Δ/𝐾 seconds, for 𝑛 ∈ {1, · · ·, 𝑁}, so that the computational task cannot be partitioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' According to the discretized time unit, the trajectory of the UAV 𝒑𝑈 (𝑡) and the position of the LEO satellite 𝒑𝐿(𝑡) is expressed as 𝒑𝑈 𝑛 = (𝑥𝑈 𝑛 , 𝑦𝑈 𝑛 , ℎ𝑈) and 𝒑𝐿 𝑛 = (𝑥𝐿 𝑛 , 𝑦𝐿 𝑛, ℎ𝑈 + ℎ𝐿), for 𝑛 ∈ N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The LEO satellite generally flies at a constant speed along its orbit and the relative positional coordinates of the LEO and UAV should vary constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For the task mission of marine IoT systems, the initial location 𝒑𝑈 𝐼 and the final location 𝒑𝑈 𝐹 of the UAV are assigned to 𝒑𝑈 1 and 𝒑𝑈 𝑁 +1, respectively, and its maximum speed constraint is given as ��𝒗𝑈 𝑛 �� = �� 𝒑𝑈 𝑛+1 − 𝒑𝑈 𝑛 �� Δ ≤ 𝑣max, (1) where the velocity vector 𝒗𝑈 𝑛 of the UAV is defined as ( 𝒑𝑈 𝑛+1 − 𝒑𝑈 𝑛 )/Δ, and 𝑣max is its maximum velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The overall system variables and parameters are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' We assume that communication channels between the IoT sensors and UAV [16], [26], and between the UAV and LEO satellite [15], [16] are dominated by line-of-sight (LoS) links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' At the 𝑛th frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' the channel gains for the IoT sensor 𝑘-UAV link and UAV-LEO link are written as 𝑔𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛( 𝒑𝑈 𝑛 ) = 𝑔0 (𝑥𝑈𝑛 − 𝑥𝐼 𝑘)2 + (𝑦𝑈𝑛 − 𝑦𝐼 𝑘)2 + ℎ𝑈 2 (2) and ℎ𝑛( 𝒑𝑈 𝑛 ) = 𝑔0𝐺 (𝑥𝐿𝑛 − 𝑥𝑈𝑛 )2 + (𝑦𝐿𝑛 − 𝑦𝑈𝑛 )2 + ℎ𝐿2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (3) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' where 𝑔0 represents the channel gain at the reference distance 1 m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' and 𝐺 is an antenna gain for the long- distance satellite communication consisting of the transmission antenna gain of the UAV and the receiver antenna gain of the LEO satellite [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In real applications, note that ℎ𝑛( 𝒑𝑈 𝑛 ) ≫ 𝑔𝑘,𝑛( 𝒑𝑈 𝑛 ) is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For communication links, an additive white Gaussian noise is considered with zero mean and power spectral density 𝑁0 [dBm/Hz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Coverage Model of the LEO Satellite In this section, we explore the beam coverage model [7], [28] of an LEO satellite that accounts for the effect of the orbit of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3, when the LEO satellite makes an orbit round, the available communication time with the UAV can be limited, which is referred to as the LEO visible time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The length of the visible time window is defined as 𝑇𝑣 = 𝐿 𝑣𝑠 = 2 (𝑟𝐸 + ℎ) 𝛾 𝑣𝑠 , (4) where 𝑣𝑠 is the speed of the LEO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿 is the arc length to define the coverage where IoT sensors can communicate with the LEO satellite, and is calculated by 𝐿 = 2 (𝑟𝐸 + ℎ) 𝛾 with 𝑟𝐸 being the radius of Earth, ℎ being the height of the LEO satellite orbit, and 𝛾 being the angle of the satellite coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In general, due to the very low altitude of a UAV in comparison to the orbit height, the same visible time window is applied to the UAV and IoT sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The maximum length of the LEO visible time window can be achieved when 𝛾 = 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The angle 𝛾 of the satellite coverage is calculated by 𝛾 = cos−1 � 𝑟𝐸 𝑟𝐸 + ℎ · cos 𝜃 � − 𝜃, (5) where 𝜃 and 𝜑 are the elevation angle and the beamwidth of the satellite, respectively, and are derived as 𝜃 = cos−1 � 𝑟𝐸+ℎ 𝑠 cos (𝜃 + 𝜑) � and 𝜑 = 𝜋/2 − (𝜃 + 𝛾) with 𝑠 indicating the distance between the IoT sensor and LEO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' We assume that the UAV can fully access the LEO satellite within the visible time window of 𝑇𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' According to the availability of LEO communication based on the coverage model, three different cases can be considered: “Always On,” “Always Off” and “Intermediate Disconnected”, the details for which are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1) “Always On” scenario (𝑇 ≤ 𝑇𝑣): The first scenario is when the UAV can communicate with the LEO satellite during the entire mission time since the total mission time is within the LEO visible time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝑇 ≤ 𝑇𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this scenario, we have 𝛼𝑘,𝑛 = 1 for all 𝑛 ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' therefore, the computation capability of the UAV determines whether the UAV or LEO will be used for computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2) “Always Off” scenario (𝑇𝑣 = 0): The second scenario is when LEO communication is not available during the entire mission time since the UAV flies outside the beam coverage of the LEO satellite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝑇𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this scenario, we have 𝛼𝑘,𝑛 = 0 for all 𝑛 ∈ N, and only the UAV computing can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Furthermore, when the offloaded data size exceeds the UAV computation capability, it is transferred to the end user via the UAV without computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3) “Intermediate Disconnected” scenario (𝑇 > 𝑇𝑣): The final scenario is when LEO connection is lost during the mission time, since the total mission time is larger than the LEO visible time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝑇 > 𝑇𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this scenario, when 𝑡 ≤ 𝑇𝑣, we have 𝛼𝑘,𝑛 = 1 for 𝑛 ∈ {1, · · ·, 𝑁𝑡}, with 𝑁𝑡 being the last frame within 𝑇𝑣, where both LEO computing and UAV computing can be performed: that is, 𝛽𝑘,𝑛 ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' When 𝑡 > 𝑇𝑣, 𝛼𝑘,𝑛 = 0 for 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁}, where only UAV computing is available: that is, 𝛽𝑘,𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For example, if the 5 TABLE II: Three different scenarios according to LEO availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Scenario 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 Available types of computing “Always On” (𝑇 ≤ 𝑇𝑣) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for all 𝑛 ∈ N 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for all 𝑛 ∈ N UAV Computing 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for all 𝑛 ∈ N LEO Computing “Always Off” (𝑇𝑣 = 0) 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for all 𝑛 ∈ N 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for all 𝑛 ∈ N UAV Computing “Intermediate Disconnected” (𝑇 > 𝑇𝑣) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑛 ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑛 ∈ {𝑁𝑡 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 } 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑛 ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑛 ∈ {𝑁𝑡 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 } UAV Computing 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑛 ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' for 𝑛 ∈ {𝑁𝑡 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 } LEO Computing → UAV Computing LEO connection is lost at 𝑇𝑣 = 𝑇/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 is defined as 𝑁/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The frame data of 𝑛 ∈ {1, · · ·, 𝑁𝑡} is computed by the LEO or UAV, while the frame data of 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁} is computed by the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The details for these three scenarios are summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Energy Consumption Model for Offloading In the proposed hierarchical architecture, IoT sensors and the UAV are battery-limited, while the available energy of the LEO satellite is much more sufficient due to its larger size and mass, which is therefore negligible for the system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' With the aim of minimizing the total energy consumption of the UAV, we cover the energy consumption model for compu- tation, communication and flying required for offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Here, the LEO satellite is assumed to have sufficient battery capacity compared to the UAV and IoT sensors [7], [13], which is not reflected in the system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1) Computation energy model: First, we define the amount of computation energy consumption at the LEO and UAV-mounted cloudlets at the 𝑛th frame for IoT sensor 𝑘 as [29], [30] 𝐸𝑑 𝑘,𝑛(𝑙𝑑 𝑘,𝑛) = 𝛾𝑑𝐶𝑑 𝑘 𝑙𝑑 𝑘,𝑛 Δ2 � 𝐾 ∑︁ 𝑘′=1 𝐶𝑑 𝑘′𝑙𝑑 𝑘′,𝑛 �2 , (6) where 𝑑 ∈ {𝐿,𝑈} with 𝐿 indicating the LEO satellite and 𝑈 indicating the UAV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑙𝑑 𝑘,𝑛 is the number of bits to be computed at the cloudlet and 𝛾𝑑 is the effective switched capacitance of the cloudlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2) Communication energy model: In the proposed system, the transmit energy consumption from the UAV to LEO at the 𝑛th frame for offloading the task of the IoT sensor 𝑘 is defined as [26], [31] 𝐸𝑈,𝐿 𝑘,𝑛 (𝐿𝑈,𝐿 𝑘,𝑛 , 𝒑𝑈 𝑛 ) = 𝑁0𝐵Δ/𝐾 ℎ𝑛( 𝒑𝑈𝑛 ) � 2 𝐿𝑈,𝐿 𝑘,𝑛 𝐵Δ/𝐾 − 1 � , (7) where 𝐿𝑈,𝐿 𝑘,𝑛 is the number of uplink bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' At the final destination of the UAV above the end user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' the downlink communication energy consumption is required so that the UAV can transmit the computing results accumulated during flying,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' which is given as 𝐸𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸 (𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝑈 𝑁 +1) = 𝑁0𝐵Δ/𝐾 𝑔𝐾+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑁 +1( 𝒑𝑈 𝑁 +1) � 2 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸 𝐵Δ/𝐾 − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (8) where 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸 is the number of downlink bits and is the same as the sum of output bits of the UAV and LEO-mounted cloudlets as follows: 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐸 = 𝑂𝑈 𝑘 𝑁 −2 ∑︁ 𝑛=1 𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1 + 𝑂𝐿 𝑘 𝑁 −4 ∑︁ 𝑛=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (9) In addition, the transmit energy consumption from the LEO and IoT sensor 𝑘 to the UAV at the 𝑛th frame is defined as 𝐸 𝐿,𝑈 𝑘,𝑛 (𝐿𝐿,𝑈 𝑘,𝑛 , 𝒑𝑈 𝑛 ) = 𝑁0𝐵Δ/𝐾 ℎ𝑛( 𝒑𝑈𝑛 ) � 2 𝐿𝐿,𝑈 𝑘,𝑛 𝐵Δ/𝐾 − 1 � (10) and 𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝐿𝐼 ,𝑈 𝑘,𝑛 , 𝒑𝑈 𝑛 ) = 𝑁0𝐵Δ/𝐾 𝑔𝑘,𝑛( 𝒑𝑈𝑛 ) � 2 𝐿𝐼,𝑈 𝑘,𝑛 𝐵Δ/𝐾 − 1 � , (11) where 𝐿𝐿,𝑈 𝑘,𝑛 is the number of downlink bits transmitted at the LEO and 𝐿𝐼 ,𝑈 𝑘,𝑛 is the number of uplink bits transmitted at the IoT sensor 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The energy consumption for reception is excluded since it is much smaller than the transmission energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3) Flying energy model: Following [32], [33], the flying energy consumption of the UAV at the 𝑛th frame is written as 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) = 𝜅∥𝒗𝑈 𝑛 ∥2, (12) where 𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5𝑀Δ and 𝑀 is the mass of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The flying energy consumption depends only on the velocity vector 𝒗𝑈 𝑛 of the UAV, and the level flight entails no change in the gravitational potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Our purpose is to minimize the total energy consumption of the UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' which must be calculated as the sum of the energy consumption of computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' communication and flying: 𝐸𝑡𝑜𝑡𝑎𝑙 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 = 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 � 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛𝐸𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝑈 𝑛 ) + (1 − 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛)𝐸𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛(𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛) � + (1 − 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛)(1 − 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛)𝐸𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛(𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛) + 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (13) where 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 and 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 are variables for the LEO availability and scheduling between LEO computing and UAV computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' which are given as 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 = � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' if LEO communication is available,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (14) 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 = � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' if LEO computing is performed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' if UAV computing is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (15) Note that the energy consumption 𝐸𝑈,𝐸 for downlink com- munication with the end user in (8) is excluded from (13) 6 since it is constant regardless of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In addition, LEO computing is considered by 𝛽𝑘,𝑛 = 1 when the input bits of the IoT sensor 𝑘 exceeds the computation capability of the UAV: that is, 𝑁 ∑︁ 𝑛=1 𝐿𝐼 ,𝑈 𝑘,𝑛 > 𝑁 ∑︁ 𝑛=1 � 𝑓 𝑈 𝑛 · Δ 𝐾 � 1 𝐶𝑈 𝑘 , (16) where 𝑓 𝑈 𝑛 [CPU cycles/s] is the CPU frequency at the UAV edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' OPTIMAL ENERGY CONSUMPTION FOR THE “ALWAYS ON” SCENARIO In this section, we formulate an optimization problem and the proposed algorithm to obtain a solution for the “Always On” scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Depending on the size of the offloaded data, either LEO computing or UAV computing is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' As mentioned above, the total UAV energy consumption 𝐸𝑡𝑜𝑡𝑎𝑙 𝑘,𝑛 in (13) is rewritten with 𝛼𝑘,𝑛 = 1, for all 𝑛 ∈ N, as 𝐸𝑡𝑜𝑡𝑎𝑙 𝑘,𝑛 = 𝛽𝑘,𝑛𝐸𝑈,𝐿 𝑘,𝑛 (𝐿𝑈,𝐿 𝑘,𝑛 , 𝒑𝑈 𝑛 ) + (1 − 𝛽𝑘,𝑛)𝐸𝑈 𝑘,𝑛(𝑙𝑈 𝑘,𝑛) + 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (17) When LEO computing is considered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝛽𝑘,𝑛 = 1, we need to jointly optimize the bit allocation of {𝐿𝐼 ,𝑈 𝑘,𝑛 }𝑛∈{1,···,𝑁 −4},𝑘 ∈K, {𝐿𝑈,𝐿 𝑘,𝑛 }𝑛∈{2,···,𝑁 −3},𝑘 ∈K, {𝑙𝐿 𝑘,𝑛}𝑛∈{3,···,𝑁 −2},𝑘 ∈K and {𝐿𝐿,𝑈 𝑘,𝑛 }𝑛∈{4,···,𝑁 −1},𝑘 ∈K along with the UAV trajectory { 𝒑𝑈 𝑛 }𝑛∈{2,···,𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' When UAV computing is performed, that is, 𝛽𝑘,𝑛 = 0, we must jointly optimize the bit allocation of {𝐿𝐼 ,𝑈 𝑘,𝑛 }𝑛∈{1,···,𝑁 −2},𝑘 ∈K and {𝑙𝑈 𝑘,𝑛}𝑛∈{2,···,𝑁 −1},𝑘 ∈K along with the UAV trajectory { 𝒑𝑈 𝑛 }𝑛∈{2,···,𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This problem is formulated with (17) as follows: min 𝐿𝐼,𝑈 𝑘,𝑛 ,𝐿𝑈,𝐿 𝑘,𝑛 ,𝐿𝐿,𝑈 𝑘,𝑛 𝑙𝑈 𝑘,𝑛,𝑙𝐿 𝑘,𝑛,𝒑𝑈 𝑛 𝐾 ∑︁ 𝑘=1 �𝑁 −4 ∑︁ 𝑛=1 𝛽𝑘,𝑛𝐸𝑈,𝐿 𝑘,𝑛+1(𝐿𝑈,𝐿 𝑘,𝑛+1, 𝒑𝑈 𝑛+1) + 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,𝑛)𝐸𝑈 𝑘,𝑛+1(𝑙𝑈 𝑘,𝑛+1) � + 𝑁 ∑︁ 𝑛=1 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) (18a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐸 𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝑈 𝑛 ) ≤ 𝜀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 ∈ N (18b) 𝑛 ∑︁ 𝑖=1 𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+1 ≤ 𝑛 ∑︁ 𝑖=1 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 − 2 (18c) 𝑛 ∑︁ 𝑖=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+1 ≤ 𝑛 ∑︁ 𝑖=1 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 − 4 (18d) 𝑛 ∑︁ 𝑖=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+2 ≤ 𝑛 ∑︁ 𝑖=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 − 4 (18e) 𝑛 ∑︁ 𝑖=1 𝐿𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+3 ≤ 𝑂𝐿 𝑘 𝑛 ∑︁ 𝑖=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 − 4 (18f) 𝑁 −4 ∑︁ 𝑛=1 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 + 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛)𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 = 𝐼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (18g) 𝑁 −4 ∑︁ 𝑛=1 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+2 + 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛)𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1 = 𝐼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (18h) 𝑁 −4 ∑︁ 𝑛=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+2 = 𝑁 −4 ∑︁ 𝑛=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (18i) 𝑁 −4 ∑︁ 𝑛=1 𝐿𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+3 = 𝑂𝐿 𝑘 𝑁 −4 ∑︁ 𝑛=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (18j) 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 ∈ N (18k) 𝒑𝑈 1 = 𝒑𝑈 𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝑈 𝑁 +1 = 𝒑𝑈 𝐹 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (18l) ��𝒗𝑈 𝑛 �� ≤ 𝑣max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑛 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (18m) where 𝜀 in (18b) represents the energy budget constraint per frame for the IoT sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The inequality constraint (18c) and (18e) ensures that the number of bits computed at the UAV and LEO-mounted cloudlet is less than or equal to the number of uplink bits transmitted from the IoT sensor and UAV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The inequality constraints (18d) and (18f) ensure that the number of uplink bits from the UAV is less than or equal to the number of uplink bits from the IoT sensor, and the number of downlink bits from the LEO is limited by the number of output bits from the LEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The equality constraints (18g) and (18h) enforce that the sum of the uplink bits of the IoT sensor and the sum of the computation bits for the LEO and UAV computing are equal to the input bits of the IoT sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The equality constraints (18i) and (18j) enforce the completion of LEO computing, while (18k) is imposed for the non-negative bit allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The constraints (18l) and (18m) represent the flying UAV’s initial and final position constraint and the maximum speed constraint, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Problem (18) is non-convex because the objective function and the energy budget constraint are non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' To address this non-convexity, we apply the SCA-based strategy [24], [25] which builds on the inner convex approximation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In particular, we develop proposed algorithm 1 by using the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Lemma 1: Given that a non-convex objective function 𝑈(𝒙) = 𝑓1(𝒙) 𝑓2(𝒙) is the product of 𝑓1 and 𝑓2 convex and non-negative for any 𝒚 in the domain of 𝑈(𝒙), a convex approximation that satisfies the conditions required by the SCA algorithm is given as ¯𝑈 (𝒙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒚) = 𝑓1(𝒙) 𝑓2(𝒚) + 𝑓1(𝒚) 𝑓2(𝒙) + 𝜏𝑖 2 (𝒙 − 𝒚)T𝑯(𝒚)(𝒙 − 𝒚), (19) where 𝜏𝑖 > 0 is a positive constant, 𝑯(𝒚) is a positive definite matrix, and (·)T indicates the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Lemma 2: Given a non-convex constraint 𝑔(𝒙1, 𝒙2) ≤ 0, where 𝑔(𝒙1, 𝒙2) = ℎ1(𝒙1)ℎ2(𝒙2) is the product of the ℎ1 and ℎ2 convex and non-negative, for any (𝒚1, 𝒚2) in the domain of 𝑔(𝒙1, 𝒙2), a convex approximation that satisfies the conditions required by the SCA algorithm is given as ¯𝑔 (𝒙1, 𝒙2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒚1, 𝒚2) Δ= 1 2 (ℎ1(𝒙1) + ℎ2(𝒙2))2 − 1 2 (ℎ12(𝒚1) + ℎ22(𝒚2)) − ℎ1(𝒚1)ℎ1 ′(𝒚1)(𝒙1 − 𝒚1) − ℎ2(𝒚2)ℎ2 ′(𝒚2)(𝒙2 − 𝒚2), (20) where the partial derivative of 𝑓 (·) is 𝑓 ′ (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 7 We set the primal variables for the formulated Problem (18) as 𝒛 = {𝒛𝑛}𝑛∈N with 𝒛𝑛 = ({𝐿𝐼 ,𝑈 𝑘,𝑛 }𝑘 ∈K, {𝐿𝑈,𝐿 𝑘,𝑛 }𝑘 ∈K, {𝐿𝐿,𝑈 𝑘,𝑛 }𝑘 ∈K, {𝑙𝑈 𝑘,𝑛}𝑘 ∈K, {𝑙𝐿 𝑘,𝑛}𝑘 ∈K, 𝒑𝑈 𝑛 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' We observe that the function 𝐸𝑈,𝐿 𝑘,𝑛 (𝒛𝑛) Δ= 𝐸𝑈,𝐿 𝑘,𝑛 (𝐿𝑈,𝐿 𝑘,𝑛 , 𝒑𝑈 𝑛 ) in (18a) is the product of two convex and non-negative functions, namely 𝑓1(𝐿𝑈,𝐿 𝑘,𝑛 ) = 𝑁0𝐵Δ/𝐾 𝑔0𝐺 � 2 𝐿𝑈,𝐿 𝑘,𝑛 𝐵Δ/𝐾 − 1 � (21) and 𝑓2( 𝒑𝑈 𝑛 ) = (𝑥𝐿 𝑛 − 𝑥𝑈 𝑛 )2 + (𝑦𝐿 𝑛 − 𝑦𝑈 𝑛 )2 + ℎ𝐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (22) Then, by using Lemma 1 and defining 𝒛𝑛(𝑣) = ({𝐿𝐼 ,𝑈 𝑘,𝑛 (𝑣)}𝑘 ∈K, {𝐿𝑈,𝐿 𝑘,𝑛 (𝑣)}𝑘 ∈K, {𝐿𝐿,𝑈 𝑘,𝑛 (𝑣)}𝑘 ∈K, {𝑙𝑈 𝑘,𝑛(𝑣)}𝑘 ∈K, {𝑙𝐿 𝑘,𝑛(𝑣)}𝑘 ∈K, 𝒑𝑈 𝑛 (𝑣))∈ X for the 𝑣th iterate within the feasible set X of (18), we obtain a strongly convex surrogate function ¯𝐸𝑈,𝐿 𝑘,𝑛 (𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) of 𝐸𝑈,𝐿 𝑘,𝑛 (𝒛𝑛) as ¯𝐸𝑈,𝐿 𝑘,𝑛 (𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) Δ= ¯𝐸𝑈,𝐿 𝑘,𝑛 (𝐿𝑈,𝐿 𝑘,𝑛 , 𝒑𝑈 𝑛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿𝑈,𝐿 𝑘,𝑛 (𝑣), 𝒑𝑈 𝑛 (𝑣)) = 𝑓1(𝐿𝑈,𝐿 𝑘,𝑛 ) 𝑓2( 𝒑𝑈 𝑛 (𝑣)) + 𝑓1(𝐿𝑈,𝐿 𝑘,𝑛 (𝑣)) 𝑓2( 𝒑𝑈 𝑛 ) + 𝜏𝐿𝑈,𝐿 𝑘,𝑛 2 (𝐿𝑈,𝐿 𝑘,𝑛 − 𝐿𝑈,𝐿 𝑘,𝑛 (𝑣))2 + 𝜏𝑥𝑈 𝑛 2 (𝑥𝑈 𝑛 − 𝑥𝑈 𝑛 (𝑣))2 + 𝜏𝑦𝑈 𝑛 2 (𝑦𝑈 𝑛 − 𝑦𝑈 𝑛 (𝑣))2, (23) where 𝜏𝐿𝑈,𝐿 𝑘,𝑛 , 𝜏𝑥𝑈 𝑛 , 𝜏𝑦𝑈 𝑛 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Also, the function 𝐸𝑈 𝑘,𝑛(𝒛𝑛) Δ= 𝐸𝑈 𝑘,𝑛(𝑙𝑈 𝑘,𝑛) in (18a) is the product of two convex and non- negative functions, namely 𝑓1(𝑙𝑈 𝑘,𝑛) = 𝛾𝑈𝐶𝑈 𝑘 𝑙𝑈 𝑘,𝑛 Δ2 (24) and 𝑓2(𝑙𝑈 𝑘′,𝑛) = � 𝐾 ∑︁ 𝑘′=1 𝐶𝑈 𝑘′𝑙𝑈 𝑘′,𝑛 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (25) As in (23), we obtain a strongly convex surrogate function ¯𝐸𝑈 𝑘,𝑛(𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) of 𝐸𝑈 𝑘,𝑛(𝒛𝑛) as ¯𝐸𝑈 𝑘,𝑛(𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) Δ= ¯𝐸𝑈 𝑘,𝑛(𝑙𝑈 𝑘,𝑛, 𝑙𝑈 𝑘′,𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑙𝑈 𝑘,𝑛(𝑣), 𝑙𝑈 𝑘′,𝑛(𝑣)) = 𝑓1(𝑙𝑈 𝑘,𝑛) 𝑓2(𝑙𝑈 𝑘′,𝑛(𝑣)) + 𝑓1(𝑙𝑈 𝑘,𝑛(𝑣)) 𝑓2(𝑙𝑈 𝑘′,𝑛) + 𝜏𝑙𝑈 𝑘,𝑛 2 (𝑙𝑈 𝑘,𝑛 − 𝑙𝑈 𝑘,𝑛(𝑣))2 + 𝜏𝑙𝑈 𝑘′,𝑛 2 (𝑙𝑈 𝑘′,𝑛 − 𝑙𝑈 𝑘′,𝑛(𝑣))2, (26) where 𝜏𝑙𝑈 𝑘,𝑛, 𝜏𝑙𝑈 𝑘′,𝑛 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For the non-convex energy budget constraint (18b), we derive a convex upper bound by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The function 𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝒛𝑛) Δ= 𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝐿𝐼 ,𝑈 𝑘,𝑛 , 𝒑𝑈 𝑛 ) is the product of two convex and non-negative functions, namely ℎ1(𝐿𝐼 ,𝑈 𝑘,𝑛 ) = 2 𝐿𝐼,𝑈 𝑘,𝑛 𝐵Δ/𝐾 − 1 (27) and ℎ2( 𝒑𝑈 𝑛 ) = (𝑥𝑈 𝑛 − 𝑥𝐼 𝑘)2 + (𝑦𝑈 𝑛 − 𝑦𝐼 𝑘)2 + ℎ𝑈 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (28) Algorithm 1 Proposed algorithm for the “Always On” scenario Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Set 𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Output: {𝐿𝐼 ,𝑈 𝑘,𝑛 }, {𝐿𝑈,𝐿 𝑘,𝑛 }, {𝐿𝐿,𝑈 𝑘,𝑛 }, {𝑙𝑈 𝑘,𝑛}, {𝑙𝐿 𝑘,𝑛}, { 𝒑𝑈 𝑛 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1: If 𝒛(𝑣) is a stationary solution of (18): STOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2: Compute ˆ𝒛 (𝒛(𝑣)) of (30) using dual decomposition or CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4: 𝑣 ← 𝑣 + 1 and go to step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Then, by using Lemma 2 and defining 𝒛𝑛(𝑣) for the 𝑣th iterate, we obtain a strongly convex surrogate function ¯𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) of 𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝒛𝑛) as ¯𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) Δ= 𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝐿𝐼 ,𝑈 𝑘,𝑛 , 𝒑𝑈 𝑛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝑣),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒑𝑈 𝑛 (𝑣)) = 𝑁0𝐵Δ/𝐾 2𝑔0 ������ � 2 𝐿𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 𝐵Δ/𝐾 − 1 + (𝑥𝑈 𝑛 − 𝑥𝐼 𝑘) 2 + (𝑦𝑈 𝑛 − 𝑦𝐼 𝑘) 2 + ℎ𝑈 2 �2 − � 2 𝐿𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝑣) 𝐵Δ/𝐾 − 1 �2 − � (𝑥𝑈 𝑛 (𝑣) − 𝑥𝐼 𝑘) 2 + (𝑦𝑈 𝑛 (𝑣) − 𝑦𝐼 𝑘) 2 + ℎ𝑈 2�2������ − 𝑁0 ln 2 𝑔0 2 𝐿𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝑣) 𝐵Δ/𝐾 � 2 𝐿𝐼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝑣) 𝐵Δ/𝐾 − 1 � � 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 − 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛 (𝑣) � − 2𝑁0𝐵Δ/𝐾 𝑔0 � (𝑥𝑈 𝑛 (𝑣) − 𝑥𝐼 𝑘) 2 + (𝑦𝑈 𝑛 (𝑣) − 𝑦𝐼 𝑘) 2 + ℎ𝑈 2� � (𝑥𝑈 𝑛 (𝑣) − 𝑥𝐼 𝑘)(𝑥𝑈 𝑛 − 𝑥𝑈 𝑛 (𝑣)) + (𝑦𝑈 𝑛 (𝑣) − 𝑦𝐼 𝑘)(𝑦𝑈 𝑛 − 𝑦𝑈 𝑛 (𝑣)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (29) Finally, the problem in Equation (18) can be transformed into the strongly convex inner approximation for a given feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as min 𝒛 𝐾 ∑︁ 𝑘=1 �𝑁 −4 ∑︁ 𝑛=1 𝛽𝑘,𝑛 ¯𝐸𝑈,𝐿 𝑘,𝑛+1(𝒛𝑛+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛+1(𝑣)) + 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,𝑛) ¯𝐸𝑈 𝑘,𝑛+1(𝒛𝑛+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛+1(𝑣)) � + 𝑁 ∑︁ 𝑛=1 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) (30a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ¯𝐸 𝐼 ,𝑈 𝑘,𝑛 (𝒛𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛(𝑣)) ≤ 𝜀, ∀𝑘 ∈ K, 𝑛 ∈ N (30b) (18c) − (18m), (30c) which has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Since Prob- lem (30) is convex, we can obtain the closed-form solutions via dual decomposition [34] or a standard convex optimization solver such as CVX [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The proposed algorithm based on the SCA method is summarized as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The sequence {𝒛(𝑣)} generated by Algorithm 1 converges if the step size 𝛾(𝑣) is chosen so that 𝛾(𝑣) ∈ (0, 1], 𝛾(𝑣) → 0, and � 𝑣 𝛾(𝑣) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Also, {𝒛(𝑣)} is bounded and every limit point of {𝒛(𝑣)} is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Furthermore, if Algorithm 1 does not stop after a finite number of steps, none of the stationary points are a local minimum of Problem (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 8 Algorithm 2 Proposed algorithm for the “Always Off” sce- nario Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Set 𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Output: {𝐿𝐼 ,𝑈 𝑘,𝑛 }, {𝑙𝑈 𝑘,𝑛}, { 𝒑𝑈 𝑛 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1: If 𝒛(𝑣) is a stationary solution of (32): STOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2: Compute ˆ𝒛 (𝒛(𝑣)) of (33) using dual decomposition or CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4: 𝑣 ← 𝑣 + 1 and go to step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' OPTIMAL ENERGY CONSUMPTION FOR THE “ALWAYS OFF” SCENARIO In this section, we find the optimal bit allocation and UAV path planning when the LEO communication is not available during the entire mission time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Therefore, the total UAV energy consumption 𝐸𝑡𝑜𝑡𝑎𝑙 𝑘,𝑛 in (13) is rewritten with 𝛼𝑘,𝑛 = 0 for all 𝑛 ∈ N, as 𝐸𝑡𝑜𝑡𝑎𝑙 𝑘,𝑛 = (1 − 𝛽𝑘,𝑛)𝐸𝑈 𝑘,𝑛(𝑙𝑈 𝑘,𝑛) + 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (31) For UAV computing with 𝛽𝑘,𝑛 = 0, the problem is given with (31) by min 𝐿𝐼,𝑈 𝑘,𝑛 ,𝑙𝑈 𝑘,𝑛,𝒑𝑈 𝑛 𝐾 ∑︁ 𝑘=1 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,𝑛)𝐸𝑈 𝑘,𝑛+1(𝑙𝑈 𝑘,𝑛+1) + 𝑁 ∑︁ 𝑛=1 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) (32a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,𝑛)𝐿𝐼 ,𝑈 𝑘,𝑛 = 𝐼𝑘, ∀𝑘 ∈ K (32b) 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,𝑛)𝑙𝑈 𝑘,𝑛+1 = 𝐼𝑘, ∀𝑘 ∈ K (32c) (18b), (18c), (18k) − (18m), (32d) where the equality constraints (32b) and (32c) guarantee that the total number of uplink bits from the IoT sensor and the total number of computation bits at the UAV must be equal to the input bits of the IoT sensor for complete offloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Always Off” case, the primal variables are defined as 𝒛 = {𝒛𝑛}𝑛∈N with 𝒛𝑛 = ({𝐿𝐼 ,𝑈 𝑘,𝑛 }𝑘 ∈K, {𝑙𝑈 𝑘,𝑛}𝑘 ∈K, 𝒑𝑈 𝑛 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Since Problem (32) is non-convex, it can be transformed into the strongly convex inner approximation, for a given a feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as min 𝒛 𝐾 ∑︁ 𝑘=1 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,𝑛) ¯𝐸𝑈 𝑘,𝑛+1(𝒛𝑛+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛+1(𝑣)) + 𝑁 ∑︁ 𝑛=1 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) (33a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (32b), (32c), (30b), (18c), (18k) − (18m), (33b) where ¯𝐸𝑈 𝑘,𝑛 of the objective function is defined equally in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Problem (33) has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)) due to its convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' As in Problem (30), the locally optimal solution can be obtained by dual decomposition or a standard convex optimization solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The proposed SCA-based algorithm is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Algorithm 3 Proposed algorithm for the “Intermediate Dis- connected” scenario Input: 𝛾(𝑣) ∈ (0, 1], 𝒛(0) = {𝒛𝑛(0)}𝑛∈N ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Set 𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Output: {𝐿𝐼 ,𝑈 𝑘,𝑛 }, {𝐿𝑈,𝐿 𝑘,𝑛 }, {𝐿𝐿,𝑈 𝑘,𝑛 }, {𝑙𝑈 𝑘,𝑛}, {𝑙𝐿 𝑘,𝑛}, { 𝒑𝑈 𝑛 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 1: If 𝒛(𝑣) is a stationary solution of (34): STOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 2: Compute ˆ𝒛 (𝒛(𝑣)) of (35) using dual decomposition or CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 3: Set 𝒛(𝑣 + 1) = 𝒛(𝑣) + 𝛾(𝑣) (ˆ𝒛 (𝒛(𝑣)) − 𝒛(𝑣)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4: 𝑣 ← 𝑣 + 1 and go to step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' OPTIMAL ENERGY CONSUMPTION FOR THE “INTERMEDIATE DISCONNECTED” SCENARIO For the “Intermediate Disconnected” case, we provide joint path planning and resource allocation when the LEO commu- nication is intermediately disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The total UAV energy consumption in this case follows (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' During the LEO computing for 𝑛 ∈ {1, · · ·, 𝑁𝑡} with 𝛼𝑘,𝑛 = 1 and 𝛽𝑘,𝑛 = 1, we jointly optimize the bit allocation {𝐿𝐼 ,𝑈 𝑘,𝑛 }𝑛∈{1,···,𝑁𝑡 },𝑘 ∈K, {𝐿𝑈,𝐿 𝑘,𝑛 }𝑛∈{2,···,𝑁𝑡+1},𝑘 ∈K, {𝑙𝐿 𝑘,𝑛}𝑛∈{3,···,𝑁𝑡+2},𝑘 ∈K and {𝐿𝐿,𝑈 𝑘,𝑛 }𝑛∈{4,···,𝑁𝑡+3},𝑘 ∈K along with the UAV trajectory { 𝒑𝑈 𝑛 }𝑛∈{2,···,𝑁𝑡+4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' During UAV com- puting for 𝑛 ∈ {1, · · ·, 𝑁𝑡} with 𝛼𝑘,𝑛 = 1 and 𝛽𝑘,𝑛 = 0 and 𝑛 ∈ {𝑁𝑡 + 1, · · ·, 𝑁} with 𝛼𝑘,𝑛 = 0 and 𝛽𝑘,𝑛 = 0, the bit allocation and the UAV path planning are jointly designed as in the UAV computing process of the “Always On” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Accordingly, we can formulate the problem as min 𝐿𝐼,𝑈 𝑘,𝑛 ,𝐿𝑈,𝐿 𝑘,𝑛 ,𝐿𝐿,𝑈 𝑘,𝑛 𝑙𝑈 𝑘,𝑛,𝑙𝐿 𝑘,𝑛,𝒑𝑈 𝑛 𝐾 ∑︁ 𝑘=1 𝑁𝑡 ∑︁ 𝑛=1 𝛼𝑘,𝑛 � 𝛽𝑘,𝑛𝐸𝑈,𝐿 𝑘,𝑛+1(𝐿𝑈,𝐿 𝑘,𝑛+1, 𝒑𝑈 𝑛+1) + �1 − 𝛽𝑘,𝑛 � 𝐸𝑈 𝑘,𝑛+1(𝑙𝑈 𝑘,𝑛+1) � + 𝐾 ∑︁ 𝑘=1 𝑁 −2 ∑︁ 𝑛=𝑁𝑡+1 �1 − 𝛼𝑘,𝑛 � (1 − 𝛽𝑘,𝑛)𝐸𝑈 𝑘,𝑛+1(𝑙𝑈 𝑘,𝑛+1) + 𝑁 ∑︁ 𝑛=1 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) (34a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 ∑︁ 𝑖=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+1 ≤ 𝑛 ∑︁ 𝑖=1 𝐿𝐼 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 (34b) 𝑛 ∑︁ 𝑖=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+2 ≤ 𝑛 ∑︁ 𝑖=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 (34c) 𝑛 ∑︁ 𝑖=1 𝐿𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+3 ≤ 𝑂𝐿 𝑘 𝑛 ∑︁ 𝑖=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑖+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑛 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' · · ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝑁𝑡 (34d) 𝑁𝑡 ∑︁ 𝑛=1 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+2 + 𝑁 −2 ∑︁ 𝑛=1 (1 − 𝛽𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛)𝑙𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1 = 𝐼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (34e) 𝑁𝑡 ∑︁ 𝑛=1 𝑙𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+2 = 𝑁𝑡 ∑︁ 𝑛=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (34f) 𝑁𝑡 ∑︁ 𝑛=1 𝐿𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑈 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+3 = 𝑂𝐿 𝑘 𝑁𝑡 ∑︁ 𝑛=1 𝐿𝑈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝐿 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='𝑛+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' ∀𝑘 ∈ K (34g) (18b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (18c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (32b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (18k) − (18m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (34h) 9 TABLE III: Simulation Parameters Parameter Value Parameter Value 𝑣𝑠 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5 km/s 𝑟𝐸 6371 km 𝜃 10 ◦ 𝑇𝑣 830 s ℎ𝑈 1 km ℎ𝐿 600 km 𝐾 10 𝑣max 50 m/s 𝑀 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='65 kg 𝑂𝐿 𝑘 , 𝑂𝑈 𝑘 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5 𝑓 𝑈 𝑛 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5 × 109 cycles/s [7] 𝐺 10 dB 𝛾𝐿, 𝛾𝑈 10−28 [29], [30] 𝐶𝐿 𝑘 , 𝐶𝑈 𝑘 1550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='7 [29], [30] 𝐵 40 MHz 𝑁0 174 dBm/Hz 𝜀 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='11 J ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' SNR 80 dB where the inequality constraints (34b)-(34d) and equality con- straints (34e)-(34g) limit the number of frames to 𝑛 = 1, ···, 𝑁𝑡 instead of 𝑛 = 1, ···, 𝑁 −4 in constraints (18d)-(18f) and (18h)- (18j), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the“Intermediate Disconnected” case, the primal vari- ables are defined the same as in the“Always On” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' By applying the SCA method,the non-convex Problem (34) can be transformed into the strongly convex inner approximation for a given a feasible 𝒛(𝑣) = {𝒛𝑛(𝑣)}𝑛∈N, as min 𝒛 𝐾 ∑︁ 𝑘=1 𝑁𝑡 ∑︁ 𝑛=1 𝛼𝑘,𝑛 � 𝛽𝑘,𝑛 ¯𝐸𝑈,𝐿 𝑘,𝑛+1(𝒛𝑛+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛+1(𝑣)) + �1 − 𝛽𝑘,𝑛 � ¯𝐸𝑈 𝑘,𝑛+1(𝒛𝑛+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛+1(𝑣)) � + 𝐾 ∑︁ 𝑘=1 𝑁 −2 ∑︁ 𝑛=𝑁𝑡+1 �1 − 𝛼𝑘,𝑛 � (1 − 𝛽𝑘,𝑛) ¯𝐸𝑈 𝑘,𝑛+1(𝒛𝑛+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 𝒛𝑛+1(𝑣)) + 𝑁 ∑︁ 𝑛=1 𝐸𝐹 𝑛 (𝒗𝑈 𝑛 ) (35a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (34b) − (34g), (30b), (18c), (32b), (18k) − (18m), (35b) which has a unique solution denoted by ˆ𝒛 (𝒛(𝑣)) to be obtained by dual decomposition or a standard convex optimization solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Algorithm 3 describes the proposed method for the “Intermediate Disconnected” scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' SIMULATION RESULTS In this section, we evaluate the performance of the proposed algorithms to jointly optimize the bit allocation and the UAV trajectory for marine IoT systems in various LEO accessible statuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For reference, we consider the following schemes: (i) No optimization: The equal bit allocation is considered for communication and computation per frame, while the UAV flies at constant velocity between the initial and final positions as 𝒑𝑈 𝑛 = 𝒑𝑈 𝐼 + (𝑛 − 1) � 𝒑𝑈 𝐹 − 𝒑𝑈 𝐼 ��𝑁, for 𝑛 ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (ii) Opti- mized bit allocation: The communication and computation bits are optimized by the proposed algorithms while considering the UAV trajectory with the constant-velocity as in (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' (iii) Optimized UAV trajectory: The path planning of the UAV is obtained by the proposed algorithms with fixed equal bit allocation per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The simulation parameters are provided in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Particularly, the space segment considers Iridium- like LEO satellite networks that provide global coverage with 66 satellites distributed in 6 polar orbits [15], where the orbit 0 1 2 3 4 5 6 7 8 9 10 x [km] 0 1 2 3 4 5 6 7 8 9 10 y [km] IoT2 IoT5 IoT3 IoT4 IoT7 IoT8 IoT9 IoT10 IoT6 IoT1 LEO UAV trajectory ("Always On") UAV trajectory ("Always Off") UAV trajectory ("Intermediate Disconnected") Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4: Optimal UAV trajectories according to the different LEO access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' height is ℎ = 601 km with the elevation angle 𝜃 = 10 ◦, and satellites in the orbit travel at a speed of around 𝑣𝑠 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' To better understand the proposed algorithms, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4 and 5 consider the partial optimization of UAV path planning or bit allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4, there are 𝐾 = 10 IoT sensors distributed randomly in a 10 km × 10 km area within the beam coverage of the central LEO satellite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝛼𝑘,𝑛 = 1, for all 𝑛 ∈ N and 𝑘 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' With the LEO visible time of 𝑇𝑣 = 830 s obtained from (4) and a bandwidth of 𝐵 = 40 MHz [15], the data size collected from each IoT sensor is randomly determined based on the computation capability of the UAV in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In our simulation, the scheduling variable 𝛽𝑘,𝑛 is defined from (16), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝛽𝑘,𝑛 = [0 0 0 1 1 1 0 1 0 0], for 𝑘 ∈ K and 𝑛 ∈ N, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The IoT sensors with 𝛽𝑘,𝑛 = 0 for UAV computing and with 𝛽𝑘,𝑛 = 1 for LEO computing are indicated by black-colored circles and green- colored circles, respectively, while the LEO satellite, indicated by a red-colored hexagram, travels along the red dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The initial and final positions of the UAV are 𝒑𝑈 𝐼 = (5, 0, 0) to 𝒑𝑈 𝐹 = (10, 5, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4 shows the optimized UAV trajectories with the fixed equal bit allocation according to the different LEO satellite access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For this experiment, the latency constraint is 𝑇 = 360 s with 𝑁 = 60 and Δ = 6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Always On” case, the optimized UAV trajectory, represented by a blue asterisk line, is designed to fly close to the IoT sensors with LEO computing until its final destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This can significantly reduce the large amount of uplink communication energy consumption induced by the long distance between the LEO satellite and UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Always Off” case, where only UAV computing is considered, the UAV flies along a straight path to a destination, which is represented by a yellow crossed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this case, the flying energy consumption must be reduced to to minimize the total UAV energy due to the fixed computation bit allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=" In the “Intermediate Disconnected” case, where the LEO communication is lost at 𝑁𝑡 = 𝑁/2, the 10 10 20 30 40 50 60 Frame number 0 1 2 IoT6's number of bits 107 LI,U 6,n lU 6,n LU,L 6,n lL 6,n LL,U 6,n (a) “Always On” scenario 10 20 30 40 50 60 Frame number 0 5 10 IoT6's number of bits 106 (b) “Always Off” scenario 10 20 30 40 50 60 Frame number 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content="5 2 IoT6's number of bits 107 (c) “Intermediate Disconnected” scenario Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5: Optimal bit allocations for IoT sensor 6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4 according to the different LEO access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' optimized UAV trajectory, represented by a purple square line, tends to fly close to the IoT sensors with LEO computing for 𝑛 = 1, · · ·, 𝑁𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Then, in the frame period of 𝑛 = 𝑁𝑡 + 1, · · ·, 𝑁 where LEO communication is disconnected, the UAV flies straight to the final destination because it performs only UAV computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5 illustrates the optimized bit allocations for IoT sensor 6 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4 with the fixed constant-velocity UAV trajectory according to different LEO access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Except for the UAV trajectory, the simulation environment is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(a), the optimal bit allocations 𝐿𝐼 ,𝑈 𝑘,𝑛 , 𝐿𝑈,𝐿 𝑘,𝑛 , 𝑙𝐿 𝑘,𝑛, 𝐿𝐿,𝑈 𝑘,𝑛 by proposed Algorithm 1 are shown for LEO computing in the “Always On” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' First, most of the uplink bits 𝐿𝐼 ,𝑈 𝑘,𝑛 are allocated between frames 20 to 35, which corresponds to the period where the UAV flies closest to IoT sensor 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The offloading bits 𝐿𝑈,𝐿 𝑘,𝑛 are allocated equally in the entire frame because the equal bit allocation can achieve the minimal communication energy from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Finally, the LEO 1 2 3 4 5 6 7 8 9 10 x [km] 0 1 2 3 4 5 6 7 8 9 10 y [km] IoT3 IoT4 IoT10 IoT6 IoT2 IoT8 IoT1 IoT7 IoT9 IoT5 LEO 2nd orbit 1st orbit 3rd orbit UAV trajectory (1st orbit) UAV trajectory (2nd orbit) UAV trajectory (3rd orbit) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 6: Optimal UAV trajectories according to different LEO satellite orbits, where the IoT sensors with LEO computing are deployed at the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' computing bits 𝑙𝐿 𝑘,𝑛 and LEO downlink bits 𝐿𝐿,𝑈 𝑘,𝑛 are mostly allocated in the latter parts between frames 50 to 60 to satisfy the inequality constraints of (18e) and (18f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(b), the optimized bit allocations 𝐿𝐼 ,𝑈 𝑘,𝑛 and 𝑙𝑈 𝑘,𝑛 obtained by proposed Algorithm 2 are shown for UAV computing of the“Always Off” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Since the UAV cannot communicate with the LEO satellite, the computing process is entirely at the UAV-mounted cloudlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The uplink bits 𝐿𝐼 ,𝑈 𝑘,𝑛 and the computing bits 𝑙𝑈 𝑘,𝑛 are assigned the same as 𝐿𝐼 ,𝑈 𝑘,𝑛 and 𝐿𝑈,𝐿 𝑘,𝑛 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' However, 𝑙𝑈 𝑘,𝑛 is dramatically reduced to 8 × 106 per frame compared to 10 × 106, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This is because the amount of data exceeding the UAV computation capability is excluded from the UAV computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(c) shows the optimization result of bit allocation attained by proposed Algorithm 3 in the “Intermediate Disconnected” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' LEO computing is performed during the first half of frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝑛 = 1, ···, 𝑁𝑡, while UAV computing is performed during the second half of frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝑛 = 𝑁𝑡 + 1, · · ·, 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The computing bits 𝑙𝐿 𝑘,𝑛 at LEO and the downlink bits 𝐿𝐿,𝑈 𝑘,𝑛 are reduced in proportion to the reduced frame duration of LEO computing compared to those shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For UAV computing, there are more computing bits 𝑙𝑈 𝑘,𝑛 allocated at the UAV than those from the case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This means that less data exceeds the computational capability of the UAV thanks to the LEO computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 6 shows the optimal UAV trajectories according to the different LEO satellite orbits in the “Always On” scenario, where the IoT sensors that need LEO computing are clustered at the corner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=', 𝛽𝑘,𝑛 = [0 1 1 0 1 1 0 0 0 0], for 𝑘 ∈ K and 𝑛 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this deployment, the three different movements of the LEO satellite in different orbital directions are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the first orbit moving from the upper right corner to the lower left corner, the UAV flies near the corner area with IoT sensors with LEO computing to its final destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the 11 400 600 800 1000 1200 1400 1600 Total time T (s) 0 1 2 3 4 5 6 7 Total UAV energy consumption (J) 106 No opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' - "Always On" No opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' - "Always Off" No opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' - "Intermediate Disconnected" Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' bit allocation - "Always On" Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' UAV trajectory - "Always On" Joint opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' - "Always On" Joint opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' - "Always Off" Joint opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' - "Intermediate Disconnected" Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 7: Comparison of the total UAV energy consumption for different optimization schemes in the three LEO satellite access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' second orbit moving from the upper left corner to the lower right corner, the UAV flies in a diagonally downward direction along its own orbit rather than the optimized UAV trajectory for the first orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the third orbit moving upwards from below the midpoint, the UAV flies in an upward direction along its own orbit rather than the optimal UAV trajectory for the first orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' From these results, we can see that the LEO movements resulting from the orbit influences the optimal UAV path so as to reduce the communication energy consumption between the UAV and the LEO satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 7 compares the total UAV energy consumption of the joint optimization scheme with reference schemes in three LEO satellite access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For this experiment, the latency constraint is 𝑇 = [360:90:1620] s with 𝑁 = [60:15:270] and Δ = 6 s, while the remaining simulation parameters are the same as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' First, the no optimization scheme consumes the highest energy in the three scenarios, among which the largest energy consumption takes place in the “Always Off” case, where only the UAV computing is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This is natural since the UAV-mounted cloudlet has a slightly larger burden in terms of the energy consumption with no support of the LEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Always On” case, for 𝑇 = 360 s, the total UAV energy consumption for the joint optimization scheme is the lowest at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='6 × 106 J, whereas the optimized UAV trajectory scheme with fixed equal bit allocation requires 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5×106 J, and the optimized bit allocation with the constant-velocity UAV and no optimization schemes requires 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='1 × 106 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This implies that the UAV path planning is more effective in terms of UAV energy consumption than bit allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Moreover, the total energy consumption in all schemes decreases as the total time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This is because the same amount of data is processed over a longer period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Compared to the total UAV energy consumption of the joint optimization scheme in the “Always Off” scenario, those of the joint optimization scheme in other scenarios 0 1/8 2/8 4/8 6/8 7/8 1 LEO satellite access time rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='9 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='4 Total UAV energy consumption (J) 106 55 60 65 70 75 80 85 90 95 100 Collected data usage rate (%) Total energy - "Always On" Total energy - "Always Off" Total energy - "Intermediate Disconnected" Data usage rate - "Always On" Data usage rate - "Always Off" Data usage rate - "Intermediate Disconnected" Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 8: Relationship between the total UAV energy consump- tion and the collected data usage rate in three LEO satellite access scenarios according to the LEO satellite access time rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' are much higher since the UAV flies straight to its final destination when the LEO satellite connection is lost, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' However, there is a trade-off between the total UAV energy consumption and the collected data usage rate for computing, which determines the amount of data executed at cloudlet, which is analyzed in the following figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 8 shows the relationship between the total UAV energy consumption and the collected data usage rate for computing in the different LEO accessibility scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Any amount of data exceeding the UAV computation capability is excluded from UAV computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For this experiment, the scheduling variables are defined as 𝛽𝑘,𝑛 = [0 0 1 1 1 1 0 1 0 0], for 𝑘 ∈ K and 𝑛 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' The UAV computation capability is applied to 226 Mbits by using the CPU frequency at the UAV server 𝑓 𝑈 𝑛 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='75 × 109 cycles/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Always On” scenario, the LEO satellite access time rate is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' At this time, the total UAV energy consumption is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='3×106 J and the collected data usage rate is 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Always Off” case, where the LEO satellite access time rate is 0, the total UAV energy consump- tion is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content='24 × 106 J and the collected data usage rate is 54%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Although the energy consumption in the “Always Off” case is dramatically reduced, the utilization rate of the collected data is also cut in half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In the “Intermediate Disconnected” case, as the LEO satellite access time rate increases, the total UAV energy consumption and the collected data usage rate increase differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' When the LEO satellite access time rate is above 6/8, the total UAV energy consumption is saturated with the total UAV energy consumption of the “Always On” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' This is because the straight flight segment of the UAV to the final destination after disconnecting with the LEO satellite matches that of the “Always On” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' Also, when the LEO satellite access time rate is more than 7/8, the collected data usage rate is more than about 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In this simulation environment, adequate data usage and energy consumption is achieved with more than a 7/8 LEO satellite access time rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' 12 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' CONCLUSIONS In this paper, a marine IoT system using hybrid LEO and UAV computing for real-time utilization of marine data has been analyzed according to the different LEO satellite access scenarios: “Always On,” “Always Off” and “Inter- mediate Disconnected”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' For each scenario, we proposed the joint optimization problem of bit allocation for computing and communication in offloading and UAV path planning to minimize the total UAV energy consumption under latency, energy budget, and UAV operational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' To solve the optimization problem, we developed an SCA-based algorithm whose performance in terms of energy efficiency was validated via numerical results compared to conventional approaches with partial optimization that design only the bit allocation or UAV trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' According to LEO satellite access time and its orbit direction, the path planning of the UAV is optimized differently for energy saving, whose impact is pronounced for the case when the LEO connectivity is unstable or discon- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE2T4oBgHgl3EQfUwcR/content/2301.03815v1.pdf'} +page_content=' In future works, different 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Reyes1 and Giovanni Maria Tomaselli2 +1Institute for Theoretical Physics and +2GRAPPA, +University of Amsterdam, Amsterdam, 1098 XH, The Netherlands +Very compact stars seem to be forbidden in General Relativity. While Buchdahl’s theorem sets +an upper bound on compactness, further no-go results rely on the existence of two light rings, the +inner of which has been associated to gravitational instabilities. However, little is known about +the role of quantum fields in these strong gravity regimes. Working in the probe approximation +where the backreaction is ignored, we show that the trapping of modes around the inner light ring +leads the renormalized stress tensor of Conformal Field Theories to diverge faster than the classical +source in the Buchdahl limit. This leads to the violation of the Null Energy Condition, as well as +the isotropy assumption used in Buchdahl’s theorem. The backreaction of quantum fields in this +regime therefore cannot be ignored. This happens as the star’s surface approaches the Buchdahl +radius 9GM/4 rather than the Schwarzschild radius, with the quantum fields having support in +a small region around the center, becoming negligible at the surface. These are generic quantum +features and do not depend on the details of the interactions. Our findings open a way for further +investigation into the role of QFT in astrophysics. +COMPACT RELATIVISTIC STARS +The General Relativistic prediction of the existence of +compact objects, such as white dwarfs and neutron stars, +has been confirmed by many observations. Their macro- +scopic properties follow from the Tolman-Oppenheimer- +Volkoff equation. However, quantum theory is essential +in understanding the physics of these stars, as it provides +the ultimate reason for their existence, namely, Fermi’s +exclusion principle. +The question regarding the maximum mass of such +compact object is crucial: it is the main criterion used to +discriminate between what we suspect is a neutron star or +a black hole. Well-known upper limits were set by Chan- +drasekhar [1] and Rhoades-Ruffini [2]. A more generic re- +sult, that is independent of the equation of state of the +matter, was established by Buchdahl [3] and gives an up- +per bound on compactness in GR. Consider an isotropic +perfect fluid star, with stress tensor +T µ +ν = diag(−ρ, p, p, p) , +(1) +on a static, spherically symmetric metric +ds2 = −f(r) dt2 + h(r) dr2 + r2(dθ2 + sin2 θ dφ2) . (2) +Assuming in addition that ρ > 0, ∂rρ ≤ 0 and that +Einstein’s equations hold, the requirement that the met- +ric is everywhere regular leads to +R ≥ 9GM/4 , +(3) +where R is the radius of the star and M is its mass. The +saturation of the bound is known as the Buchdahl limit. +Notice one can also formulate this bound in a coordinate- +independent way. +A particularly simple solution that manifestly sat- +urates Buchdahl’s second assumption is the constant- +density star or ‘Schwarzschild interior metric’. +These +configurations have uniform density ρ += +3M/4πR3 +throughout the star, and as is well known they can sat- +urate the Buchdahl limit (3). Although they are unre- +alistic models of an astrophysical object, they are the +standard example when studying the TOV equations. +The metric for this equation of state takes the form +(2), with +f(r) = +� +3 +2 +� +1 − 2GM +R +− 1 +2 +� +1 − 2GMr2 +R3 +�2 +, +(4) +h(r) = +� +1 − 2GMr2 +R3 +�−1 +, +(5) +and is matched to the usual exterior Schwarzschild vac- +uum solution at the sphere’s surface. +The simplicity of these solutions make them an excel- +lent setup to test Quantum Field Theory (QFT) in the +strong gravity regime. +A final motivation to consider +this metric is that it is conformally (Weyl) flat. In fact, +the uniform density metric above is the unique solution +to Einstein’s equations coupled to a static perfect fluid +that is conformally flat [4, 5]. This will allow us to ob- +tain explicit analytic results. +We will thus work with +this spacetime, and comment about the generality of our +results later on. +WAVE EQUATION +In order to understand the behaviour of quantum fields +in this spacetime, let us begin by first considering the +propagation of classical waves in it. The wave equation +for the uniform density background was first discussed +by Chandrasekhar and Ferrari [6]. For simplicity we take +a massless scalar Φ using the usual decomposition +Φ = +� +fωℓm , +fωℓm(x) = u(r) +r +Yℓm(θ, φ)e−iωt . (6) +arXiv:2301.00826v1 [gr-qc] 2 Jan 2023 + +2 +−30 +−20 +−10 +0 +10 +0 +0.05 +0.1 +0.15 +r∗/(GM) +V/(GM)2 +FIG. 1. Potential (for ℓ = 1) given in (9), for R/(GM) = +9/4 (blue), 2.3 (orange) and 2.4 (green). The dashed lines +mark the the values of r∗ corresponding to the centre of the +star in the two latter cases. +A discontinuous jump at the +star’s surface matches it to the exterior vacuum Schwarzschild +solution. +The wave equation □Φ += +0 can be recast in a +Schr¨odinger-like form: +−∂2 +r∗u + V (r∗)u = ω2u , +(7) +where we defined the tortoise coordinate r∗ via +dr∗ +dr = +� +h(r)/f(r) . +(8) +The potential V (r∗) takes the form +V = 1 +r ∂2 +r∗r + ℓ(ℓ + 1) +r2 +f +(9) +and is plotted in Fig. 1 for ℓ = 1 and various values of +R/(GM). The potential at r > R corresponds to the +Schwarzschild vacuum metric, and vanishes at infinity. It +connects to the interior of the star with a discontinuous +step. +As we can see from Fig. 1, when R > 9GM/4, the +tortoise coordinate has a finite minimum possible value +(dashed lines) corresponding to the centre of the star, +because the factor h/f is always regular around r = 0. +Moreover, V (r∗) reaches a local minimum greater than +zero and then increases towards the surface. +When R → 9GM/4, however, one has h/f ∼ r−2 +and therefore the domain of r∗ becomes infinite on both +sides, while V (r∗) vanishes at the centre of the star. This +closely resembles the situation for black holes, but in that +case it is the horizon that is mapped to r∗ → −∞. The +field modes can thus be trapped inside the star, leading +to a spectrum of quasi-bound states whose magnitudes +are amplified close to the origin. +These properties of the effective potential, together +with the behavior of the tortoise coordinate, suggests +that upon quantization the renormalized stress tensor +can become important in the Buchdahl limit. The rest +of our analysis will be done in a more generic way that +depends less on the specific theory considered. +QFT IN CURVED SPACETIME +QFT in curved spacetime has seen significant progress +in the last half century. In the semi-classical approxima- +tion, gravity is still treated classically and one considers +some quantum fields as another dynamical source to Ein- +stein’s equations, +Rµν − 1 +2gµνR = 8πG +� +Tµν + ⟨ ˆTµν⟩ +� +. +(10) +We shall denote by ˆTµν the operator of the QFT to dis- +tinguish it from the classical source (1). +However, most work in this field has focused on either +cosmology or black holes. Here, we will study the role +it plays for astrophysical compact stars. The question +we will address in this work is whether there exists some +generic feature of QFT, independent of the details of +the nuclear interactions and the quantum states involved, +that becomes important for very compact stars, in the +regime of strong gravitational fields. We will show that +there is indeed such an effect. +We shall focus on the effects of conformally coupled +fields where the computation is easier, taking it as a toy +model for more generic scenarios. We work in 3+1 dimen- +sions, but the generalization to even higher dimensions is +straightforward. The non-conformal case will be treated +elsewhere. +As is well known, conformally coupled classical mat- +ter has a vanishing trace of its stress tensor. However, +its quantum counterpart develops a trace anomaly. In +3 + 1 dimensions, the vacuum expectation value of the +trace of the renormalized stress tensor for quantum fields +propagating in a curved spacetime is +⟨ ˆT µ +µ ⟩ = +1 +(4π)2 [cF − aG − d□R] , +(11) +where R is the Ricci scalar, F is the square of the Weyl +tensor and G is the Gauss-Bonnet invariant. Amongst the +three real coefficients, c > 0 and a > 0 are well under- +stood and characterize the particular theory in question. +On the other hand, d is not determined by the bare La- +grangian as it depends on the renormalization scheme, +and is closely related to the quadratic corrections to the +gravity action as we review below. As such, it should +be fixed by experiments. For now we will leave d as a +fixed but undetermined constant and proceed with the +calculation. +If additionally the metric is conformally flat – as is the +case for the constant-density star – then all components + +3 +of the renormalized stress tensor are fixed [7]: +⟨ ˆT µν⟩ = +− +a +(4π)2 +� +gµν +�R2 +2 − RαβRαβ +� ++ 2RµλRν +λ − 4 +3RRµν +� ++ +d +(4π)2 +� 1 +12gµν(R2 − 4R,λ +;λ) − 1 +3(RRµν − R,µ;ν) +� +. +(12) +The quantum state chosen for (12) is the vacuum, but +this will not play an important role. It could be a state +at finite temperature or with a large number of fermions: +this would only add an extra contribution independent of +the curvature. The vacuum stress tensor for the interior +of the uniform density star is therefore given by (12). We +now proceed to evaluate it and examine its properties. +QUANTUM FIELDS IN THE BUCHDAHL LIMIT +In this section, we describe the main features of +the +quantum +stress +tensor +(12) +evaluated +on +the +Schwarzschild interior metric. In particular, we wish to +understand its behavior as we approach the Buchdahl +limit +R = (9/4 + ϵ)GM , +ϵ → 0 . +(13) +We will report the results to leading orders in ϵ. +The Buchdahl limit (13) is a finite distance above the +black hole compactness corresponding to R = 2GM. +Nevertheless, this regime is no less extreme: the Ricci +scalar R of the background metric at the center diverges +in this limit as +R(0) = − 3 +R2ϵ + O(1) . +(14) +Correspondingly, the central density and pressure of the +classical uniform density star solution behave as +ρ(0) = +1 +3πGR2 + O(ϵ) , +(15) +p(0) = +1 +8πGR2ϵ + O(1) . +(16) +Let us contrast this behavior with its quantum coun- +terpart (12). For generic ϵ, this takes the form +⟨ ˆT µ +ν ⟩ = diag(−⟨ˆρ⟩, ⟨ˆpr⟩, ⟨ˆpθ⟩, ⟨ˆpθ⟩) , +(17) +with ⟨ˆpr⟩ ̸= ⟨ˆpθ⟩. The radial dependence of the compo- +nents are illustrated in Fig. 2. In the limit ϵ → 0, their +central values scale as +⟨ˆρ(0)⟩ = +9d +(8πR2ϵ)2 + +d +6(πR2)2ϵ + O(1) , +(18) +⟨ˆpr(0)⟩ = ⟨ˆpθ(0)⟩ = − +3d +(8πR2ϵ)2 + +2a − d +(3πR2)2ϵ + O(1) . (19) +0 +0.5 +1 +1.5 +2 +0 +0.1 +0.2 +r/(GM) +⟨ˆρ⟩ +⟨ˆpr⟩ +⟨ˆpθ⟩ +FIG. 2. +Radial profile of the three components of ⟨ ˆT µ +ν ⟩, +for a = d = 1/360, ϵ = 0.003, in units where GM = 1. The +location of the inner light ring is depicted by the dashed line. +We emphasize that the pressures match only at the cen- +ter, and not elsewhere. Moreover, notice that the leading +order of ⟨ˆρ⟩ and ⟨ˆp⟩ have opposite signs. There is no con- +tribution from c because the Weyl tensor vanishes. +By comparing (15) and (16) with (18) and (19), we see +that the components of the renormalized stress tensor +scale with higher powers of ϵ than the classical contribu- +tions, and therefore cannot be ignored in the Buchdahl +limit. Furthermore, notice that the leading divergence of +the quantum terms depends only on d: in this regime the +quantum effects are dominated by the scheme-dependent +terms proportional to d, and not by c or a. +The backreaction of quantum effects cannot be ne- +glected if they become of the same order as the clas- +sical ones. +By comparing the classical and quantum +central pressures (16) and (19), this crossover happens +at ϵ ∼ |d| (ℓP /R)2, which corresponds to a pressure +p ∼ |d|−1ℓ−4 +P +and central curvature R ∼ −|d|−1ℓ−2 +P , +where ℓP is the Planck length. If d ≪ 1, this corresponds +to sub-Planckain lengths and therefore we cannot trust +our semi-classical analysis. Instead, if d ≫ 1, the QFT ef- +fects cannot be neglected in this regime. For this specific +equation of state, a different, earlier crossover is found +if the energy densities are compared instead. However, +the impact of the constant energy density on the metric +is negligible compared to that of the diverging central +pressure, in the Buchdahl limit. +We will not address the problem of full backreaction in +this work. Nevertheless, some general features of a lin- +earized approximation provide useful insight. Consider +the trace of the semi-classical equations (10), +−R = 8πG (−ρ + 3p − ⟨ˆρ⟩ + ⟨ˆpr⟩ + 2⟨ˆpθ⟩) , +(20) +evaluated at the origin, as we approach the Buchdahl +limit. +In the absence of the quantum corrections, the +right-hand side of (20) diverges as ϵ−1 as shown in (14). +However, as we see from (18) and (19), the quantum +contributions of the last three terms scale as −dϵ−2. + +4 +If d < 0, the quantum terms on the right side of (20) +grow without bound with the same sign as the classi- +cal ones. This suggests a runaway: as the curvature in- +creases, so do quantum effects, which increase the curva- +ture further and so on. Conversely, if d > 0, the quantum +contributions to the trace have the opposite sign, which +decreases the curvature. This suggests the possible exis- +tence of a backreacted solution, but only for d > 0. Such +an equilibrium would require a small but finite ϵ of the +order discussed above, so the surface of such an object +would lie very close the Buchdahl radius, and far from +the Schwarzschild radius. +ROLE OF THE LIGHT RING +Light rings (photon spheres) play a key role in our anal- +ysis. These are defined as regions where null geodesics +form circles, and they always come in pairs due to topo- +logical arguments [8]. +For 9GM/4 < R ≤ 3GM, the +above metric develops two light rings located at: +rext = 3GM , +rint = 1 +3 +� +R3 +GM +4R − 9GM +R − 2GM . +(21) +The outer ring rext, also present for black holes, corre- +sponds to the usual photon sphere outside the surface of +the star and is unstable: photons crossing it either es- +cape to infinity or spiral inwards. It has been probed by +recent observations [9–11]. +The inner ring rint lies in the interior and is a stable +attractor of null geodesics, meaning that massless fields +remain trapped around it. Notice that it shrinks to the +origin in the Buchdahl limit. As illustrated in Fig. 2, the +quantum stress tensor (12) is maximum at the center and +falls steeply around the inner light ring. Indeed, in the +Buchdahl limit the inner light ring sets the location at +which the field values have dropped roughly by one order +of magnitude, i.e. +⟨ˆρ(rint)⟩ +⟨ˆρ(0)⟩ +∼ 0.1 +(22) +and similarly for the pressures. This shows that the re- +gion inside the inner photon sphere is where the quantum +fields have most support, which is the quantum analogue +to the classical trapping of modes discussed above us- +ing the wave equation. The crossover when the classical +and quantum pressures become comparable corresponds +to an inner light ring of radius r ∼ +� +|d| ℓP . +The inner photon sphere plays yet another important +role: it is the location where the Null Energy Condition +(NEC) is violated. Given a null vector kµ, one defines an +operator by contracting the (total) stress tensor with it +NEC = +� +Tµν + ⟨ ˆTµν⟩ +� +kµkν , +(23) +where we have included both the classical and quantum +contributions. For classical matter, one expects NEC ≥ +0, while it is well known that quantum fields can violate +this. +In the star’s interior, but far from the inner light ring, +the NEC will be positive, since quantum effects there are +negligible. +In order to investigate the behavior of the +NEC in the vicinity of the inner photon sphere as we +approach the Buchdahl bound, we choose the null vector +as kµ = (1, kr, 0, 0). We then compute (23) inside the +star, in the limit ϵ → 0, keeping fixed the ratio r/rint. +This yields +NEC(r) = +2d +27π2G4R4 +r2 +int − r2 +r2 +int + r2 + O(ϵ) . +(24) +This is effectively ‘tracking’ the NEC in the region +around the inner photon sphere as the configuration ap- +proaches the Buchdahl bound, since rint → 0 in this limit. +The NEC clearly changes sign at the light ring and is +thus violated. Notice that the classical contribution is +subdominant in this limit and is contained in the sub- +leading orders. On the other hand, choosing kµ along +the (t, φ) plane does not lead to a violation. +The analysis above posits an interesting question. Sta- +ble light rings have been recently associated with gravita- +tional instabilities due to the existence of slowly decaying +modes around it [12, 13], which would rule out ultra com- +pact objects [8]. However, we have shown here that it is +precisely this feature that enhances the quantum effects +there, leading to the violation of energy conditions and +to significant backreaction. Exploring this interaction at +the non-linear level is an interesting direction. +COMMENTS ON BUCHDAHL’S THEOREM. +Buchdahl’s theorem relies on several assumptions as +stated in the introduction. Our results show that QFT +in curved spacetime violates two of these assumptions, +namely isotropy of the matter and the effective equations +of motion. +As we have seen in (12) and is illustrated in Fig. 2, the +renormalized vacuum stress tensor of the quantum fields +is not isotropic, thus violating one of the assumptions of +Buchdahl’s theorem. Anisotropic versions of Buchdahl’s +bound exist but they require extra assumptions [14–18]. +These typically take the form of energy conditions, with +the strength of the bound depending on the strength +of the conditions. Here, we have shown that quantum +fields violate energy conditions in the probe approxima- +tion. We leave it for future work to examine whether the +equations including backreaction violate the assumptions +leading to these generalized theorems. +Second, close to the compactness bound the relevant +equations of motion to solve are (10), rather than the + +5 +classical Einstein equations. +These differ by the pres- +ence of the quantum source which, as we have shown, +becomes the dominant term in the Buchdahl limit. This +contribution depends explicitly on the curvature tensors, +and therefore the differential equations to solve are of a +different nature than the purely classical ones. +This last feature has an alternative description in terms +of quadratic gravity. +For our specific background, we +have shown that among the terms that determine ⟨ ˆTµν⟩ +in (11) and (12), only those controlled by d diverge faster +than the classical Tµν as ϵ → 0. The ones associated with +a diverge with the same power as the classical terms, but +come with a coefficient that is very small for astrophysi- +cal objects. Now as anticipated, d is a scheme-dependent +parameter that can be generated by adding the countert- +erm − +d +12(4π)2 R2 to the Lagrangian. This means that our +results can also be interpreted as coming from quadratic +corrections to Einstein’s gravity. +The Weyl-flatness of +the background, then, is not essential to find the leading +terms of ⟨ ˆTµν⟩. +This two-faced interpretation is akin to Starobinsky’s +inflation [19], initially formulated in terms of the back- +reaction of quantum fields, then as R2 gravity (in the +Jordan frame) or Einstein gravity coupled to a scalar +field (in the Einstein frame). In the latter picture, the +stability of the scalar field requires the condition d > 0, +the same we found and discussed earlier. +It is worth noticing that Buchdahl’s theorem holds in +a local form as +r +Gm(r) ≥ 9 +4, where the radius and mass +of the star are replaced by an arbitrary coordinate ra- +dius r and the Misner-Sharp mass m(r) = 4π +� r +0 dr r2ρ +contained within it, provided the assumptions are met +inside that sphere. For example, the star could consist of +an incompressible dense core surrounded by an external +crust obeying a softer equation of state. Our results also +apply to this generalized scenario. +Interesting recent work has also considered quantum +fields in the Buchdahl limit [20–22] in the approxima- +tion of a two-dimensional reduction. This corresponds +to the s-wave (ℓ = 0) sector, and leaves the stress tensor +undetermined up to an arbitrary function. Our results +differ from theirs in that (12) fully captures the 3 + 1- +dimensional features, leaving no functional freedom. For +other applications of similar techniques see [23–25]. +SUMMARY +We have investigated the universal behavior of QFT +in the interior of very compact stars. A useful arena to +probe this is the strong gravity regime close to Buch- +dahl’s limit that, classically, sets an upper bound on the +compactness of static, spherically symmetric spheres in +General Relativity. As a proxy for this, we have worked +with the constant-density Schwarzschild interior solution. +Motivated by the trapping of classical waves in this +metric close to Buchdahl’s limit, we have studied quan- +tum fields propagating on this background in the approx- +imation of no backreaction. +Exploiting the conformal +flatness of this solution, we have evaluated the full renor- +malized stress tensor (12) for Conformal Field Theories. +This depends on two coefficients a and d, the latter of +which is not fixed by the theory in question. +The vacuum renormalized stress tensor (17) is not +isotropic, since the radial and angular pressures are dif- +ferent. The sign of the energy density is opposite to that +of the pressures. Its components acquire their maximum +magnitude at the origin, and fall steeply around the inner +light ring, as shown in (22). +As we approach the Buchdahl limit, the d term of +the renormalized stress tensor (18)-(19) diverges faster +than the classical source (15)-(16), meaning that quan- +tum fields respond stronger to changes in compactness +than their classical counterpart. +The crossover when +classical and quantum contributions are of the same or- +der happens when the proper radius of the inner light +ring is rint ∼ +� +|d|ℓP . The radial Null Energy Condition +– including both classical and quantum contributions – +changes sign at the inner photon sphere as shown in (24), +and is thus violated inside the star. Whether the scales +involved are Planckian or not depends on the value of d. +If d ≪ 1, we cannot trust our semi-classical analysis. On +the other hand, if d ≫ 1, the effects of the QFT cannot +be ignored in this regime. +We emphasize that the enhancement of quantum ef- +fects discussed here happens as the surface of the star +approaches the Buchdahl radius 9GM/4 instead of 2GM. +Moreover, the effect of the quantum fields is localized in +a small region around the center – the inner light ring +– and not the surface. This is different from ultra com- +pact objects close to the Schwarzschild radius. There, +the renormalized stress tensor in the Boulware vacuum +is well known to diverge at the surface as the star ap- +proaches the black hole limit [26]. +The isotropy assumption used in Buchdahl’s theorem +is violated by vacuum quantum fields. Whether the con- +ditions leading to the anisotropic generalizations of this +bound hold or not requires further investigation. +We have not attempted to solve the semi-classical +equations (10) here. Nevertheless, our results suggests +that if d > 0, quantum fields act by decreasing the cur- +vature, suggesting that a self-consistent solution to these +equations might exist that avoids curvature singularities. +It is intriguing to wonder whether quantum physics +may play yet another, unexpected, role in the determi- +nation of the maximum mass of compact stars. +Acknowledgments. +We thank Max Ba˜nados, Pablo +Bosch, Alejandra Castro, Jan de Boer and Erik Verlinde +for insightful discussions. +We also thank Daniel Bau- +mann and Vitor Cardoso for feedback on the manuscript. +We are particularly grateful to Ben Freivogel for exten- +sive discussions. + +6 +[1] S. Chandrasekhar, The maximum mass of ideal white +dwarfs, Astrophys. J. 74, 81 (1931). +[2] C. E. Rhoades, Jr. and R. Ruffini, Maximum mass of a +neutron star, Phys. Rev. Lett. 32, 324 (1974). +[3] H. A. Buchdahl, General Relativistic Fluid Spheres, +Phys. Rev. 116, 1027 (1959). +[4] H. A. 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Rep. 12, 15958 +(2022), arXiv:2110.15808 [gr-qc]. +[22] J. Arrechea, C. Barcel´o, R. Carballo-Rubio, and L. J. +Garay, Semiclassical constant-density spheres in a reg- +ularized Polyakov approximation, Phys. Rev. D 104, +084071 (2021), arXiv:2105.11261 [gr-qc]. +[23] E. Mottola and R. Vaulin, Macroscopic Effects of the +Quantum Trace Anomaly, Phys. Rev. D 74, 064004 +(2006), arXiv:gr-qc/0604051. +[24] H. Kawai and Y. Yokokura, A Model of Black Hole Evap- +oration and 4D Weyl Anomaly, Universe 3, 51 (2017), +arXiv:1701.03455 [hep-th]. +[25] P. Beltr´an-Palau, A. del R´ıo, and J. Navarro-Salas, Quan- +tum corrections to the Schwarzschild metric from vacuum +polarization, (2022), arXiv:2212.08089 [gr-qc]. +[26] C. Barcelo, S. Liberati, S. Sonego, and M. Visser, Fate of +gravitational collapse in semiclassical gravity, Phys. Rev. +D 77, 044032 (2008), arXiv:0712.1130 [gr-qc]. + diff --git a/MNAyT4oBgHgl3EQf6vpX/content/tmp_files/load_file.txt b/MNAyT4oBgHgl3EQf6vpX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0ffc6de34e493b58306ff13589b3cbd2430e32de --- /dev/null +++ b/MNAyT4oBgHgl3EQf6vpX/content/tmp_files/load_file.txt @@ -0,0 +1,382 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf,len=381 +page_content='Compact stars in Quantum Field Theory Ignacio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Reyes1 and Giovanni Maria Tomaselli2 1Institute for Theoretical Physics and 2GRAPPA, University of Amsterdam, Amsterdam, 1098 XH, The Netherlands Very compact stars seem to be forbidden in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' While Buchdahl’s theorem sets an upper bound on compactness, further no-go results rely on the existence of two light rings, the inner of which has been associated to gravitational instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, little is known about the role of quantum fields in these strong gravity regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Working in the probe approximation where the backreaction is ignored, we show that the trapping of modes around the inner light ring leads the renormalized stress tensor of Conformal Field Theories to diverge faster than the classical source in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This leads to the violation of the Null Energy Condition, as well as the isotropy assumption used in Buchdahl’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The backreaction of quantum fields in this regime therefore cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This happens as the star’s surface approaches the Buchdahl radius 9GM/4 rather than the Schwarzschild radius, with the quantum fields having support in a small region around the center, becoming negligible at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' These are generic quantum features and do not depend on the details of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Our findings open a way for further investigation into the role of QFT in astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' COMPACT RELATIVISTIC STARS The General Relativistic prediction of the existence of compact objects, such as white dwarfs and neutron stars, has been confirmed by many observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Their macro- scopic properties follow from the Tolman-Oppenheimer- Volkoff equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, quantum theory is essential in understanding the physics of these stars, as it provides the ultimate reason for their existence, namely, Fermi’s exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The question regarding the maximum mass of such compact object is crucial: it is the main criterion used to discriminate between what we suspect is a neutron star or a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Well-known upper limits were set by Chan- drasekhar [1] and Rhoades-Ruffini [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' A more generic re- sult, that is independent of the equation of state of the matter, was established by Buchdahl [3] and gives an up- per bound on compactness in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Consider an isotropic perfect fluid star, with stress tensor T µ ν = diag(−ρ, p, p, p) , (1) on a static, spherically symmetric metric ds2 = −f(r) dt2 + h(r) dr2 + r2(dθ2 + sin2 θ dφ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (2) Assuming in addition that ρ > 0, ∂rρ ≤ 0 and that Einstein’s equations hold, the requirement that the met- ric is everywhere regular leads to R ≥ 9GM/4 , (3) where R is the radius of the star and M is its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The saturation of the bound is known as the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Notice one can also formulate this bound in a coordinate- independent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' A particularly simple solution that manifestly sat- urates Buchdahl’s second assumption is the constant- density star or ‘Schwarzschild interior metric’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' These configurations have uniform density ρ = 3M/4πR3 throughout the star, and as is well known they can sat- urate the Buchdahl limit (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Although they are unre- alistic models of an astrophysical object, they are the standard example when studying the TOV equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The metric for this equation of state takes the form (2), with f(r) = � 3 2 � 1 − 2GM R − 1 2 � 1 − 2GMr2 R3 �2 , (4) h(r) = � 1 − 2GMr2 R3 �−1 , (5) and is matched to the usual exterior Schwarzschild vac- uum solution at the sphere’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The simplicity of these solutions make them an excel- lent setup to test Quantum Field Theory (QFT) in the strong gravity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' A final motivation to consider this metric is that it is conformally (Weyl) flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In fact, the uniform density metric above is the unique solution to Einstein’s equations coupled to a static perfect fluid that is conformally flat [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This will allow us to ob- tain explicit analytic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We will thus work with this spacetime, and comment about the generality of our results later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' WAVE EQUATION In order to understand the behaviour of quantum fields in this spacetime, let us begin by first considering the propagation of classical waves in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The wave equation for the uniform density background was first discussed by Chandrasekhar and Ferrari [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For simplicity we take a massless scalar Φ using the usual decomposition Φ = � fωℓm , fωℓm(x) = u(r) r Yℓm(θ, φ)e−iωt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (6) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='00826v1 [gr-qc] 2 Jan 2023 2 −30 −20 −10 0 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='15 r∗/(GM) V/(GM)2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Potential (for ℓ = 1) given in (9), for R/(GM) = 9/4 (blue), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='3 (orange) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='4 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The dashed lines mark the the values of r∗ corresponding to the centre of the star in the two latter cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' A discontinuous jump at the star’s surface matches it to the exterior vacuum Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The wave equation □Φ = 0 can be recast in a Schr¨odinger-like form: −∂2 r∗u + V (r∗)u = ω2u , (7) where we defined the tortoise coordinate r∗ via dr∗ dr = � h(r)/f(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (8) The potential V (r∗) takes the form V = 1 r ∂2 r∗r + ℓ(ℓ + 1) r2 f (9) and is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 1 for ℓ = 1 and various values of R/(GM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The potential at r > R corresponds to the Schwarzschild vacuum metric, and vanishes at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' It connects to the interior of the star with a discontinuous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 1, when R > 9GM/4, the tortoise coordinate has a finite minimum possible value (dashed lines) corresponding to the centre of the star, because the factor h/f is always regular around r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Moreover, V (r∗) reaches a local minimum greater than zero and then increases towards the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' When R → 9GM/4, however, one has h/f ∼ r−2 and therefore the domain of r∗ becomes infinite on both sides, while V (r∗) vanishes at the centre of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This closely resembles the situation for black holes, but in that case it is the horizon that is mapped to r∗ → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The field modes can thus be trapped inside the star, leading to a spectrum of quasi-bound states whose magnitudes are amplified close to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' These properties of the effective potential, together with the behavior of the tortoise coordinate, suggests that upon quantization the renormalized stress tensor can become important in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The rest of our analysis will be done in a more generic way that depends less on the specific theory considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' QFT IN CURVED SPACETIME QFT in curved spacetime has seen significant progress in the last half century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In the semi-classical approxima- tion, gravity is still treated classically and one considers some quantum fields as another dynamical source to Ein- stein’s equations, Rµν − 1 2gµνR = 8πG � Tµν + ⟨ ˆTµν⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (10) We shall denote by ˆTµν the operator of the QFT to dis- tinguish it from the classical source (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, most work in this field has focused on either cosmology or black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Here, we will study the role it plays for astrophysical compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The question we will address in this work is whether there exists some generic feature of QFT, independent of the details of the nuclear interactions and the quantum states involved, that becomes important for very compact stars, in the regime of strong gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We will show that there is indeed such an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We shall focus on the effects of conformally coupled fields where the computation is easier, taking it as a toy model for more generic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We work in 3+1 dimen- sions, but the generalization to even higher dimensions is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The non-conformal case will be treated elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As is well known, conformally coupled classical mat- ter has a vanishing trace of its stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, its quantum counterpart develops a trace anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In 3 + 1 dimensions, the vacuum expectation value of the trace of the renormalized stress tensor for quantum fields propagating in a curved spacetime is ⟨ ˆT µ µ ⟩ = 1 (4π)2 [cF − aG − d□R] , (11) where R is the Ricci scalar, F is the square of the Weyl tensor and G is the Gauss-Bonnet invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Amongst the three real coefficients, c > 0 and a > 0 are well under- stood and characterize the particular theory in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' On the other hand, d is not determined by the bare La- grangian as it depends on the renormalization scheme, and is closely related to the quadratic corrections to the gravity action as we review below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As such, it should be fixed by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For now we will leave d as a fixed but undetermined constant and proceed with the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' If additionally the metric is conformally flat – as is the case for the constant-density star – then all components 3 of the renormalized stress tensor are fixed [7]: ⟨ ˆT µν⟩ = − a (4π)2 � gµν �R2 2 − RαβRαβ � + 2RµλRν λ − 4 3RRµν � + d (4π)2 � 1 12gµν(R2 − 4R,λ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='λ) − 1 3(RRµν − R,µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (12) The quantum state chosen for (12) is the vacuum, but this will not play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' It could be a state at finite temperature or with a large number of fermions: this would only add an extra contribution independent of the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The vacuum stress tensor for the interior of the uniform density star is therefore given by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We now proceed to evaluate it and examine its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' QUANTUM FIELDS IN THE BUCHDAHL LIMIT In this section, we describe the main features of the quantum stress tensor (12) evaluated on the Schwarzschild interior metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In particular, we wish to understand its behavior as we approach the Buchdahl limit R = (9/4 + ϵ)GM , ϵ → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (13) We will report the results to leading orders in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The Buchdahl limit (13) is a finite distance above the black hole compactness corresponding to R = 2GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Nevertheless, this regime is no less extreme: the Ricci scalar R of the background metric at the center diverges in this limit as R(0) = − 3 R2ϵ + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (14) Correspondingly, the central density and pressure of the classical uniform density star solution behave as ρ(0) = 1 3πGR2 + O(ϵ) , (15) p(0) = 1 8πGR2ϵ + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (16) Let us contrast this behavior with its quantum coun- terpart (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For generic ϵ, this takes the form ⟨ ˆT µ ν ⟩ = diag(−⟨ˆρ⟩, ⟨ˆpr⟩, ⟨ˆpθ⟩, ⟨ˆpθ⟩) , (17) with ⟨ˆpr⟩ ̸= ⟨ˆpθ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The radial dependence of the compo- nents are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In the limit ϵ → 0, their central values scale as ⟨ˆρ(0)⟩ = 9d (8πR2ϵ)2 + d 6(πR2)2ϵ + O(1) , (18) ⟨ˆpr(0)⟩ = ⟨ˆpθ(0)⟩ = − 3d (8πR2ϵ)2 + 2a − d (3πR2)2ϵ + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (19) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='2 r/(GM) ⟨ˆρ⟩ ⟨ˆpr⟩ ⟨ˆpθ⟩ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Radial profile of the three components of ⟨ ˆT µ ν ⟩, for a = d = 1/360, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='003, in units where GM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The location of the inner light ring is depicted by the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We emphasize that the pressures match only at the cen- ter, and not elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Moreover, notice that the leading order of ⟨ˆρ⟩ and ⟨ˆp⟩ have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' There is no con- tribution from c because the Weyl tensor vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' By comparing (15) and (16) with (18) and (19), we see that the components of the renormalized stress tensor scale with higher powers of ϵ than the classical contribu- tions, and therefore cannot be ignored in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Furthermore, notice that the leading divergence of the quantum terms depends only on d: in this regime the quantum effects are dominated by the scheme-dependent terms proportional to d, and not by c or a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The backreaction of quantum effects cannot be ne- glected if they become of the same order as the clas- sical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' By comparing the classical and quantum central pressures (16) and (19), this crossover happens at ϵ ∼ |d| (ℓP /R)2, which corresponds to a pressure p ∼ |d|−1ℓ−4 P and central curvature R ∼ −|d|−1ℓ−2 P , where ℓP is the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' If d ≪ 1, this corresponds to sub-Planckain lengths and therefore we cannot trust our semi-classical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Instead, if d ≫ 1, the QFT ef- fects cannot be neglected in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For this specific equation of state, a different, earlier crossover is found if the energy densities are compared instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, the impact of the constant energy density on the metric is negligible compared to that of the diverging central pressure, in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We will not address the problem of full backreaction in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Nevertheless, some general features of a lin- earized approximation provide useful insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Consider the trace of the semi-classical equations (10), −R = 8πG (−ρ + 3p − ⟨ˆρ⟩ + ⟨ˆpr⟩ + 2⟨ˆpθ⟩) , (20) evaluated at the origin, as we approach the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In the absence of the quantum corrections, the right-hand side of (20) diverges as ϵ−1 as shown in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, as we see from (18) and (19), the quantum contributions of the last three terms scale as −dϵ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 4 If d < 0, the quantum terms on the right side of (20) grow without bound with the same sign as the classi- cal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This suggests a runaway: as the curvature in- creases, so do quantum effects, which increase the curva- ture further and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Conversely, if d > 0, the quantum contributions to the trace have the opposite sign, which decreases the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This suggests the possible exis- tence of a backreacted solution, but only for d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Such an equilibrium would require a small but finite ϵ of the order discussed above, so the surface of such an object would lie very close the Buchdahl radius, and far from the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' ROLE OF THE LIGHT RING Light rings (photon spheres) play a key role in our anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' These are defined as regions where null geodesics form circles, and they always come in pairs due to topo- logical arguments [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For 9GM/4 < R ≤ 3GM, the above metric develops two light rings located at: rext = 3GM , rint = 1 3 � R3 GM 4R − 9GM R − 2GM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (21) The outer ring rext, also present for black holes, corre- sponds to the usual photon sphere outside the surface of the star and is unstable: photons crossing it either es- cape to infinity or spiral inwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' It has been probed by recent observations [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The inner ring rint lies in the interior and is a stable attractor of null geodesics, meaning that massless fields remain trapped around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Notice that it shrinks to the origin in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 2, the quantum stress tensor (12) is maximum at the center and falls steeply around the inner light ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Indeed, in the Buchdahl limit the inner light ring sets the location at which the field values have dropped roughly by one order of magnitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' ⟨ˆρ(rint)⟩ ⟨ˆρ(0)⟩ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content='1 (22) and similarly for the pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This shows that the re- gion inside the inner photon sphere is where the quantum fields have most support, which is the quantum analogue to the classical trapping of modes discussed above us- ing the wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The crossover when the classical and quantum pressures become comparable corresponds to an inner light ring of radius r ∼ � |d| ℓP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The inner photon sphere plays yet another important role: it is the location where the Null Energy Condition (NEC) is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Given a null vector kµ, one defines an operator by contracting the (total) stress tensor with it NEC = � Tµν + ⟨ ˆTµν⟩ � kµkν , (23) where we have included both the classical and quantum contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For classical matter, one expects NEC ≥ 0, while it is well known that quantum fields can violate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In the star’s interior, but far from the inner light ring, the NEC will be positive, since quantum effects there are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In order to investigate the behavior of the NEC in the vicinity of the inner photon sphere as we approach the Buchdahl bound, we choose the null vector as kµ = (1, kr, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We then compute (23) inside the star, in the limit ϵ → 0, keeping fixed the ratio r/rint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This yields NEC(r) = 2d 27π2G4R4 r2 int − r2 r2 int + r2 + O(ϵ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' (24) This is effectively ‘tracking’ the NEC in the region around the inner photon sphere as the configuration ap- proaches the Buchdahl bound, since rint → 0 in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The NEC clearly changes sign at the light ring and is thus violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Notice that the classical contribution is subdominant in this limit and is contained in the sub- leading orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' On the other hand, choosing kµ along the (t, φ) plane does not lead to a violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The analysis above posits an interesting question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Sta- ble light rings have been recently associated with gravita- tional instabilities due to the existence of slowly decaying modes around it [12, 13], which would rule out ultra com- pact objects [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' However, we have shown here that it is precisely this feature that enhances the quantum effects there, leading to the violation of energy conditions and to significant backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Exploring this interaction at the non-linear level is an interesting direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' COMMENTS ON BUCHDAHL’S THEOREM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Buchdahl’s theorem relies on several assumptions as stated in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Our results show that QFT in curved spacetime violates two of these assumptions, namely isotropy of the matter and the effective equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As we have seen in (12) and is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 2, the renormalized vacuum stress tensor of the quantum fields is not isotropic, thus violating one of the assumptions of Buchdahl’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Anisotropic versions of Buchdahl’s bound exist but they require extra assumptions [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' These typically take the form of energy conditions, with the strength of the bound depending on the strength of the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Here, we have shown that quantum fields violate energy conditions in the probe approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We leave it for future work to examine whether the equations including backreaction violate the assumptions leading to these generalized theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Second, close to the compactness bound the relevant equations of motion to solve are (10), rather than the 5 classical Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' These differ by the pres- ence of the quantum source which, as we have shown, becomes the dominant term in the Buchdahl limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This contribution depends explicitly on the curvature tensors, and therefore the differential equations to solve are of a different nature than the purely classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This last feature has an alternative description in terms of quadratic gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For our specific background, we have shown that among the terms that determine ⟨ ˆTµν⟩ in (11) and (12), only those controlled by d diverge faster than the classical Tµν as ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The ones associated with a diverge with the same power as the classical terms, but come with a coefficient that is very small for astrophysi- cal objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Now as anticipated, d is a scheme-dependent parameter that can be generated by adding the countert- erm − d 12(4π)2 R2 to the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This means that our results can also be interpreted as coming from quadratic corrections to Einstein’s gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The Weyl-flatness of the background, then, is not essential to find the leading terms of ⟨ ˆTµν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This two-faced interpretation is akin to Starobinsky’s inflation [19], initially formulated in terms of the back- reaction of quantum fields, then as R2 gravity (in the Jordan frame) or Einstein gravity coupled to a scalar field (in the Einstein frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' In the latter picture, the stability of the scalar field requires the condition d > 0, the same we found and discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' It is worth noticing that Buchdahl’s theorem holds in a local form as r Gm(r) ≥ 9 4, where the radius and mass of the star are replaced by an arbitrary coordinate ra- dius r and the Misner-Sharp mass m(r) = 4π � r 0 dr r2ρ contained within it, provided the assumptions are met inside that sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For example, the star could consist of an incompressible dense core surrounded by an external crust obeying a softer equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Our results also apply to this generalized scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Interesting recent work has also considered quantum fields in the Buchdahl limit [20–22] in the approxima- tion of a two-dimensional reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This corresponds to the s-wave (ℓ = 0) sector, and leaves the stress tensor undetermined up to an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Our results differ from theirs in that (12) fully captures the 3 + 1- dimensional features, leaving no functional freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' For other applications of similar techniques see [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' SUMMARY We have investigated the universal behavior of QFT in the interior of very compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' A useful arena to probe this is the strong gravity regime close to Buch- dahl’s limit that, classically, sets an upper bound on the compactness of static, spherically symmetric spheres in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As a proxy for this, we have worked with the constant-density Schwarzschild interior solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Motivated by the trapping of classical waves in this metric close to Buchdahl’s limit, we have studied quan- tum fields propagating on this background in the approx- imation of no backreaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Exploiting the conformal flatness of this solution, we have evaluated the full renor- malized stress tensor (12) for Conformal Field Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This depends on two coefficients a and d, the latter of which is not fixed by the theory in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The vacuum renormalized stress tensor (17) is not isotropic, since the radial and angular pressures are dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The sign of the energy density is opposite to that of the pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Its components acquire their maximum magnitude at the origin, and fall steeply around the inner light ring, as shown in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' As we approach the Buchdahl limit, the d term of the renormalized stress tensor (18)-(19) diverges faster than the classical source (15)-(16), meaning that quan- tum fields respond stronger to changes in compactness than their classical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The crossover when classical and quantum contributions are of the same or- der happens when the proper radius of the inner light ring is rint ∼ � |d|ℓP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The radial Null Energy Condition – including both classical and quantum contributions – changes sign at the inner photon sphere as shown in (24), and is thus violated inside the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Whether the scales involved are Planckian or not depends on the value of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' If d ≪ 1, we cannot trust our semi-classical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' On the other hand, if d ≫ 1, the effects of the QFT cannot be ignored in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We emphasize that the enhancement of quantum ef- fects discussed here happens as the surface of the star approaches the Buchdahl radius 9GM/4 instead of 2GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Moreover, the effect of the quantum fields is localized in a small region around the center – the inner light ring – and not the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' This is different from ultra com- pact objects close to the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' There, the renormalized stress tensor in the Boulware vacuum is well known to diverge at the surface as the star ap- proaches the black hole limit [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' The isotropy assumption used in Buchdahl’s theorem is violated by vacuum quantum fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Whether the con- ditions leading to the anisotropic generalizations of this bound hold or not requires further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We have not attempted to solve the semi-classical equations (10) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Nevertheless, our results suggests that if d > 0, quantum fields act by decreasing the cur- vature, suggesting that a self-consistent solution to these equations might exist that avoids curvature singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' It is intriguing to wonder whether quantum physics may play yet another, unexpected, role in the determi- nation of the maximum mass of compact stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We thank Max Ba˜nados, Pablo Bosch, Alejandra Castro, Jan de Boer and Erik Verlinde for insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We also thank Daniel Bau- mann and Vitor Cardoso for feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' We are particularly grateful to Ben Freivogel for exten- sive discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 6 [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' Chandrasekhar, The maximum mass of ideal white dwarfs, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQf6vpX/content/2301.00826v1.pdf'} +page_content=' 74, 81 (1931).' metadata={'source': 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--- /dev/null +++ b/N9AzT4oBgHgl3EQfk_2S/content/tmp_files/2301.01541v1.pdf.txt @@ -0,0 +1,573 @@ +Condensed Matter Physics, 2022, Vol. 25, No. 4, 43710: 1–9 +DOI: 10.5488/CMP.25.43710 +http://www.icmp.lviv.ua/journal +Electric field induced polarization rotation in squaric +acid crystals revisited +A. P. Moina +Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, +1 Svientsitskii St., 79011 Lviv, Ukraine +Received July 10, 2022, in final form July 26, 2022 +Using the previously developed model we revisit the problem of the electric field induced polarization rotation +in antiferroelectric crystals of squaric acid. We test an alternative set of the model parameters, according to +which the dipole moments associated with the H2C4O4 groups are assumed to be parallel to the diagonals of +the 𝑎𝑐 plane. The 𝑇-𝐸 phase diagrams and the polarization curves 𝑃(𝐸) for the fields directed along the 𝑎 axis +and along one of the diagonals are considered. Comparison of the theoretical results with the newly published +experimental data confirm the validity of the model. The calculations reveal no apparent advantage of the new +set of the parameters over the previously used set. +Key words: polarization, electric field, phase transition, phase diagram, squaric acid +1. Introduction +The squaric acid H2C4O4 is a classical two-dimensional antiferroelectric. The crystal is tetrago- +nal, 𝐼4/𝑚, in the paraelectric phase and monoclinic, 𝑃21/𝑚, in the antiferroelectric phase. The hydrogen +bonded C4O4 groups form sheets, parallel to the 𝑎𝑐 plane and stacked along the 𝑏-axis. Below the tran- +sition at 373 K, a spontaneous polarization arises in these sheets, with the neighboring sheets polarized +in the opposite directions [1–3]. +External electric fields applied to a uniaxial antiferroelectric can switch a sublattice polarization by +180◦ and induce thereby the transition from antiferroelectric (AFE) to ferroelectric (FE) phase. The +(pseudo)tetragonal symmetry of the squaric acid crystal lattice and of its hydrogen bond networks allows +the sublattice polarizations to be directed along two perpendicular axes in the fully ordered system. +As a result, here the external field can rotate one of the sublattice polarizations by 90◦, whereupon +a noncollinear ferrielectric phase with perpendicular sublattice polarizations (NC90 [4]) is induced. +The possibility of such a rotation has been suggested by Horiuchi et al [5], and their hysteresis loop +measurements and Berry phase calculations gave evidence for it. Further calculations [6] indicated that +the 90◦ rotation is possible at different orientations of the field within the 𝑎𝑐 plane. It is also predicted [5, 6] +that application of higher fields along the diagonals of the 𝑎𝑐 plane can lead to the second rotation of the +negative sublattice polarization by 90◦ and induction of the collinear ferroelectric phase. +Recently [4, 7] we developed a deformable [8] two-sublattice proton-ordering model for a description +of squaric acid behaviour in external electric fields, applied arbitrarily within the plane of hydrogen +bonds. The model calculations confirm the two-step process of polarization reorientation [5, 6] at low +temperatures, with the negative sublattice polarization being switched twice by 90◦ at each transition, +for any orientation of the field within the 𝑎𝑐 plane, but a few exceptional directions. The exceptional +directions are those, when the field is either i) collinear to the axes of the sublattice polarization in the AFE +phase, or ii) directed at 45◦ to these axes. In the case i), the crystal behaves like a uniaxial antiferroelectric, +undergoing a single-step polarization switching to the FE phase without the intermediate noncollinear +phase, while in the case ii), the transition field from the NC90 to the FE phase goes to infinity, i.e., the +transition never occurs [7]. +This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution +of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. +43710-1 +arXiv:2301.01541v1 [cond-mat.mtrl-sci] 4 Jan 2023 + +A. P. Moina +The temperature-electric field phase diagrams of squaric acid were constructed [4, 7] for the field +𝐸1(𝐸3) directed along the 𝑎(𝑐) tetragonal axis, for the fields denoted for brevity as 𝐸1 ± 𝐸3 and directed +along the diagonals of the 𝑎𝑐 plane, as well as for the fields of the two above mentioned exceptional +directions i) and ii). Note that the 𝑇-𝐸 diagrams are identical for the fields rotated by 90◦ around the 𝑏 +axis, because of the pseudotetragonal symmetry of the model [7]. +Experimentally, the low-temperature transition between the NC90 and FE phases has not been detected +yet due to the dielectric breakdown of the samples. As follows from the model calculations [7], the field +of this transition is the lowest when its direction is close to the axis of the sublattice polarization, so it is +most likely to be experimentally observed at this field orientation. +On the other hand, for the AFE-NC90 switching, the experimental data by Horiuchi et al [5] had +been available, when our calculations were carried out. The polarization hysteresis curves at different +temperatures for the field 𝐸1 had been measured, and the temperature dependence of the switching +field had been deduced from those; for the field 𝐸1 + 𝐸3, the measurements had been performed for +one temperature only [5]. With the fitting procedure for the model being based on the data [5] for the +static dielectric permittivity, the obtained agreement between the theory and the experiment for the +switching fields and for the 𝑃(𝐸) curves was only qualitative [4, 7]. Quantitatively, the agreement was +conspicuously unsatisfactory, which led us to believe that the model used was not completely appropriate, +and that essential modifications were required [7]. Quite recently, however, the same group of Horiuchi +et al reported [9] the results of their new measurements of the polarization loops for the squaric acid +crystals of an improved dielectric strength. This permitted to increase the maximum electric field that +could be applied to the samples in the hysteresis experiments. Our preliminary calculations showed that +the new experimental data were much closer to the predictions of the model [4, 7] than the previous data +of [5], and that the doubts concerning the model validity were premature. +It was extensively discussed in [4] that the accepted set of the values of the model parameters, in +particular of the dipole moments assigned to the ground state configurations of the H2C4O4 groups, +is not unique. While the magnitude of the dipole moment vector is constrained by the fitting to the +permittivity [5], its orientation (and thereby the orientation of the ground state sublattice polarizations) +can be varied within the 𝑎𝑐 plane. With the set of the model parameters adopted in [4, 7] these vectors +are oriented at about 56◦ to the 𝑎(𝑐) axes. On the other hand, the Berry phase calculations [5, 9] indicate +that the axes of the spontaneous sublattice polarization, in fact, are very close or even coincide with +the diagonals of the 𝑎𝑐 plane. In terms of our model, this means that the crystallographic axes and the +diagonals are the above mentioned exceptional directions: the axis 𝑎(𝑐) is the direction ii), while the +diagonals of the 𝑎𝑐 plane are the direction i). The topology of the 𝑇-𝐸 diagrams and the shape of the +𝑃(𝐸) curves for the fields 𝐸1(𝐸3) and 𝐸1 ± 𝐸3 will change accordingly. The availability of the new, more +reliable experimental data [9] makes a quantitative comparison of theoretical and experimental 𝑃(𝐸) +curves meaningful and could help to ascertain the orientation of the model dipole moment vectors. +Thus, it seems worthwhile to revisit the problem of polarization rotation in squaric acid, to perform +calculations with an alternative set of the model parameters, where the sublattice polarizations are +oriented along the diagonals of the 𝑎𝑐 plane, and to compare the theoretical results with the most +recent [9] experimental data. The model [4, 7], briefly described in section 2, is used without any further +modification of the formulae. In section 3 the results of the theoretical calculations with the old and new +sets of the model parameters are compared with the experimental data. +2. The model +The model has been introduced and explicated in [4], and a concise outline is given in [7]. Below +we present a brief qualitative description of the model; all the formulae and other relevant details and +discussions can be found in the mentioned papers. +Protons on the hydrogen bonds in squaric acid move in double-well potentials, so each of the protons +can occupy one of the two sites on the bond: closer to the given C4O4 group or to the neighboring +group. The motion of protons is described by Ising pseudospins, whose two eigenvalues are assigned to +two equilibrium positions of each proton. Two interpenetrating sublattices (layers) of pseudospins are +considered. +43710-2 + +Electric field induced polarization rotation in squaric acid crystals revisited +a) +b) +a +Figure 1. (Colour online) a) The crystal structure of squaric acid as viewed along the 𝑏 axis. Two adjacent +layers are shown, with black and open circles each. The A and B type C4O4 groups are indicated (see [4, 8] +for explanation), and the hydrogen bonds are numbered, 𝑓 = 1, 2, 3, 4. b) The dipole moments assigned +to one of the four lateral proton configurations (the configuration 1 in tables 1 in [4, 7]). Directions of +the dipole moments associated with protons 𝝁𝐻 +1 = (2𝜇𝐻 , 0, 0) and with electrons 𝝁𝜋 +1 = (2𝜇𝜋 +∥ , 0, −2𝜇𝜋 +⊥) +are shown with blue and red arrows, respectively; the green arrow is the total dipole moment of the +configuration; the vector lengths are nominal. 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋 +∥ )/𝜇𝜋 +⊥ is the angle between the total +dipole moment of configuration 1 and the 𝑐 axis. Figures are taken from [4, 7, 8, 10]. +The total system Hamiltonian [4, 7] includes ferroelectric intralayer long-range interactions between +pseudospins, ensuring ferroelectric ordering within each separate layer, antiferroelectric interlayer inter- +actions responsible for alternation of polarizations in the stacked layers, and the short-range interactions, +which include also the coupling with external electric fields 𝐸1 and 𝐸3 directed along the tetragonal +(paraelectric) 𝑎 and 𝑐 axes of the crystal. +The short-range Hamiltonian describes the four-particle configurational correlations between protons +placed around each C4O4 group. The usual Slater-Takagi type scheme [4, 8, 11, 12] of 16 degenerate +levels of lateral/diagonal/single-ionized/double-ionized proton configurations is assumed. The lateral and +single-ionized configurations have dipole moments in the 𝑎𝑐 plane; the degeneracy of their energy levels +is removed by the electric fields 𝐸1 and 𝐸3, which break the equivalence of the hydrogen bonds that link +the C4O4 groups along the 𝑎 and 𝑐 axes (see tables 1 in [4, 7]). +Assignment of the dipole moments to the ground-state lateral configurations is the crucial point of the +model. We rely on the results of the Berry phase calculations [5], which have shown that the ground-state +sublattice polarization in this crystal is formed directly by displacements of protons along the hydrogen +bonds and, mostly, by the electronic contributions of switchable 𝜋-bond dipoles. +Positions of the 𝜋-bonds are determined by the proton arrangement around the given C4O4 group: in +the lateral configurations the 𝜋-bond is formed between the two neighboring carbons, near which protons +sit on the hydrogen bonds (see fig. 1b), and also between the carbons and adjacent to them oxygens, next +to which there is no proton (meaning that the protons on these H-bonds sit in the minima close to the +neighboring C4O4 groups). The field-induced polarization rotation by 90◦ or 180◦ occurs via flipping +of one or two protons in each molecule to the other sites along the same hydrogen bonds and via a +simultaneous switching of the 𝜋-bonds. For the depicted in figure 1b lateral proton configuration, the +vector of the proton contribution to the dipole moment is oriented along the 𝑎 axis, while the electronic +contribution is at the angle to this axis. The dipole moments of the three remaining lateral configurations +can be obtained from the scheme of figure 1b by rotation by a multiple of 90◦. +After going from the representation of proton configuration energies to the pseudospin representation, +the four-particle cluster approximation for the obtained short-range Hamiltonian is employed. The mean +field approximation is used for the long-range interlayer and intralayer interactions [4, 8]. The dependence +of all proton-proton interaction parameters on the diagonal components of the lattice strain tensor and +on the H-site distance, which are changed by the thermal expansion and potentially by an external stress +if such is applied, is taken into account [8]. The expression for the thermodynamic potential has been +obtained [4]; the order parameters and lattice strains are found by numerical minimization thereof. +43710-3 + +H +c +02Po,H +μiA. P. Moina +The values of all model parameters were chosen earlier [4, 7, 8]. In particular, they were required [8] +to provide the best fit to the experimental temperature curves of the order parameter at ambient pressure, +to the temperature and hydrostatic pressure dependences of the diagonal lattice strains, and to the pressure +dependence of the transition temperature 𝑇N in squaric acid. +The dielectric characteristics and other electric field effects in our model are mostly governed by +values of the dipole moments, which enter the final expressions only via the sum 𝜇𝐻 + 𝜇𝜋 +∥ and via 𝜇𝜋 +⊥. +These values are found by fitting the calculated curve of the static dielectric permittivity 𝜀11 at zero +external bias field to the experimental points of [5], while trying to get the best possible agreement +with the experiment for the values of the switching fields, corresponding to the first 90◦ rotation of the +sublattice polarization by the field 𝐸1. It can be shown that in the paraelectric phase 𝜀11 ∼ ¯𝜇2, where +¯𝜇 = +√︃ +(𝜇𝐻 + 𝜇𝜋 +∥ )2 + (𝜇𝜋 +⊥)2 +is half the magnitude of the dipole moment, assigned to the H2C4O4 groups. It means that above 𝑇N the +permittivity 𝜀11 at zero field is determined by the magnitude of the dipole moment vector only, whereas +the orientation of the vector within the 𝑎𝑐 plane can be varied. For the set, adopted in [4, 7] and presented +in table 1 as the set A, the dipole moment and the ground state sublattice polarization are oriented at the +angle 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋 +∥ )/𝜇𝜋 +⊥ ≈ 56◦ to the crystallographic axes. However, the results of the Berry +phase calculations [9] indicate that the angle should be closer to 45◦. Thus, we find an alternative set of +the dipole moment values with 𝜇𝐻 + 𝜇𝜋 +∥ = 𝜇𝜋 +⊥ and with the same ¯𝜇 as in the set A, which yields the same +fit to the permittivity in the paraelectric phase; this is the set B in table 1. In the next section, using the +set B, we construct the 𝑇-𝐸 phase diagrams and explore the 𝑃(𝐸) curves for the electric fields 𝐸1 and +𝐸1 + 𝐸3. The results are compared with the previous calculations [7] performed with the set A, as well +with the experimental data of [5, 9]. +Table 1. The adopted values of the model dipole moments. The set A is taken from [7]. The values of all +other model parameters are the same as in [4, 7, 8]. +𝜇𝐻 + 𝜇𝜋 +∥ +𝜇𝜋 +⊥ +¯𝜇 +(10−29 C m) +set A +3.16 +2.12 +3.8 +set B +2.66 +2.66 +3.8 +3. Calculations +3.1. Phase diagrams +In figure 2 we redraw the 𝑇-𝐸 diagrams of squaric acid for the fields 𝐸1 and 𝐸1 + 𝐸3, obtained earlier +in [4, 7] along with the newly available experimental points of [9]. Here, the set A of the dipole moment +values was used in the calculations. The diagrams overlap the color gradient plots of the introduced in [4] +noncollinearity angle 𝜃, which is the angle between the vectors of the sublattice polarizations. +Different phases in the diagrams are separated by the lines of the first order phase transitions I, II, +and III, and of the second order phase transitions IV. All these lines terminate at various critical points +(bicritical end points BCE, tricritical point TCP, critical end points CEP). Some of the critical points can +be artifacts of the mean field approximation, used for the long-range interactions. This was discussed +extensively in [4, 7]; we shall not dwell on this here. The phase denoted as AFE* (the red region) is +non-collinear antiferrielectric, very close to the initial AFE phase with 𝜃 ∼ 180◦. The purple region is the +collinear field-induced ferroelectric phase (FE) with 𝜃 = 0. The phase between the transition lines II, III, +and IV (green and blue) is the noncollinear ferrielectric phase NC90, where 𝜃 mostly remains close to 90◦, +only rapidly decreasing to zero near the second-order phase transition line IV. In the region NC135* +43710-4 + +Electric field induced polarization rotation in squaric acid crystals revisited +250 +300 +350 +400 +450 +0 +200 +400 +600 +800 +1000 +1200 +V +NC135* +I +BCE1 +CEP +TCP +BCE2 +IV +III + + +T (K) +E1 (kV/cm) +FE +NC90 +0 +1 +11 +22 +33 +45 +56 +67 +78 +90 +107 +115 +120 +125 +130 +146 +157 +169 +180 +AFE* +θ (deg) +II +300 +350 +400 +200 +400 +V +NC135* + + +FE +NC90 +I +IV +III +CEP2 +CEP1 +BCE3 +BCE1 +T (K) +E1+E3 (kV/cm) +BCE2 +II +AFE* +Figure 2. (Colour online) The 𝑇-𝐸 phase diagrams of the squaric acid, overlapping the 𝑇-𝐸 color +contour plots of the noncollinearity angle 𝜃. The set A is used in calculations. Solid and dashed lines +indicate the first and second order phase transitions, respectively; dotted lines are the supercritical lines, +corresponding to the loci of maxima in the field dependences of d𝑃(𝐸)/d𝐸. The open squares □, star �, +and full circles • indicate the critical end points (CEP), tricritical point (TCP), and bicritical end points +(BCE), respectively. Blue full triangles ▲, ▶, and ▼ are the experimental points of [5], the electronic +supplementary material thereto, and [9], respectively. +(orange to yellow), a crossover between the AFE* and NC90 phases occurs. Here, 𝜃 changes gradually +from ∼ 180◦ to ∼ 90◦: the negative sublattice polarization rotates continuously with increasing field and +becomes perpendicular to the positive sublattice polarization. As discussed in [4, 7], this continuous +rotation is a statistically averaged effect, possible only in presence of thermal fluctuations. +Crossovers are often marked by the lines formed by the loci of the extrema of the response functions — +second derivatives of the thermodynamic potentials. Those supercritical lines are continuations of the +first order transition lines beyond the critical points terminating them. The major drawback of this method +is that the extrema of different response functions yield different supercritical lines; moreover, the super- +critical lines formed by the extrema of the same response function taken along different thermodynamic +paths (e.g., isotherms or isofields) are different as well (see [13]). In order to compare the theory and the +experimental data derived from the field dependence of polarization, we mark the crossover between the +AFE* and NC90 phases using the lines formed by the maxima of the d𝑃(𝐸)/d𝐸 isotherms (the inflection +points of the 𝑃(𝐸) isotherms), where 𝑃 is the projection of the net polarization vector on the field axis. +These are the dotted lines V in the phase diagrams. +As one can see in the left-hand panel of figure 2, for the field 𝐸1, the most recent data obtained in [9] +for the sample with the improved dielectric strength appear to be in a much better agreement with the +theory than the earlier experimental data of [5]. The theoretical switching fields, calculated with the set +A, are much higher than the experimental values of [5] with the relative error 𝜂 = 1 − 𝐸exp/𝐸theor ≈ 0.42 +at room temperature (295 K). The error decreases down to ≈ 0.23 for the experimental points of [9], +which is still not quite satisfactory, but evidently much better. +In the case of 𝐸1 + 𝐸3, the switching fields calculated with the set A are higher than predicted for +the field 𝐸1. This is in a qualitative agreement with all available experimental observations [5, 9]. The +relative errors are about 0.3 for [5] at 324 K and 0.22 for [9] at 295 K, that is, the improvement here is +not so striking. +Now let us see how the situation changes, when the set B is used in calculations. For this set, the +ground state spontaneous polarization axis is oriented along the diagonal of the 𝑎𝑐 plane. It means that +the fields 𝐸1 + 𝐸3 and 𝐸1 are directed along this axis and at 45◦ to it, respectively, that is, along the +exceptional directions i) and ii), discussed in Introduction. It is then expected that the 𝑇-𝐸 diagrams will +be topologically different from those, depicted in figure 2. For the field 𝐸1 + 𝐸3, the crystal of squaric +acid should behave like a uniaxial antiferroelectric, exhibiting a one-step polarization rotation by 180◦ +43710-5 + +A. P. Moina +without the intermediate noncollinear phase. For the field 𝐸1, the field of switching to the ferroelectric +phase is expected to tend to infinity, and only the AFE*-NC90 transition can be observed. +300 +400 +100 +200 +300 +400 +T (K) +I +IV +CEP +BCE1 +FE +NC90 + + +0 +1 +11 +22 +33 +45 +56 +67 +78 +90 +107 +115 +120 +135 +146 +157 +169 +180 +E1 (kV/cm) +AFE* +BCE2 +II +θ (deg) +250 +300 +350 +100 +200 +300 +400 +FI +E1+E3 (kV/cm) +IV +I + +FE +TP +BCE +TCP2 +T (K) +TCP1 +AFE* +III +V +VII +238 +240 +296 +300 +304 +VI +VII +III +BCE +AFE* + + + +FE +TP +III +V +FI +Figure 3. (Colour online) The same as in figure 2. The set B is used in calculations. The open triangle +△ indicates the triple point TP. The dash-dotted line VII corresponds to the loci of minima in the field +dependences of d𝑃(𝐸)/d𝐸. The other notations are the same as in figure 2. +The 𝑇-𝐸 phase diagrams, calculated with the set B and presented in figure 3, are indeed in a total +agreement with the above described picture. As the magnitude of the dipole moment 2 ¯𝜇 is the same +for both sets, these diagrams are numerically identical to those obtained in [7] with the set A for +the exceptional directions ii) and i), respectively (see figures 8, 9 in [7]). This identity can be proved +algebraically, using the expression for the thermodynamic potential of the system [4, 7]. +For the field 𝐸1, the positions of the lines II of the AFE*-NC90 phase transitions, calculated with +the sets A and B, are very close but not the same (c.f. the left-hand panels in figures 2 and 3). The +closeness can be explained by the found in [7] dependence of this switching field at low temperatures on +the orientation of the spontaneous sublattice polarization axis 𝐸 𝐼 𝐼 ∼ 1/cos(𝛿𝜑 − π/4), where 𝛿𝜑 is the +angle between this axis and the external field 𝐸. Since 𝛿𝜑 for the sets A and B differ by about 11◦ only, +the difference between the corresponding switching fields is small as well. It is then trivial to say that for +𝐸1, the sets A and B yield about the same agreement with the experimental data for the switching field. +For the field 𝐸1 + 𝐸3, the intermediate phase NC90 is absent, and 𝜃 is always either 180◦ or zero (see +the right-hand panel of figure 3), i.e., all phases are collinear. The polarization switching occurs either as +a first order phase transition across lines III directly to the FE phase and across line VI to an intermediate +collinear ferrielectric phase FI, or gradually. In the latter case, the magnitude of one of the sublattice +polarizations decreases down to zero with increasing field, changes its sign continuously at line VII, and +then increases until the second order transition to the FE phase occurs at line IV. Interestingly, line VII, +where the angle 𝜃 changes from 180◦ to 0, is formed by the loci of the minima of the d𝑃(𝐸)/d𝐸 isotherms, +as opposed to line V, formed by the loci of the maxima. Line VI, emanating from the critical point BCE +(see the inset in figure 3), marks the crossover between AFE* and FI phases. It is to be compared with +the experimental data for the switching fields, and it yields nearly the same agreement as the set A, with +the relative errors about 0.3 for [5] at 324 K and 0.23 for [9]. +3.2. Polarization +In figure 4 we plot the field dependences of the projections of the net polarization vector on the field +direction for the fields 𝐸1 and 𝐸1 + 𝐸3. The experimental points of [5] and [9] are also presented. The +drastic changes in the experimental hysteresis curves, brought by the improvement of the sample quality +and by the increase of the maximum value of the applied field in [9], are very well seen. It is obvious that +the comparison of the theoretical polarization curves with the earlier data of [5] could be only qualitative. +As one can see, for 𝐸1, the sets A and B predict three and two plateaus of polarization, respectively. +43710-6 + +Electric field induced polarization rotation in squaric acid crystals revisited +100 +1000 +0 +10 +20 +30 +40 +200 +400 +600 +0 +10 +20 +30 +40 +III + set A + set B +E1 (kV/cm) +P1 (µC/cm2) +II +V +IV +E1+E3 (kV/cm) +P (µC/cm2) +Figure 4. (Colour online) The field dependences of polarizations at 295 K. Full triangles: experimental +points taken from [5] (▲) and [9] (▼). The arrows indicate phase transitions of the first order across lines +II, III (left-hand) and of the second order across lines IV (right-hand). The arrow and full circles (•) +indicate the crossovers at lines V (right-hand). Lines II-V are from the 𝑇-𝐸 phase diagrams, figures 2, 3. +In the physically reasonable field range, which includes the AFE*-NC90 first order phase transition (at +lines II from the phase diagrams 2, 3), the two sets yield very similar polarizations. The calculated +polarization jumps are 17.9 µC/cm2 for the set A and 20.4 µC/cm2 for the set B, in a fair agreement +with the experimental 17.2 µC/cm2 [9]. The set A also predicts the second step of polarization at a much +higher field, at the transition to the FE phase (across line III from the phase diagram, figure 2). It seems +unlikely, however, that the field of such high a magnitude could ever be applied in an experiment without +the squaric acid samples suffering the dielectric breakdown. +For the field 𝐸1 + 𝐸3, the two sets of the model parameters yield different behaviour of polarization +even at experimentally accessible fields. The 𝑃(𝐸) curve, calculated with the set A, has three smeared +plateaus, with a clear rounded step, corresponding to the AFE*- NC90 crossover across line V, and then +a cusp at the NC90-FE second order transition across line IV. The lower part of this curve, albeit being +shifted to higher fields, is in a good qualitative and quantitative agreement with the experimental points. +The polarization, calculated with the set B, on the other hand, has only two smeared plateaus: at low +fields and above the cusp at line IV. No pronounced intermediate plateau is seen. The change of concavity +at the inflection point, marked by a full circle in the figure, is hardly discernible. The agreement with +the experiment is visibly worse than for the set A. However, the switching field magnitude for 𝐸1 + 𝐸3 is +predicted [5, 7, 9] to be higher than for the field 𝐸1. It means that, despite the increased dielectric strength +of the samples, the maximum applied fields 𝐸1 +𝐸3 [9] could still be insufficient to obtain correct data for +polarization. A potential further improvement of the crystal quality (if such is still possible) may change +the measured values of polarization and switching field for the diagonally directed field in the same way, +as such an improvement did in the case of the field 𝐸1 in [9] as compared to [5], which is well illustrated +in the left-hand panel of figure 4. Then, the agreement with the theoretical curves can be reexamined. +4. Concluding remarks +Using the previously developed [4] deformable two-sublattice proton ordering model, we revisit +the problem of polarization rotation in antiferroelectric crystals of squaric acid under the influence +of external electric fields. The unique structure of the two-dimensional hydrogen bond networks in +squaric acid permits 90◦ rotation of the sublattice polarization. The model predicts [4, 7] that except +for some particular directions of the field, the polarization reorientation at low temperatures is a two- +step process: first, to the noncollinear phase with perpendicular sublattice polarizations and then to the +collinear ferroelectric phase. However, when the field is directed along the axis of spontaneous sublattice +polarizations, the intermediate noncollinear phase is absent; when the field is at 45◦ to this axis, the field +of the transition to the ferroelectric phase tends to infinity. +The previously obtained 𝑇-𝐸 phase diagrams and newly calculated polarization curves are compared +43710-7 + +A. P. Moina +with the most recent experimental data [9], measured using the crystal samples of the increased dielectric +strength. We also test an alternative set of the model parameters, for which the dipole moments assigned +to the H2C4O4 groups are of the same magnitude as in the previous calculations, but oriented along the +diagonals of the 𝑎𝑐 plane. +The new experimental data [9] are in a drastically better agreement with the theory than the earlier +results [5], especially for the polarization curves, as well as for the switching fields. It shows that the +simplicity of the model was not the major reason of the earlier [4, 7] disagreement between theory and +experiment and gives a strong evidence for the model validity. +Results of testing the new set of the model parameters are inconclusive. Overall, the comparison of the +theoretical polarization curves with the experimental data seems to slightly favor the previous set [4, 7], +according to which the axes of the spontaneous sublattice polarization are close, but not exactly parallel +to the diagonals of the 𝑎𝑐 plane. Further experimental studies may shed some light on this problem. +As far as a further verification of the model is concerned, the appropriateness of the mean field +approximation, used for the long-range interactions, may be addressed. This approximation is, most +likely, the origin of the artifact splitting [4, 7] of some tricritical points in the 𝑃-𝐸 phase diagrams into +the systems of bicritical and critical endpoints and also of the appearance of the intermediate FI phase, +seen in the right-hand panels of figures 2, 3. Monte Carlo calculations may be used to construct more +accurate diagrams. +References +1. Semmingsen D., Tun Z., Nelmes R. J., McMullan R. K., Koetzle T. F., Z. Kristallogr. Cryst. Mater., 1995, 210, +934–947, doi:10.1524/zkri.1995.210.12.934. +2. Semmingsen D., Hollander F. J., Koetzle T. F., J. Chem. Phys., 1977, 66, 4405–4412, doi:10.1063/1.433745 +3. Hollander F. J., Semmingsen D., Koetzle T. F., J. Chem. Phys., 1977, 67, 4825–4831, doi:10.1063/1.434686. +4. Moina A. P., Phys. Rev. B, 2021, 103, 214104, doi:10.1103/PhysRevB.103.214104. +5. Horiuchi S., Kumai R., Ishibashi S., Chem. Sci., 2018, 9, 425–432, doi:10.1039/C7SC03859C. +6. Ishibashi S., Horiuchi S., Kumai R., Phys. Rev. B, 2018, 97, 184102, doi:10.1103/PhysRevB.97.184102. +7. Moina A. P., Condens. Matter Phys., 2021, 24, No. 4, 43703, doi:10.5488/CMP.24.43703. +8. Moina A. P., Condens. Matter Phys., 2020, 23, No. 3, 33704, doi:10.5488/CMP.23.33704. +9. Horiuchi S., Ishibashi S., Chem. Phys., 2021, 12, 14198–14206, doi:10.1039/d1sc02729h. +10. Semmingsen D., Feder J., Solid State Commun., 1974, 15, 1369–1372, doi:10.1016/0038-1098(74)91382-9. +11. Matsushita E., Yoshimitsu K., Matsubara T., Progr. Theor. Phys., 1980, 64, No. 4, 1176–1192, +doi:10.1143/PTP.64.1176. +12. Matsushita E., Matsubara T., Progr. Theor. Phys., 1982, 68, No. 6, 1811–1826, doi:10.1143/PTP.68.1811. +13. Schienbein P., Marx D., Phys. Rev. E, 2018, 98, 022104, doi:10.1103/PhysRevE.98.022104. +43710-8 + +Electric field induced polarization rotation in squaric acid crystals revisited +Ще раз про обертання поляризацiї електричним полем в +кристалах квадратної кислоти +А. П. Моїна +Iнститут фiзики конденсованих систем Нацiональної академiї наук України +79011, м. Львiв, вул. Свєнцiцького, 1 +З використанням запропонованої ранiше моделi розглядаються процеси обертання поляризацiї зовнiшнi- +ми електричними полями в антисегнетоелектричних кристалах квадратної кислоти. Обчислення також +проводяться з альтернативним набором параметрiв теорiї, в якому дипольнi моменти, якi приписуються +групам H2C4O4, паралельнi до дiагоналей площини 𝑎𝑐. Дослiджено фазовi дiаграми 𝑇-𝐸 та кривi поля- +ризацiї 𝑃(𝐸) для полiв, прикладених уздовж осi 𝑎 та уздовж дiагоналi площини 𝑎𝑐. Порiвняння теорети- +чних результатiв з нещодавно опублiкованими експериментальними даними пiдтверджує правильнiсть +запропонованої моделi. Не виявлено суттєвої переваги нового набору параметрiв моделi перед тим, що +використовувався в попереднiх розрахунках. +Ключовi слова: поляризацiя, електричне поле, фазовий перехiд, антисегнетоелектрик, фазова дiаграма, +квадратна кислота +43710-9 + + diff --git a/N9AzT4oBgHgl3EQfk_2S/content/tmp_files/load_file.txt b/N9AzT4oBgHgl3EQfk_2S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a048457471eb560947d687b658f96b4fc28c214 --- /dev/null +++ b/N9AzT4oBgHgl3EQfk_2S/content/tmp_files/load_file.txt @@ -0,0 +1,389 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf,len=388 +page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 4, 43710: 1–9 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='43710 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='ua/journal Electric field induced polarization rotation in squaric acid crystals revisited A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Moina Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', 79011 Lviv, Ukraine Received July 10, 2022, in final form July 26, 2022 Using the previously developed model we revisit the problem of the electric field induced polarization rotation in antiferroelectric crystals of squaric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' We test an alternative set of the model parameters, according to which the dipole moments associated with the H2C4O4 groups are assumed to be parallel to the diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The 𝑇-𝐸 phase diagrams and the polarization curves 𝑃(𝐸) for the fields directed along the 𝑎 axis and along one of the diagonals are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Comparison of the theoretical results with the newly published experimental data confirm the validity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The calculations reveal no apparent advantage of the new set of the parameters over the previously used set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Key words: polarization, electric field, phase transition, phase diagram, squaric acid 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Introduction The squaric acid H2C4O4 is a classical two-dimensional antiferroelectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The crystal is tetrago- nal, 𝐼4/𝑚, in the paraelectric phase and monoclinic, 𝑃21/𝑚, in the antiferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The hydrogen bonded C4O4 groups form sheets, parallel to the 𝑎𝑐 plane and stacked along the 𝑏-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Below the tran- sition at 373 K, a spontaneous polarization arises in these sheets, with the neighboring sheets polarized in the opposite directions [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' External electric fields applied to a uniaxial antiferroelectric can switch a sublattice polarization by 180◦ and induce thereby the transition from antiferroelectric (AFE) to ferroelectric (FE) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The (pseudo)tetragonal symmetry of the squaric acid crystal lattice and of its hydrogen bond networks allows the sublattice polarizations to be directed along two perpendicular axes in the fully ordered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As a result, here the external field can rotate one of the sublattice polarizations by 90◦, whereupon a noncollinear ferrielectric phase with perpendicular sublattice polarizations (NC90 [4]) is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The possibility of such a rotation has been suggested by Horiuchi et al [5], and their hysteresis loop measurements and Berry phase calculations gave evidence for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Further calculations [6] indicated that the 90◦ rotation is possible at different orientations of the field within the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It is also predicted [5, 6] that application of higher fields along the diagonals of the 𝑎𝑐 plane can lead to the second rotation of the negative sublattice polarization by 90◦ and induction of the collinear ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Recently [4, 7] we developed a deformable [8] two-sublattice proton-ordering model for a description of squaric acid behaviour in external electric fields, applied arbitrarily within the plane of hydrogen bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The model calculations confirm the two-step process of polarization reorientation [5, 6] at low temperatures, with the negative sublattice polarization being switched twice by 90◦ at each transition, for any orientation of the field within the 𝑎𝑐 plane, but a few exceptional directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The exceptional directions are those, when the field is either i) collinear to the axes of the sublattice polarization in the AFE phase, or ii) directed at 45◦ to these axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In the case i), the crystal behaves like a uniaxial antiferroelectric, undergoing a single-step polarization switching to the FE phase without the intermediate noncollinear phase, while in the case ii), the transition field from the NC90 to the FE phase goes to infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', the transition never occurs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 43710-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='01541v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='mtrl-sci] 4 Jan 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Moina The temperature-electric field phase diagrams of squaric acid were constructed [4, 7] for the field 𝐸1(𝐸3) directed along the 𝑎(𝑐) tetragonal axis, for the fields denoted for brevity as 𝐸1 ± 𝐸3 and directed along the diagonals of the 𝑎𝑐 plane, as well as for the fields of the two above mentioned exceptional directions i) and ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Note that the 𝑇-𝐸 diagrams are identical for the fields rotated by 90◦ around the 𝑏 axis, because of the pseudotetragonal symmetry of the model [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Experimentally, the low-temperature transition between the NC90 and FE phases has not been detected yet due to the dielectric breakdown of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As follows from the model calculations [7], the field of this transition is the lowest when its direction is close to the axis of the sublattice polarization, so it is most likely to be experimentally observed at this field orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' On the other hand, for the AFE-NC90 switching, the experimental data by Horiuchi et al [5] had been available, when our calculations were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The polarization hysteresis curves at different temperatures for the field 𝐸1 had been measured, and the temperature dependence of the switching field had been deduced from those;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' for the field 𝐸1 + 𝐸3, the measurements had been performed for one temperature only [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' With the fitting procedure for the model being based on the data [5] for the static dielectric permittivity, the obtained agreement between the theory and the experiment for the switching fields and for the 𝑃(𝐸) curves was only qualitative [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Quantitatively, the agreement was conspicuously unsatisfactory, which led us to believe that the model used was not completely appropriate, and that essential modifications were required [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Quite recently, however, the same group of Horiuchi et al reported [9] the results of their new measurements of the polarization loops for the squaric acid crystals of an improved dielectric strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' This permitted to increase the maximum electric field that could be applied to the samples in the hysteresis experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Our preliminary calculations showed that the new experimental data were much closer to the predictions of the model [4, 7] than the previous data of [5], and that the doubts concerning the model validity were premature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It was extensively discussed in [4] that the accepted set of the values of the model parameters, in particular of the dipole moments assigned to the ground state configurations of the H2C4O4 groups, is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' While the magnitude of the dipole moment vector is constrained by the fitting to the permittivity [5], its orientation (and thereby the orientation of the ground state sublattice polarizations) can be varied within the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' With the set of the model parameters adopted in [4, 7] these vectors are oriented at about 56◦ to the 𝑎(𝑐) axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' On the other hand, the Berry phase calculations [5, 9] indicate that the axes of the spontaneous sublattice polarization, in fact, are very close or even coincide with the diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In terms of our model, this means that the crystallographic axes and the diagonals are the above mentioned exceptional directions: the axis 𝑎(𝑐) is the direction ii), while the diagonals of the 𝑎𝑐 plane are the direction i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The topology of the 𝑇-𝐸 diagrams and the shape of the 𝑃(𝐸) curves for the fields 𝐸1(𝐸3) and 𝐸1 ± 𝐸3 will change accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The availability of the new, more reliable experimental data [9] makes a quantitative comparison of theoretical and experimental 𝑃(𝐸) curves meaningful and could help to ascertain the orientation of the model dipole moment vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Thus, it seems worthwhile to revisit the problem of polarization rotation in squaric acid, to perform calculations with an alternative set of the model parameters, where the sublattice polarizations are oriented along the diagonals of the 𝑎𝑐 plane, and to compare the theoretical results with the most recent [9] experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The model [4, 7], briefly described in section 2, is used without any further modification of the formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In section 3 the results of the theoretical calculations with the old and new sets of the model parameters are compared with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The model The model has been introduced and explicated in [4], and a concise outline is given in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Below we present a brief qualitative description of the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' all the formulae and other relevant details and discussions can be found in the mentioned papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Protons on the hydrogen bonds in squaric acid move in double-well potentials, so each of the protons can occupy one of the two sites on the bond: closer to the given C4O4 group or to the neighboring group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The motion of protons is described by Ising pseudospins, whose two eigenvalues are assigned to two equilibrium positions of each proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Two interpenetrating sublattices (layers) of pseudospins are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 43710-2 Electric field induced polarization rotation in squaric acid crystals revisited a) b) a Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' (Colour online) a) The crystal structure of squaric acid as viewed along the 𝑏 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Two adjacent layers are shown, with black and open circles each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The A and B type C4O4 groups are indicated (see [4, 8] for explanation), and the hydrogen bonds are numbered, 𝑓 = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' b) The dipole moments assigned to one of the four lateral proton configurations (the configuration 1 in tables 1 in [4, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Directions of the dipole moments associated with protons 𝝁𝐻 1 = (2𝜇𝐻 , 0, 0) and with electrons 𝝁𝜋 1 = (2𝜇𝜋 ∥ , 0, −2𝜇𝜋 ⊥) are shown with blue and red arrows, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' the green arrow is the total dipole moment of the configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' the vector lengths are nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋 ∥ )/𝜇𝜋 ⊥ is the angle between the total dipole moment of configuration 1 and the 𝑐 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Figures are taken from [4, 7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The total system Hamiltonian [4, 7] includes ferroelectric intralayer long-range interactions between pseudospins, ensuring ferroelectric ordering within each separate layer, antiferroelectric interlayer inter- actions responsible for alternation of polarizations in the stacked layers, and the short-range interactions, which include also the coupling with external electric fields 𝐸1 and 𝐸3 directed along the tetragonal (paraelectric) 𝑎 and 𝑐 axes of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The short-range Hamiltonian describes the four-particle configurational correlations between protons placed around each C4O4 group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The usual Slater-Takagi type scheme [4, 8, 11, 12] of 16 degenerate levels of lateral/diagonal/single-ionized/double-ionized proton configurations is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The lateral and single-ionized configurations have dipole moments in the 𝑎𝑐 plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' the degeneracy of their energy levels is removed by the electric fields 𝐸1 and 𝐸3, which break the equivalence of the hydrogen bonds that link the C4O4 groups along the 𝑎 and 𝑐 axes (see tables 1 in [4, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Assignment of the dipole moments to the ground-state lateral configurations is the crucial point of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' We rely on the results of the Berry phase calculations [5], which have shown that the ground-state sublattice polarization in this crystal is formed directly by displacements of protons along the hydrogen bonds and, mostly, by the electronic contributions of switchable 𝜋-bond dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Positions of the 𝜋-bonds are determined by the proton arrangement around the given C4O4 group: in the lateral configurations the 𝜋-bond is formed between the two neighboring carbons, near which protons sit on the hydrogen bonds (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 1b), and also between the carbons and adjacent to them oxygens, next to which there is no proton (meaning that the protons on these H-bonds sit in the minima close to the neighboring C4O4 groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The field-induced polarization rotation by 90◦ or 180◦ occurs via flipping of one or two protons in each molecule to the other sites along the same hydrogen bonds and via a simultaneous switching of the 𝜋-bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the depicted in figure 1b lateral proton configuration, the vector of the proton contribution to the dipole moment is oriented along the 𝑎 axis, while the electronic contribution is at the angle to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The dipole moments of the three remaining lateral configurations can be obtained from the scheme of figure 1b by rotation by a multiple of 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' After going from the representation of proton configuration energies to the pseudospin representation, the four-particle cluster approximation for the obtained short-range Hamiltonian is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The mean field approximation is used for the long-range interlayer and intralayer interactions [4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The dependence of all proton-proton interaction parameters on the diagonal components of the lattice strain tensor and on the H-site distance, which are changed by the thermal expansion and potentially by an external stress if such is applied, is taken into account [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The expression for the thermodynamic potential has been obtained [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' the order parameters and lattice strains are found by numerical minimization thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 43710-3 H c 02Po,H μiA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Moina The values of all model parameters were chosen earlier [4, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In particular, they were required [8] to provide the best fit to the experimental temperature curves of the order parameter at ambient pressure, to the temperature and hydrostatic pressure dependences of the diagonal lattice strains, and to the pressure dependence of the transition temperature 𝑇N in squaric acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The dielectric characteristics and other electric field effects in our model are mostly governed by values of the dipole moments, which enter the final expressions only via the sum 𝜇𝐻 + 𝜇𝜋 ∥ and via 𝜇𝜋 ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' These values are found by fitting the calculated curve of the static dielectric permittivity 𝜀11 at zero external bias field to the experimental points of [5], while trying to get the best possible agreement with the experiment for the values of the switching fields, corresponding to the first 90◦ rotation of the sublattice polarization by the field 𝐸1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It can be shown that in the paraelectric phase 𝜀11 ∼ ¯𝜇2, where ¯𝜇 = √︃ (𝜇𝐻 + 𝜇𝜋 ∥ )2 + (𝜇𝜋 ⊥)2 is half the magnitude of the dipole moment, assigned to the H2C4O4 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It means that above 𝑇N the permittivity 𝜀11 at zero field is determined by the magnitude of the dipole moment vector only, whereas the orientation of the vector within the 𝑎𝑐 plane can be varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the set, adopted in [4, 7] and presented in table 1 as the set A, the dipole moment and the ground state sublattice polarization are oriented at the angle 𝜑0 = arctan(𝜇𝐻 + 𝜇𝜋 ∥ )/𝜇𝜋 ⊥ ≈ 56◦ to the crystallographic axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' However, the results of the Berry phase calculations [9] indicate that the angle should be closer to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Thus, we find an alternative set of the dipole moment values with 𝜇𝐻 + 𝜇𝜋 ∥ = 𝜇𝜋 ⊥ and with the same ¯𝜇 as in the set A, which yields the same fit to the permittivity in the paraelectric phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' this is the set B in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In the next section, using the set B, we construct the 𝑇-𝐸 phase diagrams and explore the 𝑃(𝐸) curves for the electric fields 𝐸1 and 𝐸1 + 𝐸3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The results are compared with the previous calculations [7] performed with the set A, as well with the experimental data of [5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The adopted values of the model dipole moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The set A is taken from [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The values of all other model parameters are the same as in [4, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 𝜇𝐻 + 𝜇𝜋 ∥ 𝜇𝜋 ⊥ ¯𝜇 (10−29 C m) set A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='8 set B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='66 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Calculations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Phase diagrams In figure 2 we redraw the 𝑇-𝐸 diagrams of squaric acid for the fields 𝐸1 and 𝐸1 + 𝐸3, obtained earlier in [4, 7] along with the newly available experimental points of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Here, the set A of the dipole moment values was used in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The diagrams overlap the color gradient plots of the introduced in [4] noncollinearity angle 𝜃, which is the angle between the vectors of the sublattice polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Different phases in the diagrams are separated by the lines of the first order phase transitions I, II, and III, and of the second order phase transitions IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' All these lines terminate at various critical points (bicritical end points BCE, tricritical point TCP, critical end points CEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Some of the critical points can be artifacts of the mean field approximation, used for the long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' This was discussed extensively in [4, 7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' we shall not dwell on this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The phase denoted as AFE* (the red region) is non-collinear antiferrielectric, very close to the initial AFE phase with 𝜃 ∼ 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The purple region is the collinear field-induced ferroelectric phase (FE) with 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The phase between the transition lines II, III, and IV (green and blue) is the noncollinear ferrielectric phase NC90, where 𝜃 mostly remains close to 90◦, only rapidly decreasing to zero near the second-order phase transition line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In the region NC135* 43710-4 Electric field induced polarization rotation in squaric acid crystals revisited 250 300 350 400 450 0 200 400 600 800 1000 1200 V NC135* I BCE1 CEP TCP BCE2 IV III T (K) E1 (kV/cm) FE NC90 0 1 11 22 33 45 56 67 78 90 107 115 120 125 130 146 157 169 180 AFE* θ (deg) II 300 350 400 200 400 V NC135* FE NC90 I IV III CEP2 CEP1 BCE3 BCE1 T (K) E1+E3 (kV/cm) BCE2 II AFE* Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' (Colour online) The 𝑇-𝐸 phase diagrams of the squaric acid, overlapping the 𝑇-𝐸 color contour plots of the noncollinearity angle 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The set A is used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Solid and dashed lines indicate the first and second order phase transitions, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' dotted lines are the supercritical lines, corresponding to the loci of maxima in the field dependences of d𝑃(𝐸)/d𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The open squares □, star �, and full circles • indicate the critical end points (CEP), tricritical point (TCP), and bicritical end points (BCE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Blue full triangles ▲, ▶, and ▼ are the experimental points of [5], the electronic supplementary material thereto, and [9], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' (orange to yellow), a crossover between the AFE* and NC90 phases occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Here, 𝜃 changes gradually from ∼ 180◦ to ∼ 90◦: the negative sublattice polarization rotates continuously with increasing field and becomes perpendicular to the positive sublattice polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As discussed in [4, 7], this continuous rotation is a statistically averaged effect, possible only in presence of thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Crossovers are often marked by the lines formed by the loci of the extrema of the response functions — second derivatives of the thermodynamic potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Those supercritical lines are continuations of the first order transition lines beyond the critical points terminating them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The major drawback of this method is that the extrema of different response functions yield different supercritical lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' moreover, the super- critical lines formed by the extrema of the same response function taken along different thermodynamic paths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', isotherms or isofields) are different as well (see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In order to compare the theory and the experimental data derived from the field dependence of polarization, we mark the crossover between the AFE* and NC90 phases using the lines formed by the maxima of the d𝑃(𝐸)/d𝐸 isotherms (the inflection points of the 𝑃(𝐸) isotherms), where 𝑃 is the projection of the net polarization vector on the field axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' These are the dotted lines V in the phase diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As one can see in the left-hand panel of figure 2, for the field 𝐸1, the most recent data obtained in [9] for the sample with the improved dielectric strength appear to be in a much better agreement with the theory than the earlier experimental data of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The theoretical switching fields, calculated with the set A, are much higher than the experimental values of [5] with the relative error 𝜂 = 1 − 𝐸exp/𝐸theor ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='42 at room temperature (295 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The error decreases down to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='23 for the experimental points of [9], which is still not quite satisfactory, but evidently much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In the case of 𝐸1 + 𝐸3, the switching fields calculated with the set A are higher than predicted for the field 𝐸1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' This is in a qualitative agreement with all available experimental observations [5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The relative errors are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='3 for [5] at 324 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='22 for [9] at 295 K, that is, the improvement here is not so striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Now let us see how the situation changes, when the set B is used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For this set, the ground state spontaneous polarization axis is oriented along the diagonal of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It means that the fields 𝐸1 + 𝐸3 and 𝐸1 are directed along this axis and at 45◦ to it, respectively, that is, along the exceptional directions i) and ii), discussed in Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It is then expected that the 𝑇-𝐸 diagrams will be topologically different from those, depicted in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the field 𝐸1 + 𝐸3, the crystal of squaric acid should behave like a uniaxial antiferroelectric, exhibiting a one-step polarization rotation by 180◦ 43710-5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Moina without the intermediate noncollinear phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the field 𝐸1, the field of switching to the ferroelectric phase is expected to tend to infinity, and only the AFE*-NC90 transition can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 300 400 100 200 300 400 T (K) I IV CEP BCE1 FE NC90 0 1 11 22 33 45 56 67 78 90 107 115 120 135 146 157 169 180 E1 (kV/cm) AFE* BCE2 II θ (deg) 250 300 350 100 200 300 400 FI E1+E3 (kV/cm) IV I FE TP BCE TCP2 T (K) TCP1 AFE* III V VII 238 240 296 300 304 VI VII III BCE AFE* FE TP III V FI Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' (Colour online) The same as in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The set B is used in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The open triangle △ indicates the triple point TP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The dash-dotted line VII corresponds to the loci of minima in the field dependences of d𝑃(𝐸)/d𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The other notations are the same as in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The 𝑇-𝐸 phase diagrams, calculated with the set B and presented in figure 3, are indeed in a total agreement with the above described picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As the magnitude of the dipole moment 2 ¯𝜇 is the same for both sets, these diagrams are numerically identical to those obtained in [7] with the set A for the exceptional directions ii) and i), respectively (see figures 8, 9 in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' This identity can be proved algebraically, using the expression for the thermodynamic potential of the system [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the field 𝐸1, the positions of the lines II of the AFE*-NC90 phase transitions, calculated with the sets A and B, are very close but not the same (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' the left-hand panels in figures 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The closeness can be explained by the found in [7] dependence of this switching field at low temperatures on the orientation of the spontaneous sublattice polarization axis 𝐸 𝐼 𝐼 ∼ 1/cos(𝛿𝜑 − π/4), where 𝛿𝜑 is the angle between this axis and the external field 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Since 𝛿𝜑 for the sets A and B differ by about 11◦ only, the difference between the corresponding switching fields is small as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It is then trivial to say that for 𝐸1, the sets A and B yield about the same agreement with the experimental data for the switching field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the field 𝐸1 + 𝐸3, the intermediate phase NC90 is absent, and 𝜃 is always either 180◦ or zero (see the right-hand panel of figure 3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', all phases are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The polarization switching occurs either as a first order phase transition across lines III directly to the FE phase and across line VI to an intermediate collinear ferrielectric phase FI, or gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In the latter case, the magnitude of one of the sublattice polarizations decreases down to zero with increasing field, changes its sign continuously at line VII, and then increases until the second order transition to the FE phase occurs at line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Interestingly, line VII, where the angle 𝜃 changes from 180◦ to 0, is formed by the loci of the minima of the d𝑃(𝐸)/d𝐸 isotherms, as opposed to line V, formed by the loci of the maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Line VI, emanating from the critical point BCE (see the inset in figure 3), marks the crossover between AFE* and FI phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It is to be compared with the experimental data for the switching fields, and it yields nearly the same agreement as the set A, with the relative errors about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='3 for [5] at 324 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='23 for [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Polarization In figure 4 we plot the field dependences of the projections of the net polarization vector on the field direction for the fields 𝐸1 and 𝐸1 + 𝐸3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The experimental points of [5] and [9] are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The drastic changes in the experimental hysteresis curves, brought by the improvement of the sample quality and by the increase of the maximum value of the applied field in [9], are very well seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It is obvious that the comparison of the theoretical polarization curves with the earlier data of [5] could be only qualitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As one can see, for 𝐸1, the sets A and B predict three and two plateaus of polarization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 43710-6 Electric field induced polarization rotation in squaric acid crystals revisited 100 1000 0 10 20 30 40 200 400 600 0 10 20 30 40 III set A set B E1 (kV/cm) P1 (µC/cm2) II V IV E1+E3 (kV/cm) P (µC/cm2) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' (Colour online) The field dependences of polarizations at 295 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Full triangles: experimental points taken from [5] (▲) and [9] (▼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The arrows indicate phase transitions of the first order across lines II, III (left-hand) and of the second order across lines IV (right-hand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The arrow and full circles (•) indicate the crossovers at lines V (right-hand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Lines II-V are from the 𝑇-𝐸 phase diagrams, figures 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' In the physically reasonable field range, which includes the AFE*-NC90 first order phase transition (at lines II from the phase diagrams 2, 3), the two sets yield very similar polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The calculated polarization jumps are 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='9 µC/cm2 for the set A and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='4 µC/cm2 for the set B, in a fair agreement with the experimental 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='2 µC/cm2 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The set A also predicts the second step of polarization at a much higher field, at the transition to the FE phase (across line III from the phase diagram, figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It seems unlikely, however, that the field of such high a magnitude could ever be applied in an experiment without the squaric acid samples suffering the dielectric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' For the field 𝐸1 + 𝐸3, the two sets of the model parameters yield different behaviour of polarization even at experimentally accessible fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The 𝑃(𝐸) curve, calculated with the set A, has three smeared plateaus, with a clear rounded step, corresponding to the AFE*- NC90 crossover across line V, and then a cusp at the NC90-FE second order transition across line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The lower part of this curve, albeit being shifted to higher fields, is in a good qualitative and quantitative agreement with the experimental points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The polarization, calculated with the set B, on the other hand, has only two smeared plateaus: at low fields and above the cusp at line IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' No pronounced intermediate plateau is seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The change of concavity at the inflection point, marked by a full circle in the figure, is hardly discernible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The agreement with the experiment is visibly worse than for the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' However, the switching field magnitude for 𝐸1 + 𝐸3 is predicted [5, 7, 9] to be higher than for the field 𝐸1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It means that, despite the increased dielectric strength of the samples, the maximum applied fields 𝐸1 +𝐸3 [9] could still be insufficient to obtain correct data for polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' A potential further improvement of the crystal quality (if such is still possible) may change the measured values of polarization and switching field for the diagonally directed field in the same way, as such an improvement did in the case of the field 𝐸1 in [9] as compared to [5], which is well illustrated in the left-hand panel of figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Then, the agreement with the theoretical curves can be reexamined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Concluding remarks Using the previously developed [4] deformable two-sublattice proton ordering model, we revisit the problem of polarization rotation in antiferroelectric crystals of squaric acid under the influence of external electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The unique structure of the two-dimensional hydrogen bond networks in squaric acid permits 90◦ rotation of the sublattice polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The model predicts [4, 7] that except for some particular directions of the field, the polarization reorientation at low temperatures is a two- step process: first, to the noncollinear phase with perpendicular sublattice polarizations and then to the collinear ferroelectric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' However, when the field is directed along the axis of spontaneous sublattice polarizations, the intermediate noncollinear phase is absent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' when the field is at 45◦ to this axis, the field of the transition to the ferroelectric phase tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The previously obtained 𝑇-𝐸 phase diagrams and newly calculated polarization curves are compared 43710-7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Moina with the most recent experimental data [9], measured using the crystal samples of the increased dielectric strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' We also test an alternative set of the model parameters, for which the dipole moments assigned to the H2C4O4 groups are of the same magnitude as in the previous calculations, but oriented along the diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' The new experimental data [9] are in a drastically better agreement with the theory than the earlier results [5], especially for the polarization curves, as well as for the switching fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' It shows that the simplicity of the model was not the major reason of the earlier [4, 7] disagreement between theory and experiment and gives a strong evidence for the model validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Results of testing the new set of the model parameters are inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Overall, the comparison of the theoretical polarization curves with the experimental data seems to slightly favor the previous set [4, 7], according to which the axes of the spontaneous sublattice polarization are close, but not exactly parallel to the diagonals of the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Further experimental studies may shed some light on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' As far as a further verification of the model is concerned, the appropriateness of the mean field approximation, used for the long-range interactions, may be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' This approximation is, most likely, the origin of the artifact splitting [4, 7] of some tricritical points in the 𝑃-𝐸 phase diagrams into the systems of bicritical and critical endpoints and also of the appearance of the intermediate FI phase, seen in the right-hand panels of figures 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Monte Carlo calculations may be used to construct more accurate diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Semmingsen D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', Ishibashi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=', 2021, 12, 14198–14206, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='1039/d1sc02729h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content='022104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' 43710-8 Electric field induced polarization rotation in squaric acid crystals revisited Ще раз про обертання поляризацiї електричним полем в кристалах квадратної кислоти А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Моїна Iнститут фiзики конденсованих систем Нацiональної академiї наук України 79011, м.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Львiв, вул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Свєнцiцького, 1 З використанням запропонованої ранiше моделi розглядаються процеси обертання поляризацiї зовнiшнi- ми електричними полями в антисегнетоелектричних кристалах квадратної кислоти.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Обчислення також проводяться з альтернативним набором параметрiв теорiї, в якому дипольнi моменти, якi приписуються групам H2C4O4, паралельнi до дiагоналей площини 𝑎𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Дослiджено фазовi дiаграми 𝑇-𝐸 та кривi поля- ризацiї 𝑃(𝐸) для полiв, прикладених уздовж осi 𝑎 та уздовж дiагоналi площини 𝑎𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Порiвняння теорети- чних результатiв з нещодавно опублiкованими експериментальними даними пiдтверджує правильнiсть запропонованої моделi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Не виявлено суттєвої переваги нового набору параметрiв моделi перед тим, що використовувався в попереднiх розрахунках.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} +page_content=' Ключовi слова: поляризацiя, електричне поле, фазовий перехiд, антисегнетоелектрик, фазова дiаграма, квадратна кислота 43710-9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AzT4oBgHgl3EQfk_2S/content/2301.01541v1.pdf'} diff --git a/N9FAT4oBgHgl3EQfyR7D/content/tmp_files/2301.08692v1.pdf.txt b/N9FAT4oBgHgl3EQfyR7D/content/tmp_files/2301.08692v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5f204f5d5d41b92ab521fa93f83697622e53457 --- /dev/null +++ b/N9FAT4oBgHgl3EQfyR7D/content/tmp_files/2301.08692v1.pdf.txt @@ -0,0 +1,2656 @@ +MNRAS 000, 1–20 (2023) +Preprint 23 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Constraints on S8 from a full-scale and full-shape analysis +of redshift-space clustering and galaxy–galaxy lensing in +BOSS +Johannes U. Lange1,2,3,4⋆, Andrew P. Hearin5, Alexie Leauthaud2, Frank C. van den Bosch6, +Enia Xhakaj2, Hong Guo7, Risa H. Wechsler1 and Joseph DeRose8 +1Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, CA 94305, USA +2Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA +3Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA +4Leinweber Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA +5Argonne National Laboratory, Argonne, IL 60439, USA +6Department of Astronomy, Yale University, New Haven, CT 06511, USA +7Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Shanghai 200030, China +8Berkeley Center for Cosmological Physics, University of California, Berkeley, CA 94720, USA +Accepted xxx. Received xxx +ABSTRACT +We present a novel simulation-based cosmological analysis of galaxy–galaxy lensing and galaxy redshift-space clus- +tering. Compared to analysis methods based on perturbation theory, our simulation-based approach allows us to +probe a much wider range of scales, 0.4 h−1 Mpc to 63 h−1 Mpc, including highly non-linear scales, and marginalises +over astrophysical effects such as assembly bias. We apply this framework to data from the Baryon Oscillation Spec- +troscopic Survey LOWZ sample cross-correlated with state-of-the-art gravitational lensing catalogues from the Kilo +Degree Survey and the Dark Energy Survey. We show that gravitational lensing and redshift-space clustering when +analysed over a large range of scales place tight constraints on the growth-of-structure parameter S8 = σ8 +� +Ωm/0.3. +Overall, we infer S8 = 0.792 ± 0.022 when analysing the combination of galaxy–galaxy lensing and projected galaxy +clustering and S8 = 0.771±0.027 for galaxy redshift-space clustering. These findings highlight the potential constrain- +ing power of full-scale studies over studies analysing only large scales, and also showcase the benefits of analysing +multiple large-scale structure surveys jointly. Our inferred values for S8 fall below the value inferred from the CMB, +S8 = 0.834 ± 0.016. While this difference is not statistically significant by itself, our results mirror other findings in +the literature whereby low-redshift large scale structure probes infer lower values for S8 than the CMB, the so-called +S8-tension. +Key words: cosmology: large-scale structure of Universe – cosmology: cosmological parameters – cosmology: dark +energy – cosmology: dark matter +1 INTRODUCTION +The large-scale structure (LSS) distribution in the low- +redshift Universe has emerged as one of the primary probes +of cosmology. LSS surveys such as the Baryon Oscillation +Spectroscopic Survey (BOSS; +Reid et al. 2016; Ahumada +et al. 2020), the Kilo Degree Survey (KiDS; +Giblin et al. +2021) and the Dark Energy Survey (DES; Gatti et al. 2021) +have provided some of the most stringent constraints on the +parameters of the cosmological standard model. In the com- +ing decade, LSS surveys such as the Dark Energy Spectro- +scopic Instrument (DESI; Abareshi et al. 2022) survey, the +Legacy Survey of Space and Time (LSST; The LSST Dark +⋆ email: julange.astro@pm.me +Energy Science Collaboration et al. 2018) on the Vera C. Ru- +bin Observatory, and the Nancy Grace Roman Space Tele- +scope (Spergel et al. 2015) will build upon this success and +provide even more powerful constraints on the cosmological +model. LSS surveys are particularly sensitive to Ωm, the frac- +tion of the energy density in matter, and σ8, the amplitude +of matter fluctuations on scales of 8 h−1 Mpc. There are two +promising avenues to constrain Ωm and σ8 from LSS observa- +tions: the deflection of light, so-called gravitational lensing, +by the LSS mass distribution and the clustering of matter +in redshift-space which is sensitive to peculiar velocities via +redshift-space distortions (RSDs). +Recently, several LSS probes of the low-redshift Universe +have reported tensions with respect to cosmological param- +eters preferred by the analysis of the high-redshift cosmic +microwave background (CMB) under the canonical Λ Cold +© 2023 The Authors +arXiv:2301.08692v1 [astro-ph.CO] 20 Jan 2023 + +2 +J. U. Lange et al. +Dark Matter (ΛCDM) model. Most often this tension is ex- +pressed in constraints on the cosmological parameter S8 = +σ8 +� +Ωm/0.3 and, hence, is called the “S8-tension”. Similar +to the well-known Hubble tension (Knox & Millea 2020), the +S8-tension could potentially point to new physics beyond the +standard ΛCDM cosmological model. Under ΛCDM, observa- +tions of the CMB by Planck Collaboration et al. (Planck2020, +2020) infer S8 = 0.834 ± 0.016 (TT,TE,EE+lowE), a value +that is 5 − 10% higher than the value preferred by LSS data. +Currently, the tension is 2 − 4σ significant with respect to +several LSS studies utilising gravitational lensing, while the +significance is lower for redshift-space clustering studies (see +Abdalla et al. 2022, for a review). While no study by itself can +claim a > 5σ tension between the low-redshift LSS and the +high-redshift CMB, the similarity of the findings of multiple, +independent LSS surveys using different techniques suggest +that the S8-tension might be a genuine cosmological tension. +Most often, cosmology studies of the LSS distribution fo- +cus on large scales where the statistical properties of the +matter distribution can be calculated analytically. Further- +more, on large scales, the relationship between the observed +galaxy distribution and the underlying dark matter density +field can be characterised by a small number of bias factors. +However, methods applicable to large, linear scales typically +break down on highly non-linear scales. In one approach to +extending LSS predictions to small scales, an analytical halo +model is used to make predictions for basic summary statis- +tics of the density field such as the matter power spectrum, +and the abundance and clustering of dark matter halos (Sel- +jak 2000). When augmented with additional ingredients for +the halo occupation statistics of galaxies, the halo model be- +comes a prediction pipeline for the galaxy distribution on +non-linear scales (Cooray & Sheth 2002; van den Bosch et al. +2013; Krause & Eifler 2017). One of the biggest challenges +faced by this approach is meeting the stringent demands for +percent-level accuracy with an analytic model (e.g., Tinker +et al. 2008; Hayashi & White 2008; Fedeli et al. 2014; Miy- +atake et al. 2022a; Mahony et al. 2022); however, steady im- +provements have been made over the last decade (Mead et al. +2015; Garc´ıa et al. 2021; Mead et al. 2021), and by now nu- +merous analyses have used such analytical methods to de- +rive constraints on cosmology from LSS measurements in the +non-linear regime (Cacciato et al. 2013; Reddick et al. 2014; +Tr¨oster et al. 2022). We will generically refer to this type of +analysis as a “full-scale” study, in contrast to analyses that +restrict attention to large scales only (see also Brieden et al. +2021, for the term “full-shape”). +In recent years, there has been tremendous progress in com- +putational power and statistical methods (Parejko et al. 2013; +Kwan et al. 2015; DeRose et al. 2019; Zhai et al. 2019; Lange +et al. 2019b; Nishimichi et al. 2019; Wibking et al. 2019, +2020; Lange et al. 2022; Miyatake et al. 2022b) such that +one can now directly compare predictions from simulations +against LSS observations for multiple cosmological models +and perform a rigorous Bayesian quantification of cosmolog- +ical parameters. These simulation-based methods differ from +conventional analytical halo models in that cosmological sim- +ulations are directly populated with synthetic galaxies, and +summary statistics are predicted using statistical estimators +of the resulting synthetic data. These simulation-based ap- +proaches simplify the task of incorporating systematic effects +such as galaxy assembly bias (e.g., Hearin et al. 2016) and +velocity bias (e.g., Guo et al. 2015a), and at the same time +enable a full-scale analysis to obtain stringent cosmological +constraints (Wibking et al. 2019; Zhai et al. 2019; Lange et al. +2022; Salcedo et al. 2022). Thus, simulation-based full-scale +studies have potential to exhaust the information content of +LSS two-point correlation functions using analyses with a de- +gree of complexity that is difficult to achieve with an analyt- +ical model. +Simulation-based full-scale studies have matured in recent +years to the point that several have been applied to actual +LSS data, demonstrating that the long-forecasted constrain- +ing power of full-scale studies can be realised. For exam- +ple, Wibking et al. (2020) use a full-scale approach to infer +S8 ≈ 0.712 ± 0.031 from a combination of galaxy clustering +and galaxy–galaxy lensing. In a similar study using updated +lensing data from the Hyper Suprime-Cam (HSC) survey, +Miyatake et al. (2022b) infer S8 = 0.795+0.049 +−0.042. Closely re- +lated to these works are studies of the so-called “lensing is +low” effect: when assuming the best-fit Planck2020 cosmol- +ogy and fitting a model for galaxy clustering, the measured +galaxy–galaxy lensing amplitude is overpredicted (Leauthaud +et al. 2017). It was shown that the measured lensing am- +plitude under Planck CMB parameters is around 15 − 35% +lower than predicted, particularly on small scales(Leauthaud +et al. 2017; Lange et al. 2021; Amon et al. 2023) . Since +the predicted lensing amplitude at fixed clustering is corre- +lated with S8 (Yoo et al. 2006), these findings can also be +seen as evidence of an S8-tension. Furthermore, several re- +cent studies (Lange et al. 2022; Chapman et al. 2022; Zhai +et al. 2022; Yuan et al. 2022) have applied a simulation-based +modelling framework to the analysis of redshift-space clus- +tering of galaxies. All four studies, using largely independent +galaxy samples, find a preference for lower values for cosmic +structure growth than the best-fit Planck Collaboration et al. +(2020) ΛCDM model predicts. +For completeness, we point out that already in 2013 a com- +bined full-scale analysis of projected galaxy clustering and +galaxy-galaxy lensing based on SDSS data by Cacciato et al. +(2013) yielded constraints on Ωm and σ8 in excellent agree- +ment with these more recent studies. However, at that point +in time, those constraints where consistent with the then +best-fit CMB constraints provided by the WMAP7 data (Ko- +matsu et al. 2011), and thus did not signal any tension. Ad- +ditionally, this study was based on an approximate analytical +halo model and did not capture the effects of assembly bias. +Similarly, Reid et al. (2014) performed a simulation-based +full-scale analysis of redshift-space distortions in BOSS and +also found a preference for a low structure growth amplitude. +However, this study relied on re-scaling velocities in a single +cosmological simulation which has been argued to lead to +inaccurate results (Zhai et al. 2019). +In this work, we built upon previous efforts by modelling +the redshift-space clustering and galaxy–galaxy lensing of +BOSS LOWZ galaxies. This work extends previous studies +in several ways. Compared to the lensing studies of Wibking +et al. (2020) and Miyatake et al. (2022b), we incorporate a +model for galaxy assembly bias which has been shown to be +important for modelling lensing on non-linear scales (Lange +et al. 2019a; Yuan et al. 2020). Furthermore, we measure and +analyse updated high-precision galaxy–galaxy lensing mea- +surements from the state-of-the-art DES Y3 and KiDS-1000 +data sets with reduced systematic uncertainties. Compared +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +3 +to the redshift-space clustering-only analysis of Lange et al. +(2022), the present work doubles the number of BOSS LOWZ +galaxies, analysing nearly the entire BOSS LOWZ sample. +Furthermore, we introduce and apply a new blinding (mask- +ing) methodology. Most importantly, the present study is the +first simulation-based full-scale and full-shape joint cosmolog- +ical analysis of galaxy-galaxy lensing and redshift-space clus- +tering. +Our paper is organised as follows. We start by introduc- +tion the observational data set and observabeles in section 2. +Our modelling approach is described in section 3 and veri- +fied on mock catalogues in section 4. The masking strategy is +described and tested in section 5. In section 6, we apply our +analysis technique to observations and present the cosmolog- +ical constraints. Finally, we discuss our results in section 7 +and section 8 presents our conclusions. +2 OBSERVATIONS +Here, we describe the observational data and the calculation +of the summary statistics used to derive cosmological con- +straints. +2.1 Observational data sets +Our primary data set is the BOSS LOWZ galaxy sample. +The clustering of BOSS LOWZ galaxies will be used as a +summary statistic in our modelling. Additionally, we measure +the galaxy–galaxy lensing effect around LOWZ galaxies using +gravitational lensing catalogues from KiDS and DES. +2.1.1 Baryon Oscillation Spectroscopic Survey +Galaxies in BOSS LOWZ are targeted for spectroscopic ob- +servations if they fulfil a series of magnitude cuts aimed at +selecting luminous red galaxies (LRGs) in the redshift range +0.15 ⩽ z ⩽ 0.5. In the following, cuts on apparent magnitudes +use cmodel magnitudes whereas colours are calculated using +model magnitudes. The cuts are as follows: +rcmod +< +13.5 + c∥ /0.3 +(1) +|c⊥| +< +0.2 +(2) +16 < +rcmod +< 19.6 , +(3) +where +c∥ = 0.7(gmod − rmod) + 1.2(rmod − imod − 0.18) +(4) +and +c⊥ = rmod − imod − (gmod − rmod)/4.0 − 0.18 . +(5) +The above colour cuts, which are based on apparent pho- +tometric magnitudes, select galaxies that primarily fall in +the redshift range, 0.15 ⩽ z ⩽ 0.5. However, because the +LOWZ target selection is based on apparent magnitudes, +LOWZ galaxies are not uniformly selected in redshift. Par- +ticularly, galaxies of similar intrinsic luminosities and colours +will have different apparent magnitudes depending on the +redshift. However, such a selection would contradict our mod- +elling approach, which implicitly assumes a volume-limited +sample of red galaxies. Thus, additional selection cuts are +needed to arrive at approximately volume-limited samples of +Property +Sample A +Sample B +Sample C +zref +0.25 +0.40 +0.40 +min. z +0.18 +0.30 +0.36 +max. z +0.30 +0.36 +0.43 +max. Mr +−20.412 +−20.558 +−21.208 +min. c0 +⊥ +−0.216 +−0.154 +−0.166 +max. c0 +⊥ +0.172 +0.234 +0.126 +max. Mr − c0 +∥/0.3 +−25.874 +−26.729 +−26.901 +volume V [Gpc3 h−3] +0.26/0.11 +0.22/0.10 +0.35/0.15 +ngal [10−4 Mpc−3 h3] +3.11/3.37 +3.13/3.14 +1.38/1.35 +Table 1. Definitions and properties of the samples analysed in this +work. All three samples are designed to be subsets of the BOSS +LOWZ sample that are roughly volume-limited in their respective +redshift ranges. In the above table, Mr is the absolute r-band +magnitude and the superscript 0 indicates rest-frame colours. Both +absolute magnitudes and colours are k-corrected to zref. The final +two rows indicate the volumes and galaxy number densities, split +by NGC and SGC areas of the sky. +galaxies. We follow Lange et al. (2022) and construct three +subsamples of the BOSS LOWZ galaxy sample in the redshift +ranges, 0.18 < z ⩽ 0.30, 0.30 < z ⩽ 0.36 and 0.36 < z ⩽ 0.43, +whose cuts are based on absolute magnitudes and rest-frame +colours, k-corrected to a reference redshift zref. The samples +and basic properties are described in Table 1. Sample A in the +Northern Galactic Cap (NGC) area is almost identical to the +low-redshift sample in Lange et al. (2022). Conversely, sam- +ples B and C correspond to the single high-redshift sample +in Lange et al. (2022) but combined they have roughly 60% +more galaxies per sky area. Additionally, compared to Lange +et al. (2022), we also analyse galaxies from the Southern +Galactic Cap (SGC) region. Taken together, these changes +roughly double the sample size compared to Lange et al. +(2022) to roughly 3×105 galaxies altogether. Note that due to +the slightly different photometric zero-points in the NGC and +SGC regions, we opt to model and analyse galaxy samples +from these two hemispheres separately. Finally, we point out +that we do not use the so-called LOWZE2 and LOWZE3 sam- +ples in the NGC that have relied on an incorrect star–galaxy +separation criterion (Reid et al. 2016). +2.1.2 Kilo Degree Survey +We use galaxies imaged by KiDS as so-called source galaxies +when measuring the galaxy–galaxy lensing effect. Specifically, +we use the KiDS-1000 data set covering roughly 1000 deg2 of +the extra-galactic sky, roughly half of which overlaps with the +BOSS survey footprint, with a source density ∼ 6 arcmin−2 +(Giblin et al. 2021). Galaxies in KiDS-1000 are grouped into +5 broad tomographic redshift bins. As described in Hilde- +brandt et al. (2021), the source redshift distribution n(z) of +each of the tomographic redshift bins has been derived using +self-organizing maps. We use the tabulated n(z) to convert +gravitational tangential shears into estimates of the excess +surface density ∆Σ, as described in section 2.3. +2.1.3 Dark Energy Survey +In addition to KiDS-1000, we also use the DES Y3 weak +lensing shape catalogues. DES Y3 covers ∼ 4000 deg2, out +of which roughly 800 deg2 overlap with BOSS (Amon et al. +MNRAS 000, 1–20 (2023) + +4 +J. U. Lange et al. +2023), with an effective source density of 6 arcmin−2 (Gatti +et al. 2021). Similar to KiDS-1000, DES Y3 source galaxies +are grouped into 4 tomographic redshift bins. Source red- +shift distributions n(z) are derived from a combination of +self-organising maps, small-scale shear ratios, clustering red- +shifts, (Myles et al. 2021) and take into account blending +effects in the photometry (MacCrann et al. 2022). +2.2 Galaxy clustering measurements +Galaxy clustering is characterised by the two-point correla- +tion function ξ(s, µ), which measures the excess probabil- +ity of having a pair of galaxies separated by s, the three- +dimensional separation and µ, the cosine of the angle between +s and the line of sight. We use the Landy—Szalay estimator +(Landy & Szalay 1993) and correct for fibre collisions using +the methodology presented in Guo et al. (2012). +Most of the information contained in the two-point corre- +lation function can be described by its multipole moments, +ξℓ(s) = 2ℓ + 1 +2 +1 +� +−1 +Lℓ(µ)ξ(s, µ)dµ , +(6) +where Lℓ represents the Legendre polynomial of order ℓ. We +use ℓ = 0, 2 and 4, the so-called monopole, quadrupole, and +hexadecapole moments of the redshift-space correlation func- +tion, respectively. The multipole moments contain informa- +tion about galaxy peculiar velocities due to redshift-space +distortions that change the apparent positions of galaxies +along the line of sight. Thus, the redshift-space clustering +of galaxies itself, even without gravitational lensing, contains +information about cosmic structure growth. We also use the +projected correlation function wp +wp(rp) = ++πmax +� +−πmax +ξ (s, π/s) dπ , +(7) +where πmax = 80 h−1 Mpc and s = +� +π2 + r2p. The projected +correlation function is nearly independent of galaxy pecu- +liar velocities (van den Bosch et al. 2013). By combining the +projected correlation function with the galaxy–galaxy lens- +ing amplitude we can probe cosmological constraints that are +practically independent of peculiar velocities. +Both ξ(s) and wp(rp) are measured in 14 comoving loga- +rithmic bins going from 0.1 h−1 Mpc to 101.8 ≈ 63 h−1 Mpc. +Uncertainties on the correlation function were derived from +jackknife-resampling of 100 roughly equal-area patches of the +BOSS LOWZ sample, separately for the NGC and SGC ar- +eas. The cross-covariances between different clustering mea- +sures at different radial bins as well as different multipole +moments of the redshift-space correlation function are taken +into account. We apply the smoothing procedure described +and tested in Lange et al. (2022) in order to suppress noise +in the resulting covariance matrix estimate. +2.3 Gravitational lensing measurements +In addition to the clustering properties of BOSS LOWZ +galaxies, we also measure the galaxy–galaxy lensing effect +around them. This is done by analysing the mean tangential +ellipticities et of source galaxies from KiDS and DES around +BOSS LOWZ lens galaxies. The mean tangential ellipticity +is related to the so-called excess surface density ∆Σ around +lens galaxies, defined via +∆Σ(rp) = ⟨Σ(< rp)⟩ − Σ(rp) , +(8) +where Σ denotes the surface mass density, rp the projected +distance in the frame of the lens galaxy, and the ⟨Σ(< rp)⟩ is +the mean surface density inside a circle of radius rp centred +on the lenses. The induced tangential ellipticity depends on +∆Σ and Σcrit defined as +Σcrit(zl, zs) = +c2 +4πG +1 +(1 + zl)2 +DA(zs) +DA(zl)DA(zl, zs) , +(9) +where zl and zs are the redshifts of the lens and source galaxy, +respectively, and DA denotes the angular diameter distance. +In the weak lensing regime, ∆Σ ≪ Σcrit, gravitational lensing +induces a tangential shear component, +γt = ∆Σ +Σcrit . +(10) +Galaxies have intrinsic ellipticities, such that et is a stochastic +measure of γt and one needs to stack a large number of lens– +source pairs. We use the following estimator for the mean +excess surface density of galaxies: +� +∆Σ = M +−1 [∆Σl − ∆Σr] . +(11) +In the above equation, M is an estimate of the mean mul- +tiplicative bias of the tangential ellipticity, ∆Σl is the un- +corrected estimate for ∆Σ around lens galaxies and ∆Σr the +analogous quantity around random points. In the absence of +systematics, the excess surface density around random points +is zero on average but subtracting this estimate also reduces +the variance of the ∆Σ estimator (Singh et al. 2017). The +mean multiplicative shear bias differs slightly between KiDS +and DES due to different shape measurement algorithms em- +ployed. For KiDS, our multiplicative shear bias estimate is +MKiDS = 1 + +� +ls wlsms +� wls +, +(12) +where m is the multiplicative shear bias that depends on the +source tomographic redshift bin (Giblin et al. 2021), the sum +� +ls goes over all suitable lens–source pairs separated by rp +and wls is a weight associated with each galaxy pair. For +DES, the shear bias correction factor is given by +MDES = +� +1 + +� +ls wlsms +� wls +� � +1 + +� +ls wls[RT + Rsel] +� wls +� +. +(13) +In the above equation, m is the multiplicative shear bias in- +duced by blending (MacCrann et al. 2022). Furthermore, RT +and Rsel are the tangential component of the metacalibra- +tion shear response and the selection response, respectively +(Huff & Mandelbaum 2017). Finally, the raw ∆Σ estimator +for lens galaxies is defined as +∆Σl = +� +ls wlset�Σcrit,ls +� +ls wls +, +(14) +where �Σcrit is an estimator of the intrinsic critical surface +density of each lens–source pair. We do not know the pre- +cise redshift for each individual source galaxy and, instead, +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +5 +have estimates of the normalised source redshift distribution +nˆb(z)1 in each tomographic source bin ˆb (Hildebrandt et al. +2021; Myles et al. 2021). In order to obtain unbiased lensing +amplitudes, we use +�Σcrit,ls = +� +� +∞ +� +0 +Σ−1 +crit(zl, zs)nˆb(zs)dzs +� +� +−1 +, +(15) +where nˆb(zs) are the normalised intrinsic redshift distribu- +tions in each tomographic photometric redshift bin (Hilde- +brandt et al. 2021; Myles et al. 2021). Finally, the lens–source +weights are designed to minimise shape noise, +wls = +ws +�Σ−2 +crit,ls +, +(16) +where ws is the source weight provided in the KiDS and DES +shape catalogues. All lensing calculations are performed with +the dsigma galaxy–galaxy lensing package (Lange & Huang +2022) version 0.7.0. Note that we do not apply a lens weight +wl and instead correct for fibre collisions using a nearest- +neighbour correction (Miyatake et al. 2015). We find that +the alternative weighting by wl = wnoz + wcp − 1 (see e.g. +Leauthaud et al. 2017) results in a ∼ 1% lower lensing signal +than our default choice. Thus, the choice of fibre correction +scheme does not significantly affect our conclusions. +We measure ∆Σ(rp) in 14 logarithmic bins going from +0.1 to ∼ 63 h−1 Mpc. Covariance matrices for the lensing +measurements are derived from jackknife re-sampling of 50 +roughly equal-area patches of overlap regions of BOSS LOWZ +with KiDS and DES. For the BOSS NGC area, we only use +the KiDS shape catalogue, i.e. ignoring the small overlap of +BOSS and DES. For the SGC area, we only use the DES +catalogues. We employ the same smoothing procedure as for +the clustering measurements to suppress noise in our covari- +ance matrix estimate. We do not take into account the cross- +covariance between clustering and lensing measuremens since +those are expected to be negligible (Taylor & Markoviˇc 2022), +especially when considering that the overlap regions of BOSS +with KiDS and DES constitute only a small part of the to- +tal BOSS footprint. Finally, we neglect systematic uncertain- +ties in the lensing measurements stemming from photometric +redshift calibration and multiplicative shear bias corrections. +These uncertainties are at most 1.5% (Amon et al. 2023) and +would translate into a ∼ 1% systematic uncertainty on S8, +significantly below statistical uncertainties presented in sec- +tion 6. +3 MODELLING +In this section, we describe our simulation-based modelling +framework which closely follows the one presented in Lange +et al. (2022). Thus, we only describe the most salient points +here and refer the reader to Lange et al. (2022) for a more +in-depth discussion. +1 By default, the DES Y3 redshift distributions are not normalised +in order to incorporate blending effects. In this work, we absorb +the normalisation into the multiplicative shear bias m, instead. +3.1 Cosmological simulations +We use the Aemulus cosmological simulations (DeRose et al. +2019) to make predictions for galaxy clustering and galaxy– +galaxy lensing. Aemulus is a suite of 40 simulations with dif- +ferent cosmological parameters. Each simulation traces struc- +ture formation in a wCDM Universe using (1400)3 particles +in a (1050 h−1 Mpc)3 cubic volume. As discussed in DeRose +et al. (2019), the resolution of these simulations is sufficient +to be used in the study of non-linear clustering of LRGs. +Dark matter haloes in the simulations are identified with the +Rockstar halo finder (Behroozi et al. 2013). In the follow- +ing, we will only use parent haloes, i.e. no subhaloes, with a +mass M at or above 100 times the particle mass mp, where +mp = 3.51 × 1010(Ωm/0.3)h−1M⊙. +3.2 Galaxy–halo connection model +We use a Halo Occupation Distribution (HOD; +Berlind & +Weinberg 2002; Bullock et al. 2002; Zheng et al. 2007) model +as the basis for our galaxy–halo connection. In particular, +we parameterise the average number of central and satellite +galaxies expected in a halo as a function of its mass M and +maximum circular velocity Vmax. The average number of cen- +tral galaxies as a function of halo mass M is given by +⟨Ncen|M⟩ = fΓ +2 +� +1 + erf +�log M − log Mmin +σlog M +�� +, +(17) +where fΓ, log Mmin and σlog M are free parameters. The pa- +rameter fΓ models incompleteness in the selection of LRGs +(Leauthaud et al. 2016). The average number of satellites is +given by +⟨Nsat|M⟩ = +�M − M0 +M1 +�α +(18) +with M0, M1, and α being free parameters. The depen- +dence on Vmax is modelled via the decorated HOD frame- +work (Hearin et al. 2016), a natural extension to the standard, +mass-only HOD approach. This is necessary to model the im- +pact of galaxy assembly bias (Gao et al. 2005; Wechsler et al. +2006; Zentner et al. 2014) and its degeneracy with cosmo- +logical parameters (Lange et al. 2019a,b; Yuan & Eisenstein +2019). In short, we perturb the average number of expected +galaxies in a halo of mass M based on whether Vmax is above +or below average of haloes at that mass. The perturbation is +described by +⟨Ncen|M, Vmax⟩ = ⟨Ncen|M⟩ ± Acen +�1 +2 − +���� +1 +2 − ⟨Ncen|M⟩ +���� +� +, +(19) +for centrals and +⟨Nsat|M, Vmax⟩ = (1 ± Asat)⟨Nsat|M⟩, +(20) +for satellites. In both cases, ± indicates + when Vmax is above +the median at that mass and − otherwise. This adds two +more free parameters, Acen and Asat, varying in the range +[−1, +1]. Once average galaxy numbers are specified, the dis- +tribution of galaxy numbers follow Bernoulli and Poisson dis- +tributions for centrals and satellites, respectively. We note +that several recent theoretical and observational studies in- +dicate the possibility of non-Poisson satellite number distri- +butions (see, e.g., Dvornik et al. 2022; Chaves-Montero et al. +MNRAS 000, 1–20 (2023) + +6 +J. U. Lange et al. +Parameter +Description +Range +log Mmin +low-mass cut-off for ⟨Ncen⟩ +[12.5, 14.0] +σlog M +low-mass transition for ⟨Ncen⟩ +[0.1, 1.0] +fΓ +incompleteness for ⟨Ncen⟩ +[0.5, 1.0] +log M0 +low-mass cut-off for ⟨Nsat⟩ +[12.0, 15.0] +log M1 +characteristic halo mass for ⟨Nsat⟩ +[13.5, 15.0] +α +power-law index for ⟨Nsat⟩ +[0.5, 2.0] +Acen +central galaxy assembly bias +[−1.0, 1.0] +Asat +satellite galaxy assembly bias +[−1.0, 1.0] +log η +satellite spatial bias +[−0.5, 0.5] +αc +central velocity bias +[0.0, 0.4] +αs +satellite velocity bias +[0.8, 1.2] +Table 2. All galaxy–halo connection parameters modelled in this +work together with a short description and flat prior ranges used +for fitting. +2022). In Lange et al. (2022), we showed that the assumption +of a non-Poisson satellite distribution did not significantly af- +fect cosmological constraints from redshift-space clustering. +Similarly, non-Poisson numbers are not expected to have a +substantial impact on the galaxy–galaxy lensing predictions +in the two-halo regime that we are modelling (see, e.g., Zu +2020; Lange et al. 2022; Chaves-Montero et al. 2022). +Centrals and satellites have different phase-space coordi- +nates. Central galaxies coincide spatially with the halo centre +but have additional random Gaussian velocities along the line +of sight (Reid et al. 2014; Guo et al. 2015a,b) with scatter σ, +σ = αcVvir +√ +3 +. +(21) +This simulates the velocities of central galaxies with respect +to the halo centre due to the unrelaxed state of the halo (Ye +et al. 2017). The free parameter αc is commonly known as +the central velocity bias parameter. Satellite galaxies follow +a Navarro–Frenk–White (NFW; Navarro et al. 1997) profile +with respect to the halo centre, +n(r) ∝ +1 +r/rs (1 + r/rs)2 . +(22) +The concentration parameter csat = rs/rh, where rh is the +host halo radius, is allowed to be different than that of the +dark matter distribution in the same halo, cdm, +csat = ηcdm . +(23) +Similar to central galaxies, satellites follow the halo velocity +on average. We add additional velocities with respect to the +halo by drawing from a Gaussian distribution with scatter de- +rived from the spherically symmetric, anisotropy-free Jeans +equation. The derived satellite velocities are likely an approx- +imation since satellite populations will not be perfectly spher- +ically symmetric, without velocity anisotropy or dynamically +relaxed. Thus, we apply an additional free scaling factor αs +(the satellite velocity bias parameter) to the velocity scatter +derived from the Jeans equation. Overall, the phase-space co- +ordinates of galaxies add an additional three free parameters, +αc, αs, and η. Ultimately, our model for galaxies has 11 free +parameters, which we list in Table 2 for reference, that we +allow to vary when fitting for cosmology. +3.3 Data likelihood +We use Halotools (Hearin et al. 2017) to compute galaxy +clustering and lensing observables for a given simulation with +cosmological parameters C and galaxy–halo connection pa- +rameters G. For sample A, we model the observables using +simulation outputs at redshift 0.25 whereas for samples B +and C, we use z = 0.40 snapshots2. We make use of the dis- +tant observer approximation and in all cases project galaxy +populations onto each of the three simulation axes. When +making predictions for galaxy clustering, we take into ac- +count the Alcock–Paczynski effect (Alcock & Paczynski 1979) +by correcting simulation coordinates for the assumed cosmol- +ogy when obtaining the clustering measurements in section 2. +Similarly, we correct the ∆Σ predictions for the assumed cos- +mology following the methodology described in More (2013) +and approximating ∆Σ ∝ r−1 +p . To speed up the calculation of +the clustering and lensing predictions for a given Aemulus +simulation, we make use of a tabulation method for galaxy +correlation functions (Zheng & Guo 2016) as implemented in +TabCorr3 version 1.0.0. For a given clustering and lensing +prediction, the likelihood L of the observed data vector D is +computed as +ln L(D|C, G) = 1 +2 +� +(ngal − ˆngal)2σ−2 +ngal + (ξ − ˆξ)TΣ−1 +ξ (ξ − ˆξ) ++(∆Σ − � +∆Σ)TΣ−1 +∆Σ(∆Σ − � +∆Σ) +� +, +(24) +where ξ denotes the combination of all galaxy clustering mea- +surements, e.g. ξ0, ξ2 and ξ4 or just wp. +We use three different choices of data sets D. The first set, +which we call “RSD-only” consists of the three redshift-space +clustering multipole moments, ξ0, ξ2, and ξ4, and no lensing +measurements. The second set labelled “wp + ∆Σ” combines +the projected correlation function wp with galaxy–galaxy +lensing ∆Σ. Finally, the set called “RSD + ∆Σ” consists of +the redshift-space clustering multipole moments and the lens- +ing amplitude. All clustering measurements, ξ0, ξ2, ξ4, and +wp are fitted on scales larger than 0.4h−1 Mpc, as discussed +and motivated in Lange et al. (2022). Contrary, for lensing, +we only consider scales 2.5 h−1 Mpc < rp < 25h−1 Mpc. The +lower limit is chosen to avoid biases associated with baryonic +feedback (Leauthaud et al. 2017; Lange et al. 2019a; Amodeo +et al. 2021) which we do not model in this analysis. The up- +per limit is chosen in order to avoid biases in the covariance +matrix estimate related to the size of the jackknife fields (Shi- +rasaki et al. 2017). +We note that for all three choices of two-point correlation +functions, we use the number density of galaxies, ngal, as a +constraint. This observable tightly limits one dimension of +2 For sample B, there is an apparent mismatch between the mean +redshift of the sample, z = 0.33, and the redshift of the simulation +output used to analyse it, z = 0.40. As shown in Lange et al. +(2022), this should have a negligible impact on studies using RSDs +since these observations are primarily sensitive to f(z)σ8(z), which +evolves very slowly with redshift. However, the predicted lensing +signal at fixed clustering roughly scales as σ8(z) (Yoo et al. 2006). +Thus, to account for the redshift mismatch, we multiply lensing +predictions for sample B by σ8(0.33)/σ8(0.40) = 1.04. +3 https://github.com/johannesulf/TabCorr +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +7 +the HOD parameter space due to the requirement that to- +tal number density of galaxies matched the observed one. At +the same time, the galaxy–halo connection model has sig- +nificant flexibility to change the predicted number density +without affecting the clustering predictions. For example, the +predictions for the two-point correlation functions are unaf- +fected if the number of galaxies in all halos was changed by +constant factor. Consequently, if we replace fΓ → x fΓ and +M1 → x−1/α M1, the number density prediction changes to +ˆngal → x ˆngal while leaving the clustering and lensing pre- +dictions unaffected4. Ultimately, we do not expect that the +number density constraint significantly affects the cosmology +result since neither fΓ nor log M1 are tightly constrained by +the observational data or pile up against the prior ranges, i.e. +fΓ ⩽ 1. +Equation (24) describes the likelihood of any simulation +and its underlying cosmology C as a function of galaxy pa- +rameters G. However, ultimately, we want to obtain a con- +straint on C alone for which we need L(D|C). Therefore, we +first have to marginalise the data likelihood over the galaxy +parameters G. In Lange et al. (2019b) and Lange et al. (2022), +L(D|C) is the evidence Z(D|C), +Z(D|C) = +� +L(D|C, G)P(G)dG . +(25) +As an alternative, one can also use the profile likelihood, +Lp(D|C) = max +G +L(D|C, G) , +(26) +i.e. the maximum likelihood obtained over all galaxy–halo +connection parameters, as a replacement for the evidence. +The advantage of this approach is that L(D|C) becomes inde- +pendent of poorly motivated priors on the galaxy model. We +will compare the performance of both approaches in section +4. To calculate the evidence, maximum likelihood and poste- +riors on galaxy–halo connection parameters for each simula- +tion, we employ the nautilus5 sampler version 0.2.1 (Lange, +in prep.). We use 3000 live points, discard points during the +exploration phase and run the sampler until an effective sam- +ple size of 100, 000 is achieved. +3.4 Cosmological inference +Once we have computed the summary statistic, either the ev- +idence Z or the profile likelihood Lp, we have to characterise +the dependence of those summary statistics on the cosmol- +ogy of each simulation. Full-scale redshift-space clustering is +sensitive to f(z)σ8(z) (Lange et al. 2022), where f is the lin- +ear growth rate. Conversely, projected galaxy clustering and +galaxy–galaxy lensing are sensitive to both Ωm and σ8(z) +(Yoo et al. 2006). Since our analysis exhibits joint depen- +dence upon these three cosmological parameters, we elect to +model the summary statistic also as a function of three pa- +rameters that fully specify f(z), σ8(z), and Ωm. Here, we +choose these three cosmological parameters to be S8, Ωm, +and w, the Dark Energy equation of state. We assume that +the dependence of Z and Lp on S8, Ωm, and w can be mod- +elled as a multi-variate skew-normal distribution (Azzalini & +4 This is strictly true only for Acen = 0. However, observations do +not tightly constrain Acen and it is always consistent with 0. +5 https://github.com/johannesulf/Nautilus +Valle 1996). For reference, in one dimension, the probability +distribution function (PDF) of a skew-normal distribution is +given by, +f(x) = 2 +σ φ +�x − µ +σ +� +Φ +� +λx − µ +σ +� +, +(27) +where φ and Φ are the PDF and cumulative distribution func- +tion (CDF) of the standard normal, respectively. This func- +tional form is motivated empirically (Lange et al. 2019b) and +is a natural extension of the assumption of a Gaussian pos- +terior. A three-dimensional skew-normal has 12 free param- +eters: three means µ, three standard deviations ˆσ = σ/(1 + +σ) ∈ [0, 1], three skew parameters δ = λ/ +√ +1 + λ2 ∈ [−1, +1] +as well as three parameters for the off-diagonals of the co- +variance matrix, i.e. rS8,Ωm, rS8,w, and rΩm,w ∈ [−1, +1]. +We obtain our full posterior constraints on S8, Ωm, and +w as follows. Beginning with the collection of 406 values +for Z(D|C) or Lp(D|C) (depending on which we use for the +summary statistic), we approximate the distribution of these +values with a skew-normal distribution and use an MCMC +to derive full posterior distributions on the 12 skew-normal +hyper-parameters plus one additional free parameter for the +likelihood scatter of each simulation. Each point in this 13- +dimensional hyper-parameter space represents a particular +skew-normal approximation to the cosmology-dependence of +our summary statistic. We draw N samples of our hyper- +parameters based on the posteriors on these quantities, and +we derive our constraints on S8, Ωm and w, i.e. P(S8, Ωm, w) +based on the superposition of the resulting collection of nor- +malised skew-normals. As in Lange et al. (2022), we place ex- +plicit flat priors on cosmological parameters, in this case S8, +Ωm and w. This is done to take into account that we cannot +reliably extrapolate the dependence of Z and Lp on parts of +the cosmological parameter space not probed by the Aemu- +lus simulations. The prior is determined by placing a three- +dimensional minimum-volume enclosing ellipsoid around the +parameter combinations of the Aemulus simulations. Com- +binations of S8, Ωm, and w outside this ellipse are assigned 0 +prior probability. We refer the reader to Lange et al. (2022) +for a detailed discussion and motivation of the fitting proce- +dure described here. In order to verify that our conclusions +are robust to our assumption that the cosmology-dependence +of Z and Lp is well-described by a skew-normal form, in +appendix A we conduct an alternate analysis in which we +alternatively use a Gaussian Process regression to approxi- +mate these distributions, finding very similar results. Finally, +we note that our final constraints on cosmology should not +be regarded as being model-independent or valid outside the +wCDM cosmological parameter space probed by the Aemu- +lus simulations. +4 VERIFICATION ON MOCK CATALOGUES +Before applying our analysis method to the observational +data described in section 2, we test our methods on mock +6 In practice, we always exclude the simulation Box023 since it +has an unusually low S8 = 0.595, well below the second lowest +value of 0.703. The motivation is that this simulation is almost +always confidently excluded by observations and we do not want +its very low likelihood to influence the fit to L(D|C) in regions of +cosmological parameter space allowed by observations. +MNRAS 000, 1–20 (2023) + +8 +J. U. Lange et al. +catalogues to ensure that our cosmological inferences do not +suffer from significant biases. +4.1 Mock observations +The mock catalogues used here are described in more de- +tail in Lange et al. (2022). They are created from the UNIT +simulations (Chuang et al. 2019) and the associated Rock- +star halo catalogues. We populate haloes in the z = 0.25 +outputs of each of the four UNIT simulations with galaxies +using the subhalo abundance matching framework (SHAM; +Vale & Ostriker 2004; Conroy et al. 2006). In essence, we +place galaxies at the centres of the haloes with the highest +α log Vpeak + (1 − α) log Vvir, where Vvir is the halo virial ve- +locity at the epoch of peak halo mass, Vpeak the correspond- +ing value of Vmax at that epoch and α = 0.73 (Lehmann +et al. 2017). We populate haloes until the number density of +galaxies matches that of the LOWZ 0.18 ⩽ z ⩽ 0.30 sample. +Haloes in this procedure include both parent haloes as well as +subhaloes such that satellite galaxies are naturally accounted +for. As described in Lange et al. (2022), small modifications +to the SHAM procedure are applied to account for scatter +between galaxy and halo properties. Once mock galaxy cat- +alogues are created, we compute mock observables assuming +a distant observer approximation. Since we have four simu- +lations of 1 Gpc3 h−3 volume each, and we consider all three +simulation axes as the line of sight, the mock measurements +have an effective volume of ∼ 12 Gpc3 h−3 (Smith et al. 2021). +The mock observations for galaxy clustering and galaxy– +galaxy lensing are displayed in Fig. 1. Error bars represent +the observational uncertainties of the 0.18 ⩽ z ⩽ 0.30 NGC +(SGC) sample for galaxy clustering (galaxy–galaxy lensing). +In the following, we will use these same observational uncer- +tainties, i.e. covariance matrix, to fit our model to the mock +data set. The best-fit model predictions, marginalised over all +galaxy parameters and all 40 simulations, are shown in Fig. +1 by the solid lines. Overall, our HOD-based galaxy model, +which is an empirical model that is founded upon different as- +sumptions than the SHAM model used to produce the mock +catalogues, can produce good fits to the mock data. +4.2 Cosmology results +Cosmological parameter constraints for the simulated dataset +are shown in Figs. 2 and 3. In each of the figures, red lines +indicate posterior constraints when fitting both RSDs and +lensing, i.e. ξ0, ξ2, ξ4 and ∆Σ. Blue lines indicate posterior +constraints when only redshift-space clustering is considered +in the fit, and purple lines indicate results obtained without +redshift-space clustering. Since galaxy–galaxy lensing is only +sensitive to cosmology in combination with galaxy clustering +(Yoo et al. 2006), we also include the projected two-point +correlation function, wp, in that particular fit. Since wp is +obtained by projecting the redshift-space two-point correla- +tion function along the line of sight, it is mostly insensitive to +redshift-space distortions. As shown in Lange et al. (2022), +wp by itself has little cosmological information. +The difference between Figs. 2 and 3 is that the former +uses the evidence Z as the summary statistic, whereas the +latter uses the profile likelihood Lp. We see that our analysis +places strong constraints on S8, as expected. Both redshift- +space clustering and galaxy–galaxy lensing obtain roughly +0 +20 +40 +r1.5 ξ [h−1.5 Mpc1.5] +ξ0 ξ2 ξ4 +Mock +0.1 +1.0 +10 +s [h−1 Mpc] +0 +2 +δξ +8 +10 +12 +rp ∆Σ [106M⊙/pc] +Mock +0.1 +1.0 +10 +rp [h−1 Mpc] +-3 +0 ++3 +δ∆Σ +Figure 1. Mock observations of galaxy clustering in redshift space +(top) and galaxy–galaxy lensing (bottom). Dots indicate the mock +measurements themselves, error bars the assumed observational +uncertainty, and the solid line is the best-fit model prediction. The +lower panels show the difference between mock observations and +best-fit model predictions in units of the observational uncertainty. +Grey backgrounds indicate the ranges of the small-scale data that +are not included in the fit. +comparable constraints on S8 with slightly different degen- +eracies with respect to the other cosmological parameters. +On the other hand, neither RSDs nor lensing lead to strong +constraints on Ωm, i.e. the posterior PDF is very similar to +the prior. Finally, we nominally get noteworthy constraints +on w. However, a significant fraction of this constraint may +originate indirectly from the constraint on S8. Specifically, +our assumed prior implies a strong correlation between S8 +and w such that any constraint on S8 also leads to a strong +constraint on w, as is evident from the lower left panels in +each of the two figures. +We find that both approaches, either employing the evi- +dence or the profile likelihood, are successful in recovering +the input cosmology of the UNIT simulations. Additionally, +the fit using the evidence as the summary statistic produces +quantitatively similar results to the analysis using the pro- +file likelihood. As described before, the effective volume of +the mock observations is roughly ∼ 12 Gpc3 h−3, a factor +of ∼ 40 larger than the observational volume from which +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +9 +wp + ∆Σ +RSD-only +RSD + ∆Σ +truth +prior +0.28 +0.32 +0.36 +Ωm +0.60 +0.75 +0.90 +S8 +−1.2 +−0.9 +−0.6 +w +0.28 +0.32 +0.36 +Ωm +−1.2 +−0.9 +−0.6 +w +Figure 2. Posterior constraints on cosmological parameters when +analysing the mock data set derived from the UNIT simulations. +We show the results from analysis of projected clustering and +galaxy–galaxy lensing (purple), the study of redshift-space mul- +tipoles (blue), and the combination of the redshift-space cluster- +ing and galaxy–galaxy lensing (red). In all panels we highlight the +cosmological parameters of the UNIT simulations (yellow) we seek +to recover. Finally, the grey lines indicate our prior: in the off- +diagonal panels it shows the range and in the diagonal elements +the implicit one-dimensional prior implied by projecting the vol- +ume of a three-dimensional ellipsoid onto one axis. For this figure, +the evidence Z was used as the summary statistic for each of the +40 Aemulus simulations. +wp + ∆Σ +RSD-only +RSD + ∆Σ +truth +prior +0.28 +0.32 +0.36 +Ωm +0.60 +0.75 +0.90 +S8 +−1.2 +−0.9 +−0.6 +w +0.28 +0.32 +0.36 +Ωm +−1.2 +−0.9 +−0.6 +w +Figure 3. Similar to Fig. 2 but with the profile likelihood Lp +instead of the evidence used as the summary statistic. +the assumed observational uncertainties come from. Thus, +we expect our results to reproduce the input to much bet- +ter than the assumed observational uncertainty. With this in +mind, it is noteworthy that the posterior constraints on S8 +are slightly off-centred when fitting the evidence for either +the case of redshift-space clustering or galaxy–galaxy lens- +ing in isolation; this is not the case for the fit involving the +profile likelihood. Moreover, the profile likelihood has a com- +paratively stronger theoretical motivation since it is indepen- +dent of the arbitrary priors on the galaxy–halo connection. +For these reasons, we will choose the profile likelihood as the +default summary statistic when analysing the real data in +section 6. +5 MASKING STRATEGY +For scientific studies, we want to protect the experiment and +analysis design from confirmation bias of the scientists lead- +ing the analysis. A common strategy to avoid this issue is +to “blind” the analysis with respect to the key scientific re- +sults while still being able to perform critical null tests. In +our example, we wish to make analysis choices insensitive to +the cosmological constraints, i.e. S8, while still being able to +judge, for example, the goodness of fit of our model. Through- +out this work, we will use the term “masking” instead of the +more commonly used term “blinding”. +5.1 Proposed method +A masking procedure can in principle happen at various +stages of the analysis. In the case of posterior masking, one +would simply hide or randomly perturb the final cosmological +posterior. Data vector masking involves perturbing the sum- +mary statistics, i.e. ξ and ∆Σ, in ways that also randomly +perturb the final cosmological result. Finally, masking can be +implemented at the catalogue level such that both summary +statistics and cosmological posterior are affected. Data vec- +tor and catalogue level masking are harder to implement than +posterior masking but are more robust in the sense that they +are less prone to accidental unmasking. We refer the reader +to Muir et al. (2020) for a detailed discussion and motiva- +tion behind different approaches. In this work, we will apply +a data vector masking procedure. +The method employed here is a variation of the data vector +masking procedure described in Muir et al. (2020). Let us +assume that we have an unperturbed data vector D and a +model for that data vector �D(θ) where θ represents the model +parameters we wish to constrain with the analysis. Muir et al. +(2020) propose perturbing D by +∆D(∆θ) = �D(θfid + ∆θ) − �D(θfid) , +(28) +where ∆θ is an offset in the model and θfid are suitably chosen +set of fiducial model parameters. Under idealised conditions, +adding ∆D(∆θ) to the unperturbed data vector will shift the +posterior of θ by ∆θ while leaving the goodness of fit, i.e. the +minimum χ2, unchanged. In practice, these idealised condi- +tions are not perfectly met, such that the above statements +are only approximately true and the masking procedure needs +to be validated with simulations first (Muir et al. 2020). +In our analysis, we wish to mask the final constraints on S8. +Constraints on the galaxy–halo connection are of less interest +MNRAS 000, 1–20 (2023) + +10 +J. U. Lange et al. +10−1 +100 +101 +Distance s [h−1 Mpc] +−20 +−10 +0 +10 +20 +30 +(d �D/dS8)/σD +ξ0 +ξ2 +ξ4 +∆Σ +wp +Figure 4. Estimates of the derivatives of the best-fit predictions +as a function of S8. The derivatives are derived from the mock +data RSD fits presented in Lange et al. (2022). +in this study and could be described as nuisance parameters. +The original method proposed in Muir et al. (2020) would +calculate ∆D(∆θ) by looking at the difference in predictions +for two cosmologies with different S8 values while keeping the +galaxy–halo connection parameters fixed. We may choose to +perturb S8 by up to ∆S8 = 0.1, which, as we will show later, +would correspond to up to 4σ shifts in the final S8 posterior. +However, the impact of cosmology and galaxy–halo connec- +tion parameters on the data vector is often degenerate. For +example, the large-scale clustering amplitude is sensitive to +bσ8 where b is the galaxy bias. As a result, implementing the +method described in Muir et al. (2020) could result in data +vector shifts that are large with respect to the observational +uncertainties, i.e. significantly larger than 4σ. +Here, we propose a slight variation of the method described +in Muir et al. (2020). When calculating ∆D, instead of only +changing S8 while keeping all other model parameters fixed +we instead change S8 and then vary all other model parame- +ters such that the difference between �D(θfid+∆θ) and �D(θfid) +is minimised. Here, we define the difference between the data +vectors as their χ2 difference. This ensures that the shift in +the data vector is as small as possible while ensuring the de- +sired shift in S8. For example, a 4σ shift in S8 would result +in a roughly 4σ significant shift of the data vector. Overall, +this procedure seems more likely than the original Muir et al. +(2020) method to preserve the goodness of fit after masking. +5.2 Specific implementation +We now want to apply the above mentioned method to +our application by masking the value of S8. However, in +simulation-based modelling, we can only make predictions +for a handful of cosmologies and the predictions are inher- +ently noisy due to finite simulation sizes. To overcome these +problems, we slightly modify the method introduced above +while keeping the main idea. In the mock analysis in Lange +et al. (2022), we fit our model to a mock RSD vector and get +best-fit model predictions �D for all 40 simulations. We then +use the 40 predictions to fit for a linear relation between �D +and the input S8 value to estimate d �D/dS8. +0.6 +0.7 +0.8 +0.9 +1.0 +S8 +Posterior +∆S8 +χ2 Change +0.2 → 0.1 +4.4 → 6.3 +5.0 → 6.6 +wp + ∆Σ +RSD-only +RSD + ∆Σ +Figure 5. Change in the S8 posterior constraints from the mock +data set when applying the masking procedure outlined in sec- +tion 5. We show the original input S8 value (yellow dashed verti- +cal line) and the expected S8 shift ∆S8 = −0.1 (black arrow). As +expected, the posterior constraints obtained from the masked data +(solid lines) are shifted by roughly −0.1 compared to the same con- +straints obtained from the unmasked data (dashed lines). In the +upper right corner, we also indicate the change in the goodness of +fit due to applying the masking. +The resulting derivatives are shown in Fig. 4 and discussed +further in section 7.3. As expected, the figure indicates that +S8 and ∆Σ predictions are positively correlated. To mask S8, +we add the following offset to the data vector D: +∆D(∆S8) = d �D +dS8 × ∆S8 . +(29) +We test the above masking procedure on the mocks in the +previous section. We choose a target shift of ∆S8 = −0.1. In +Fig. 5, we show the shift in the S8 posterior induced by the +this choice. As expected, irrespective of whether we analyse +wp and ∆Σ, the redshift-space correlation function or the +combination of lensing and redshift-space clustering, the S8 +posterior shifts by roughly −0.1 in S8. Furthermore, as shown +in the same figure, the goodness of fit is close to unaffected by +the masking procedure. These findings on mock catalogues +indicate that the masking procedure is expected to give a +good performance in practical applications such as the one +in the present work. +We then proceed to apply the masking procedure to the +data. For each of the three redshift bins, we choose a random +∆S8 that is drawn from a uniform distribution in the range +[−0.075, +0.075]. In principle, larger ranges would be ideal +to erase any meaningful correlation between the masked and +unmasked result. However, for our simulation-based analy- +sis, we need to ensure that the value of S8 that the masked +data prefers is covered by the simulations. Thus, we cannot +make the range for ∆S8 arbitrarily large. The range chosen +here was selected as a compromise. Note that we apply the +same S8 masking shift for all analyses within each redshift +bin. Therefore, even with the masking, we were able to judge +the consistency between constraints coming from RSDs and +lensing as well as between results from the NGC and the +SGC data. The entire analysis in the next section was first +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +11 +performed and checked with the masked data and only un- +masked after all authors agreed on all analysis choices. +6 RESULTS +We now proceed to apply the modelling framework to the ob- +servational galaxy clustering and galaxy–galaxy lensing data. +Here, we concentrate on posterior constraints on cosmology. +Constraints on the galaxy–halo connection are presented and +discussed in appendix B. +6.1 Model fits +In Fig. 6, we show the best-fit model predictions for each +of the six samples. Each model was fitted to the redshift- +space clustering and galaxy–galaxy lensing data jointly. The +fits were marginalised over both the galaxy–halo connection +parameters as well as over the 40 Aemulus simulations, i.e. +cosmology. The minimum χ2 values are 23.0, 39.2 and 26.2 +(32.5, 18.5 and 31.0) for the three bins from low to high red- +shift in the NGC (SGC) with 39 data points. When fitting all +data with a single simulation, as shown in Fig. 6, the best-fit +χ2 is 185. Although our HOD model has 11 free parameters, +the number of effective degrees of freedom of the galaxy–halo +connection is smaller. We numerically determine this number +by taking model predictions, randomly perturbing them ac- +cording to the observational uncertainty and minimising χ2 +over HOD parameter space. We find that the number of ef- +fective degrees of freedom of the HOD model with respect to +fitting redshift-space clustering and lensing jointly is ∼ 8.5. +Similarly, while we have seven cosmological parameters that +are varied in the Aemulus simulations, the likelihood seems +to be only a function of around two. Thus, we estimate to +have around ∼ 2 effective degrees of freedom in cosmology. +Overall, when fitting redshift-space clustering and lensing, +the number of effective degrees of freedom Ndof is approxi- +mately 39.0 − 8.5 − 2.0 = 28.5 when fitting a single sample +of galaxies and 6 × (39.0 − 8.5) − 2.0 = 181 when fitting +all six galaxy samples. In Fig. 7, we show the distribution +of best-fit χ2 values for the mocks when fitting all six sam- +ples to redshift-space clustering and galaxy–galaxy lensing +together with the best-fit χ2 from the data. Overall, we find +χ2 +ν = χ2/Ndof ≈ 1 for all samples. The largest χ2 +ν value, +39.2/28.5, has a p-value of 0.09. Thus, overall, our model +provides a good fit to all the available data. +6.2 Cosmology +Fig. 8 shows our constraints on the cosmological parame- +ter S8. Results are presented for all six samples individu- +ally. We also distinguish between redshift-space clustering- +only fits, the combination of projected clustering and galaxy– +galaxy lensing, and joint redshift-space clustering and lens- +ing fits. Overall, the different samples show good agreement +for the value of S8. Furthermore, the RSD-only fits and +the joint projected clustering plus lensing fits are also in +good agreement. When combining all six galaxy samples, we +obtain S8 = 0.792 ± 0.022 from gravitational lensing and +S8 = 0.771 ± 0.027 from redshift-space clustering. We also +investigate how the redshift-space clustering result depends +on the scales considered. By default, we consider all scales +sample +wp + ∆Σ +RSD-only +RSD + ∆Σ +A NGC +0.815 ± 0.049 +0.813 ± 0.043 +0.807 ± 0.038 +A SGC +0.818 ± 0.041 +0.783 ± 0.048 +0.810 ± 0.035 +B NGC +0.735 ± 0.052 +0.774 ± 0.037 +0.754 ± 0.035 +B SGC +0.776 ± 0.049 +0.750 ± 0.056 +0.761 ± 0.045 +C NGC +0.746 ± 0.060 +0.789 ± 0.048 +0.763 ± 0.043 +C SGC +0.802 ± 0.042 +0.727 ± 0.058 +0.798 ± 0.045 +combined +0.792 ± 0.022 +0.771 ± 0.027 +0.779 ± 0.020 +Table 3. Posterior constraints on the cosmological parameter S8 +as a function of the galaxy sample analysed (different rows) and +the observational constraints (different columns). +larger than 400 h−1 kpc. When only analysing scales larger +than 6.3 h−1 Mpc, we obtain S8 = 0.806 ± 0.042. Finally, +when looking at the combination of reshift-space clustering +and lensing for all six samples, we obtain S8 = 0.779 ± 0.020. +The derived constraints on S8 are also listed in Table 3, for +convenience. In Fig. 9, we show the full cosmology constraints +on S8, Ωm and w when considering all six samples. In ad- +dition to constraints on S8, we infer w = −0.915 ± 0.113, +−0.967 ± 0.076 and −0.963 + / − 0.069 for observations of +wp + ∆Σ, RSD-only and RSD +∆Σ, respectively. On the +other hand, we do not obtain strong constraints on Ωm and +are instead dominated by the Aemulus prior. +6.3 Lensing amplitudes derived from different +lensing data sets +Here, we compare the measured galaxy–galaxy lensing am- +plitudes between KiDS and DES. Additionally, we contrast +them with galaxy–galaxy lensing measurements obtained +with SDSS lensing catalogues (Singh et al. 2019). In this sec- +tion, we use lensing measurements that extend to smaller +radial scales than used in the fiducial cosmology analysis. +The lensing amplitude ∆Σ is a physical quantity that should +only depend on lens properties and be independent of the +shape catalogue used. In a recent study, Leauthaud et al. +(2022) used this insight to compare the results of different +lensing catalogues, including SDSS, KiDS and DES, finding +good agreement between all of them within the statistical and +systematic uncertainties. However, the findings of Leauthaud +et al. (2022) are based on older DES Y1 and KiDS-450 data +whereas our new DES and KiDS lensing measurements have +substantially more statistical constraining power and reduced +systematics. We compare the SDSS, DES and KiDS lensing +measurements in Fig. 10. The lensing measurements used for +our cosmology analysis are limited to scales where boost fac- +tors corrections due to physical lens–source associations are +unimportant. Here, we extend the measurements to smaller +radial scales and, thus, include boost factor corrections. Over- +all, we see that the DES Y3 lensing measurements tend to be +higher than the SDSS and KiDS measurements. To estimate +the statistical significance of the difference in the lensing am- +plitudes, we follow the approach of Leauthaud et al. (2022). +In particular, we first determine an overall lensing normal- +isation A by fitting the observed lensing amplitude ∆Σobs +with a template ∆Σtemplate. Afterwards, we compare the nor- +malisations between the different lensing amplitudes. For the +template, we choose the best-fit lensing prediction when ana- +lyzing the combination of galaxy redshift-space clustering and +MNRAS 000, 1–20 (2023) + +12 +J. U. Lange et al. +0 +20 +40 +r1.5 ξ [h−1.5 Mpc1.5] +A-N +ξ0 ξ2 ξ4 +B-N +ξ0 ξ2 ξ4 +C-N +ξ0 ξ2 ξ4 +-3 +0 ++3 +δξ +0 +20 +40 +r1.5 ξ [h−1.5 Mpc1.5] +A-S +ξ0 ξ2 ξ4 +B-S +ξ0 ξ2 ξ4 +C-S +ξ0 ξ2 ξ4 +0.1 +1.0 +10 +s [h−1 Mpc] +-3 +0 ++3 +δξ +0.1 +1.0 +10 +s [h−1 Mpc] +0.1 +1.0 +10 +s [h−1 Mpc] +0 +5 +10 +15 +rp ∆Σ [106M⊙/pc] +A-N +B-N +C-N +-3 +0 ++3 +δ ∆Σ +0 +5 +10 +15 +rp ∆Σ [106M⊙/pc] +A-S +B-S +C-S +0.1 +1.0 +10 +rp [h−1 Mpc] +-3 +0 ++3 +δ ∆Σ +0.1 +1.0 +10 +rp [h−1 Mpc] +0.1 +1.0 +10 +rp [h−1 Mpc] +Figure 6. Data measurements and best-fit model predictions when analysing galaxy clustering and galaxy–galaxy lensing jointly. Upper +panels show the clustering measurements and fits whereas the lower panels display the lensing measurements and models. Each panel +indicates the sample displayed, e.g. “A-S” denotes sample A in the SGC area. For each set, the upper panels show the absolute measure- +ments (data points) and predictions (solid lines) and the lower panels show the difference between the best-fit model predictions and the +data in units of the observational uncertainty. As in Fig. 1, grey backgrounds indicate the ranges of the data that are not included in the +fit. All fits use the Aemulus simulation box B04 which provides the best fit to the combination of all observations. +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +13 +125 +150 +175 +200 +225 +250 +275 +χ2 +min +Distribution +Mocks +χ2-fit +Data +Figure 7. The distribution of best-fit χ2-values in perturbed +mocks when fitting all data (solid blue), an analytic χ2-distribution +with the same mean as the mocks (dashed blue) and the best-fit χ2 +from the actual data (black). For each of the perturbed mocks, we +fully marginalised over all 11 galaxy–halo connection parameters. +galaxy–galaxy lensing with the simulation box B04, though +the exact choice of template does not strongly affect the re- +sults. +The results are shown in Fig 11. When comparing DES +and SDSS on scales rp > 1h−1 Mpc, we find that SDSS lens- +ing amplitudes are (20 ± 6)%, (14 ± 8)% and (23 ± 10)% +lower on average than the DES amplitudes for samples A, +B and C, respectively. When doing the same comparison for +KiDS, we find (−13 ± 9)%, (14 ± 16)% and (13 ± 19)%, i.e. +KiDS and SDSS agree well on large scales. We note that the +quoted uncertainties are the statistical uncertainties only and +do not account for systematic uncertainties. The latter should +be dominated by the SDSS photometric redshift calibration +and is of order 6% (Singh et al. 2019). Taking into account +this systematic uncertainty, none of the offsets at large scales +are statistically significant. On the other hand, differences +on smaller scales have a stronger significance but are subject +to uncertainties regarding boost factor estimates (Leauthaud +et al. 2017). We leave an in-depth comparison of lensing am- +plitudes measured with different lensing data sets to a future +study with new lenses from the DESI survey. +7 DISCUSSION +To our knowledge, the current work represents the first +full-scale combined analysis of redshift-space clustering and +galaxy–galaxy lensing. Using this approach and combining all +LOWZ samples, we obtain a ∼ 2.5% constraint on S8, one of +the most stringent constraints on S8 to date (Abdalla et al. +2022). +7.1 S8-tension +In this work, we present two roughly independent constraints +on the growth of structure amplitude S8 based on the analysis +of redshift-space clustering and the combination of projected +clustering and galaxy–galaxy lensing, S8 = 0.771 ± 0.027 +and 0.792 ± 0.022, respectively. By contrast, the most re- +cent Planck2020 CMB analysis prefers S8 = 0.834 ± 0.016. +This represents a ∼ 2σ discrepancy in both cases. While +this offset is not statistically significant, our results follow +a long-standing trend whereby low-redshift probes of cosmic +structure growth infer a lower value for S8 than studies of +the CMB. In Fig. 12, we compare our results against CMB +results (Planck Collaboration et al. 2020; Aiola et al. 2020) +and other low-redshift large-scale structure studies, including +cosmic shear (Hikage et al. 2019; Asgari et al. 2021; Amon +et al. 2022), the combination of projected galaxy clustering +and lensing cross-correlation (Krolewski et al. 2021; Porredon +et al. 2022; Miyatake et al. 2022b), so-called 3 × 2pt stud- +ies (Heymans et al. 2021; Abbott et al. 2022), redshift-space +clustering (Ivanov et al. 2020; Philcox & Ivanov 2022), the +combination of redshift-space clustering and lensing cross- +correlation (Chen et al. 2022) as well as cluster counts (Mantz +et al. 2015; Planck Collaboration et al. 2016; Bocquet et al. +2019; Abbott et al. 2020). Overall, our results are in excel- +lent agreement with other low-redshift probes which also pre- +fer S8 ∼ 0.76. Our analysis is the first to analyse BOSS +LOWZ galaxies cross-correlated with KiDS-1000 and DES +Y3 to fit for cosmology. Additionally, our constraints from +redshift-space clustering are primarily derived from scales not +analysed in conventional large scale-only RSD studies (e.g., +Ivanov et al. 2020; Philcox & Ivanov 2022), which we have +achieved through a simulation-based model that marginalises +over uncertainties in galaxy assembly bias and other astro- +physical effects. We note that two other recent full-scale RSD +studies of the BOSS CMASS (Zhai et al. 2022) and the +eBOSS LRG samples (Chapman et al. 2022) also find a lower +growth of structure amplitude than predicted by Planck2020. +Overall, our results add further evidence for the existence of +an S8-tension between low redshift and CMB data and to +the evidence that this discrepancy is not limited to studies +involving gravitational lensing. +7.2 Lensing is low +For a given set of cosmological parameters, the observed +galaxy clustering amplitude makes precise predictions for +the galaxy–galaxy lensing amplitude, even after marginal- +ising over uncertainties in the galaxy–halo connection (e.g. +Yoo et al. 2006; Cacciato et al. 2009; Leauthaud et al. 2017). +Recently, several studies have shown that the lensing ampli- +tude is significantly over-predicted when assuming the best- +fit Planck2020 cosmological parameters. This discrepancy is +most significant on small scales, rp < 5 h−1 Mpc where the +difference is around 30% (Leauthaud et al. 2017; Lange et al. +2019a; Yuan et al. 2020; Lange et al. 2021; Amon et al. +2023). However, the matter distribution on such small scales +is also affected by baryonic feedback effects which are often +not explicitly modelled. This may contribute to the apparent +lensing-is-low effect on small scales (Leauthaud et al. 2017; +Lange et al. 2019a; Amodeo et al. 2021), which we do not +analyse here. +On large scales, the lensing measurements have increased +uncertainties and therefore it is less clear to what extent a +similar lensing-is-low tension exists at those scales, as well. +Recently, Lange et al. (2021), using BOSS LOWZ cluster- +ing and SDSS galaxy–galaxy lensing measurements, also find +a statistically significant difference of ∼ 30 − 35% on large +MNRAS 000, 1–20 (2023) + +14 +J. U. Lange et al. +0.6 +0.7 +0.8 +0.9 +S8 +0 +10 +20 +Posterior dp/dS8 +RSD-only +0.18 ≤ z < 0.30 +0.30 ≤ z < 0.36 +0.36 ≤ z < 0.43 +0.6 +0.7 +0.8 +0.9 +S8 +wp + ∆Σ +NGC +SGC +0.6 +0.7 +0.8 +0.9 +S8 +RSD + ∆Σ +combined +Figure 8. Posterior constraints on S8 from the combination of wp and ∆Σ (left), redshift-space distortions (middle) and the combination +of redshift-space clustering and lensing (right). In each panel, we show constraints from different redshift bins (different coloured lines) +from the NGC (solid) and SGC (dashed) area. Additionally, we show constraints when combining all six LOWZ data samples (black +solid). +wp + ∆Σ +RSD-only +RSD + ∆Σ +prior +0.28 +0.32 +0.36 +Ωm +0.60 +0.75 +0.90 +S8 +−1.2 +−0.9 +−0.6 +w +0.28 +0.32 +0.36 +Ωm +−1.2 +−0.9 +−0.6 +w +Figure 9. Cosmological constraints from BOSS LOWZ when +analysing all six samples in this work jointly. We show con- +straints derived from a combination of projected galaxy cluster- +ing and galaxy–galaxy lensing (purple), redshift-space clustering +(blue) and the combination of redshift-space clustering and lensing +(red). We also show the prior imposed by the Aemulus simulations +(grey). +scales. Recall that the lensing predictions at fixed projected +clustering roughly scale as S8 (Yoo et al. 2006). Thus, studies +cross-correlating BOSS LOWZ galaxies with SDSS shape cat- +alogues and inferring S8 ∼ 0.71 ± 0.03 (Wibking et al. 2020) +and ∼ 0.71 ± 0.04 (Singh et al. 2020), considerably below the +Planck CMB prediction, also corroborate the existence of a +lensing-is-low problem. More recently, Amon et al. (2023) re- +evaluated the lensing-is-low tension using updated DES and +KiDS lensing measurements, finding a difference at the level +of 15% for LOWZ on large scales. Overall, their results are +offset with respect to with the Planck CMB prediction at the +∼ 2σ level. Finally, our results also imply a mild, 2σ tension +with the Planck2020 results but not at the level reported in +earlier studies relying on SDSS lensing measurements (Singh +et al. 2020; Wibking et al. 2020; Lange et al. 2021). +As shown in section 6.3, we find that for LOWZ galaxies +the SDSS lensing catalogues imply lower lensing amplitudes +for the same lens samples than the DES Y3 catalogues. We +leave a detailed investigation of the statistical significance of +this finding to future work. Nonetheless, this difference helps +explain why earlier studies based on SDSS lensing measure- +ments (Wibking et al. 2020; Singh et al. 2020; Lange et al. +2021) find stronger levels of tension with Planck CMB pre- +dictions than Amon et al. (2023) and the present study. +7.3 Full-scale studies +Using a full-scale approach, growth of structure constraints +from redshift-space distortions become competitive with lead- +ing gravitational lensing studies. Recent large scale-only, full- +shape studies using BOSS LOWZ and CMASS achieve a +∼ 0.045 uncertainty on S8 (Ivanov et al. 2020; Philcox & +Ivanov 2022), whereas here we obtain a 0.027 constraint. Fur- +thermore, in this work we only use the BOSS LOWZ sam- +ple and do not analyse the two times larger BOSS CMASS +sample, unlike the aforementioned large scale-only studies. +In Fig. 4, we show the derivative of the RSD multipoles with +respect to changes in S8 after fully marginalising over galaxy– +halo connection parameters. Thus, this figure indicates where +cosmological constraints from full-scale studies are coming +from. It indeed suggests that significant information on S8 de- +rives from scales smaller than 10 h−1 Mpc. At the same time, +the figure also confirms the finding in Lange et al. (2022) +that little to no cosmological information is contained for +RSD multipoles below s ∼ 1 h−1 Mpc. We also repeat the +RSD analysis using only scales above 6.3 h−1 Mpc, finding +that our constraints on S8 degrade by 60%. Additionally, we +find that excluding the smallest scales from our analysis leads +to a constraint on S8 that, while being higher (also see Lange +et al. 2022; Chapman et al. 2022; Zhai et al. 2022), is in good +agreement with the result from the default analysis. Overall, +our results suggest that full-scale studies have the potential of +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +15 +10−1 +100 +101 +Projected Radius rp [h−1 Mpc] +0 +5 +10 +15 +Lensing rp∆Σ [106M⊙/pc] +0.18 ≤ zl <0.30 +DES +KiDS +SDSS +10−1 +100 +101 +Projected Radius rp [h−1 Mpc] +0.30 ≤ zl <0.36 +10−1 +100 +101 +Projected Radius rp [h−1 Mpc] +0.36 ≤ zl <0.43 +Figure 10. Galaxy–galaxy lensing measurements from cross-correlating BOSS LOWZ targets with lensing catalogues from DES (red), +KiDS (purple), and SDSS (orange). Different panels correspond to the three different redshift-binned samples analysed in this work. +Measurements are slightly offset in the x-direction for clarity. +DES +KiDS +SDSS +< 1 Mpc/h +> 1 Mpc/h +all scales +0.8 +1.0 +A/ADES +0.18 ≤ zl <0.30 +< 1 Mpc/h +> 1 Mpc/h +all scales +0.8 +1.0 +A/ADES +0.30 ≤ zl <0.36 +< 1 Mpc/h +> 1 Mpc/h +all scales +0.6 +0.8 +1.0 +A/ADES +0.36 ≤ zl <0.43 +Figure 11. Lensing amplitudes averaged over different scales as +a function of the lensing data set, the lens galaxy sample and +the scale range. Three different redshift bins are shown from low +redshift (top) to higher redshift (bottom). +improving cosmological constraints from redshift-space clus- +tering by a factor of 2−3 over traditional large scale-only full- +shape studies, even after marginalising over complex galaxy– +halo connection models (also see Lange et al. 2022). In ad- +dition to placing independent, high-precision constraints on +the S8-tension, analyses of the non-linear regime could also +enable precise tests of modifications to gravity in the future +(Blake et al. 2020; Alam et al. 2021). +Regarding the combination of projected clustering and +galaxy–galaxy lensing, we model the lensing signal down to +2.5 h−1 Mpc. Nominally, this is not a significant improvement +to, for example, the recent analysis by Porredon et al. (2022), +where scales down to 6 h−1 Mpc are considered. However, the +analysis of Porredon et al. (2022) marginalises over a free +point-mass term, which scales as ∆Σ ∝ r−2 +p +and is designed +to remove systematics in the modelling of non-linear scales. +As described in Prat et al. (2022), this reduces the signal-to- +noise ratio of the lensing measurements analysed in Porredon +et al. (2022) from 67 to 32 and primarily affects small scales. +Thus, Porredon et al. (2022) and similar analyses marginal- +ising over a point-mass term, e.g. Singh et al. (2020) and +Wibking et al. (2020), are by design less sensitive to the pre- +dicted lensing amplitude on small scales relative to studies +without point-mass marginalisation. In this work, we do not +marginalise over a point-mass term since we argue that our +simulation-based modelling framework and complex galaxy– +halo connection model should already accurately model scales +down to 2.5 h−1 Mpc, thereby allowing us to leverage this ad- +ditional information without the sacrifice in signal-to-noise +that results from point-mass marginalisation. +Currently we do not explore even smaller scales in order to +avoid the need to model baryonic feedback processes (Lange +et al. 2019a; Amodeo et al. 2021). However, the lensing mea- +surements on scales down to 0.1 h−1 Mpc have a signal-to- +noise ratio of 57 compared to 26 when limiting the anal- +ysis to scales > 2.5 h−1 Mpc. Thus, if we would be able +to empirically constrain and marginalise over baryonic feed- +back processes, we could potentially get even more stringent +growth-of-structure constraints. Unfortunately, galaxy clus- +tering alone is unlikely to place strong constraints on baryonic +feedback, thus marginalising over flexible baryonic feedback +models with the present data is unlikely to result in more +stringent S8 constraints. However, combining galaxy cluster- +ing and lensing with data on the baryon distribution around +galaxies, such as measurements of the Sunyaev–Zel’dovich +(Schaan et al. 2021; Amodeo et al. 2021), may break that de- +generacy, and could unlock the constraining power of small- +scale lensing measurements for cosmology. +The constraining power of full-scale studies might be fur- +MNRAS 000, 1–20 (2023) + +16 +J. U. Lange et al. +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +S8 = σ8 +� +Ωm,0/0.3 +Cosmic Microwave Background +Planck +Planck20 +ACT+WMAP +Aiola+20 +Cosmic Shear +DES +Amon+22 +HSC +Hikage+19 +KiDS +Asgari+21 +Projected Clustering and Lensing +DES×DES +Porredon+22 +BOSS×HSC +Miyatake+21 +unWISE×Planck-κ +Krolewski+21 +LOWZ×DES/KiDS +this work +Shear, Clustering and Lensing +DES×DES +Abbott+22 +BOSS/2dFLenS×KiDS +Heymans+21 +Redshift-Space Clustering +BOSS +Ivanov+20 +BOSS +Philcox+21 +LOWZ +this work +Redshift-Space Clustering and Lensing +BOSS×Planck-κ +Chen+22 +LOWZ×DES/KiDS +this work +Cluster Counts +ROSAT +Mantz+15 +Planck-SZ +Planck16 +SPT +Bocquet+19 +DES +Abbott+20 +Figure 12. Comparison of different literature constraints on S8 +against the results derived in this work (results from Mantz et al. +2015; Planck Collaboration et al. 2016; Hikage et al. 2019; Bocquet +et al. 2019; Planck Collaboration et al. 2020; Aiola et al. 2020; +Ivanov et al. 2020; Abbott et al. 2020; Asgari et al. 2021; Krolewski +et al. 2021; Porredon et al. 2022; Miyatake et al. 2022b; Heymans +et al. 2021; Amon et al. 2022; Philcox & Ivanov 2022; Abbott +et al. 2022; Chen et al. 2022). Blue band indicates the prediction +of Planck2020. +ther improved by using larger simulations with reduced sam- +ple variance. The current Aemulus simulations have a vol- +ume of (∼ 1 h−1 Gpc3) each, roughly the same as the total +LOWZ volume analysed here. Since our analysis approach +incorporates uncertainties in simulation predictions, con- +straints would likely become more stringent with larger sim- +ulations such as the AbacusSummit simulation suite (Mak- +simova et al. 2021). On the observational side, most cur- +rent spectroscopic large-scale structure surveys are designed +to be cosmic variance limited on large scales, i.e. up to +k ∼ 0.2 h Mpc−1. Due to this design choice, highly non-linear +scales are currently mostly dominated by shot noise. Future +galaxy surveys with a higher sampling density might improve +the constraining power of non-linear scales further for the +same observational volume (Dawson et al. 2022). Such high- +density surveys would also be ideal targets for multi-tracer +studies, another method to reduce the impact of cosmic vari- +ance (McDonald & Seljak 2009). +At the same time, simulations and modelling frameworks +need to be developed further. For example, current gravity +solvers predict an up to 1% different halo (matter) clustering +amplitude at k ∼ 1(10) h Mpc−1 respectively at low redshifts +(Grove et al. 2022). With increasing precision of large-scale +structure measurements, these differences might soon domi- +nate uncertainties in the cosmological interpretation of non- +linear scales. Furthermore, more work needs to be done to en- +sure that the galaxy–halo connection models used to analyse +the data are sufficiently general and do not bias cosmology +results. While several cosmology recovery tests (Lange et al. +2022; Zhai et al. 2022) have been performed on SHAM mock +galaxy catalogues, including in this work, tests on additional, +more complex galaxy models, including semi-analytic mod- +els and hydrodynamic simulations (see Wechsler & Tinker +2018, for a review), are needed. At the same time, marginal- +ising over more physically-motivated galaxy–halo connection +models than HODs such as the UniverseMachine (Behroozi +et al. 2019) or Emerge (Moster et al. 2018) might reduce +cosmological uncertainties. We leave such investigations to +future work. +8 CONCLUSION +In this work, we present a novel simulation-based joint +cosmological analysis of galaxy redshift-space clustering +and galaxy–galaxy lensing. Our work extends previous +simulation-based full-scale studies (e.g. Wibking et al. 2020; +Miyatake et al. 2022b; Lange et al. 2022; Zhai et al. 2022) +in several directions. For example, this is the first full- +scale cosmological study involving gravitational lensing that +also models galaxy assembly bias. Furthermore, this is the +first simulation-based cosmological full-scale study that uses +the most recent DES Y3 and KiDS-1000 gravitational lens- +ing measurements. Most importantly, we also present the +first joint full-scale analysis of redshift-space clustering and +galaxy–galaxy lensing. +Our analysis incorporates a complex galaxy–halo connec- +tion model, including the effects of galaxy assembly bias as +well as central and satellite velocity bias. Our best-fit model +is able to provide a good fit to the observed galaxy clustering +and lensing amplitudes over a wide range of scales, down to +2.5 h−1 Mpc for lensing and 0.4 h−1 Mpc for clustering. Our +main result are new, highly competitive constraints on the +cosmic growth of structure, particularly S8. These findings +can be summarised as follows. +• When analysing the combination of projected clustering +and galaxy–galaxy lensing, we infer S8 = 0.792 ± 0.022 while +for redshift-space clusetering we infer S8 = 0.771 ± 0.027. Fi- +nally, combining redshift-space clustering and galaxy–galaxy +lensing, we find S8 = 0.779 ± 0.020. +• We find good agreement regarding S8 between multiple +independent galaxy samples. Similarly, constraints derived +only from redshift-space clustering are consistent with those +relying on gravitational lensing. +• When repeating our analysis of redshift-space clustering +using only larger scales above s > 6.3 h−1 Mpc instead of +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +17 +400 h−1 kpc, we achieve statistically consistent results, but +our constraining power on S8 degrades by 60%. +• Our results favour a value for S8 below the best-fit value +inferred by the Planck2020 CMB analysis, S8 = 0.834. This +is in agreement with results from other analyses of the low- +redshift Universe, the so-called S8-tension (see Abdalla et al. +2022, for a review). +• Similar to other recent full-scale studies of galaxy +redshift-space clustering (Chapman et al. 2022; Zhai et al. +2022; Yuan et al. 2022), we find evidence for an S8-tension +with predictions from Planck2020, even without incorporat- +ing gravitational lensing data. +Our analysis also highlights the strong constraining power +of full-scale studies over similar analyses targeting only large +scales. Particularly, our constraints on S8 using only redshift- +space clustering are a factor of two more stringent than recent +large scale-only studies of BOSS galaxy redshift-space clus- +tering (Ivanov et al. 2020; Philcox & Ivanov 2022), despite +only using a small fraction of the data compared to these +other works. Similarly, our constraints derived from the com- +bination of projected galaxy clustering and galaxy–galaxy +lensing are significantly more stringent than, for example, +a recent DES Y3 analysis using those observables (Porredon +et al. 2022). We attribute part of this improvement to con- +sidering scales down to 2.5 h−1 Mpc for the lensing signal. +Furthermore, we argue that our complex modelling frame- +work alleviates the need to marginalise over a point mass, +further increasing sensitivity to high-S/N non-linear scales. +The current study represents a new advance in full-scale +cosmological studies. We anticipate building upon this in the +future in several respects. In the present study, we do not +use scales below 2.5 h−1 Mpc since our galaxy model model +does not include baryonic feedback (Leauthaud et al. 2017; +Lange et al. 2019a). However, the signal-to-noise ratio of the +lensing amplitude more than doubles when considering scales +down to rp = 0.1 h−1 Mpc. Observations of the Sunyaev– +Zel’dovich effect can place independent constraints on the +strength of baryonic feedback (Schaan et al. 2021; Amodeo +et al. 2021); thus, a combined analysis of clustering, lensing +and the Sunyaev–Zel’dovich effect could improve constraining +power even further. We aim to apply the full-scale approach +to upcoming data from the DESI survey using the high- +resolution, large-volume AbacusSummit simulations (Maksi- +mova et al. 2021) instead of Aemulus for the modelling. This +is also expected to improve cosmological constraints. At the +same time, in light of this increased constraining power, more +tests on complex, highly-realistic galaxy mock catalogues are +needed to verify the robustness of full-scale constraints. An- +other avenue for full-scale studies is to provide tight priors on +galaxy bias models to be used with large-scale hybrid effec- +tive field theory cosmology studies (see Kokron et al. 2022, +for a recent example). +ACKNOWLEDGEMENTS +We thank the Aemulus collaboration and the UnitSims +team for making their simulations publicly available as well +as Sandy Yuan, Jeremy Tinker, and Zhongxu Zhai for in- +teresting discussions on several aspects of this analysis. We +also thank Sukhdeep Singh for providing the SDSS lensing +measurements discussed in section 7.2. +We acknowledge use of the lux supercomputer at UC Santa +Cruz, funded by NSF MRI grant AST 1828315. This work was +partially supported by the U.S. Department of Energy, Office +of Science, Office of High Energy Physics under Award Num- +ber DE-SC0019301. JUL received support from a fellowship +from the Leinweber Center for Theoretical Physics and from +a Stanford-Santa Cruz Fellowship including support from the +Kavli Institute for Particle Astrophysics and Cosmology. AL +acknowledges support from the David and Lucille Packard +foundation, and from the Alfred P. Sloan foundation. HG ac- +knowledges the support from the National Natural Science +Foundation of China (Nos. 11833005, 11922305). Work done +by APH was supported by the U.S. Department of Energy, +Office of Science, Office of Nuclear Physics, under contract +DE-AC02-06CH11357. FvdB is supported by the National +Aeronautics and Space Administration through Grant No. +19-ATP19-0059 issued as part of the Astrophysics Theory +Program. +This work made use of the following software packages: +matplotlib (Hunter 2007), SciPy, NumPy (van der Walt +et al. 2011), Astropy (Astropy Collaboration et al. 2013), +Colossus (Diemer 2015), halotools (Hearin et al. 2017), +MultiNest (Feroz & Hobson 2008; Feroz et al. 2009, 2019), +PyMultiNest (Buchner et al. 2014), scikit-learn (Pe- +dregosa et al. 2011), emcee (Foreman-Mackey et al. 2013), +UltraNest (Buchner 2021), Spyder and Setzer. +This project used public archival data from the Dark En- +ergy Survey (DES). Funding for the DES Projects has been +provided by the U.S. Department of Energy, the U.S. Na- +tional Science Foundation, the Ministry of Science and Edu- +cation of Spain, the Science and Technology FacilitiesCouncil +of the United Kingdom, the Higher Education Funding Coun- +cil for England, the National Center for Supercomputing Ap- +plications at the University of Illinois at Urbana-Champaign, +the Kavli Institute of Cosmological Physics at the Univer- +sity of Chicago, the Center for Cosmology and Astro-Particle +Physics at the Ohio State University, the Mitchell Institute +for Fundamental Physics and Astronomy at Texas A&M Uni- +versity, Financiadora de Estudos e Projetos, Funda¸c˜ao Car- +los Chagas Filho de Amparo `a Pesquisa do Estado do Rio de +Janeiro, Conselho Nacional de Desenvolvimento Cient´ıfico e +Tecnol´ogico and the Minist´erio da Ciˆencia, Tecnologia e In- +ova¸c˜ao, the Deutsche Forschungsgemeinschaft, and the Col- +laborating Institutions in the Dark Energy Survey. +The Collaborating Institutions are Argonne National Lab- +oratory, the University of California at Santa Cruz, the Uni- +versity of Cambridge, Centro de Investigaciones Energ´eticas, +Medioambientales y Tecnol´ogicas-Madrid, the University of +Chicago, University College London, the DES-Brazil Consor- +tium, the University of Edinburgh, the Eidgen¨ossische Tech- +nische Hochschule (ETH) Z¨urich, Fermi National Accelerator +Laboratory, the University of Illinois at Urbana-Champaign, +the Institut de Ci`encies de l’Espai (IEEC/CSIC), the Institut +de F´ısica d’Altes Energies, Lawrence Berkeley National Lab- +oratory, the Ludwig-Maximilians Universit¨at M¨unchen and +the associated Excellence Cluster Universe, the University of +Michigan, the National Optical Astronomy Observatory, the +University of Nottingham, The Ohio State University, the +OzDES Membership Consortium, the University of Pennsyl- +vania, the University of Portsmouth, SLAC National Acceler- +ator Laboratory, Stanford University, the University of Sus- +sex, and Texas A&M University. +MNRAS 000, 1–20 (2023) + +18 +J. U. Lange et al. +Based in part on observations at Cerro Tololo Inter- +American Observatory, National Optical Astronomy Obser- +vatory, which is operated by the Association of Universi- +ties for Research in Astronomy (AURA) under a cooperative +agreement with the National Science Foundation. +Based on observations made with ESO Telescopes at the La +Silla Paranal Observatory under programme IDs 177.A-3016, +177.A-3017, 177.A-3018 and 179.A-2004, and on data prod- +ucts produced by the KiDS consortium. The KiDS produc- +tion team acknowledges support from: Deutsche Forschungs- +gemeinschaft, ERC, NOVA and NWO-M grants; Target; the +University of Padova, and the University Federico II (Naples). +We use the gold sample of weak lensing and photomet- +ric redshift measurements from the fourth data release of +the Kilo-Degree Survey (Kuijken et al. 2019; Wright et al. +2020; Hildebrandt et al. 2021; Giblin et al. 2021) (Kuijken et +al. 2019), hereafter referred to as KiDS-1000. Cosmological +parameter constraints from KiDS-1000 have been presented +in (Asgari et al. 2021, cosmic shear), (Heymans et al. 2021, +3 × 2pt) and (Tr¨oster et al. 2021, beyond ΛCDM), with the +methodology presented in Joachimi et al. (2021). +Funding for SDSS-III has been provided by the Alfred +P. Sloan Foundation, the Participating Institutions, the Na- +tional Science Foundation, and the U.S. Department of En- +ergy Office of Science. The SDSS-III web site is http://www. +sdss3.org/. +SDSS-III is managed by the Astrophysical Research Con- +sortium for the Participating Institutions of the SDSS-III +Collaboration including the University of Arizona, the Brazil- +ian Participation Group, Brookhaven National Laboratory, +Carnegie Mellon University, University of Florida, the French +Participation Group, the German Participation Group, Har- +vard University, the Instituto de Astrofisica de Canarias, +the Michigan State/Notre Dame/JINA Participation Group, +Johns Hopkins University, Lawrence Berkeley National Lab- +oratory, Max Planck Institute for Astrophysics, Max Planck +Institute for Extraterrestrial Physics, New Mexico State Uni- +versity, New York University, Ohio State University, Pennsyl- +vania State University, University of Portsmouth, Princeton +University, the Spanish Participation Group, University of +Tokyo, University of Utah, Vanderbilt University, University +of Virginia, University of Washington, and Yale University. +DATA AVAILABILITY +The Aemulus and UNIT simulations used in this ar- +ticle are publicly available at https://aemulusproject. +github.io/ and http://www.unitsims.org/, respectively. +The DES, KiDS, and SDSS data sets analysed are available at +https://www.darkenergysurvey.org/, http://kids.strw. +leidenuniv.nl/, and https://www.sdss.org/, respectively. +All derived data generated in this research as well as code +used will be shared on reasonable request to the correspond- +ing author. +REFERENCES +Abareshi B., et al., 2022, AJ, 164, 207 +Abbott T. 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P., van den Bosch F. C., 2014, MNRAS, +443, 3044 +Zhai Z., et al., 2019, ApJ, 874, 95 +Zhai Z., et al., 2022, arXiv e-prints, p. arXiv:2203.08999 +Zheng Z., Guo H., 2016, MNRAS, 458, 4015 +Zheng Z., Coil A. L., Zehavi I., 2007, ApJ, 667, 760 +Zu Y., 2020, arXiv e-prints, p. arXiv:2010.01143 +van den Bosch F. C., More S., Cacciato M., Mo H., Yang X., 2013, +MNRAS, 430, 725 +van der Walt S., Colbert S. C., Varoquaux G., 2011, Computing +in Science and Engineering, 13, 22 +APPENDIX A: GAUSSIAN PROCESS +MODELLING +Our analysis method requires generalising the dependence of +the maximum likelihood or evidence as a function of cosmol- +ogy for the 40 Aemulus simulations to arbitrary cosmologies. +By default, we utilise the method described in section 3.4 that +fits the results for the 40 simulations with a multi-dimensional +skew-normal distribution in the (S8, Ωm, w)-plane. Here, we +test an alternative approach using Gaussian Process (GP) +emulation. This follows similar applications of GP emulation +in the literature (Zhai et al. 2019; Yuan et al. 2020). The +main difference to the aforementioned works is that we are +emulating only a single summary statistic, Lmax, as a func- +MNRAS 000, 1–20 (2023) + +20 +J. U. Lange et al. +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +S8 +Posterior +wp + ∆Σ +RSD-only +RSD + ∆Σ +Figure A1. Inferred posterior constraints on S8 from the un- +masked mock catalogues. We compare the results from the de- +fault procedure to model the likelihood as a function of cosmology +(solid) using skew-normals to an alternative approach employing +Gaussian Process fitting (dashed). +tion of cosmology instead of all observables as a function of +galaxy and cosmology parameters (Lange et al. 2019b). +We apply the alternative GP emulation technique to the +mock analyses described in section 4. The cosmological pos- +terior is based on Lmax, similar to the results in Fig. 3 and +the cosmological parameters considered are S8, Ωm and w. +GP interpolation is performed using the publicly available +GPy package7. To train a GP one needs to first determine +the kernel that best constrains the covariance matrix of the +training set. We first perform a k-fold cross validation over a +set of kernels such as polynomial, exponential, RBF, Mat´ern +3/2 and Mat´ern 5/2 to determine the kernel with maximum +predictive power. k-fold cross validation involves splitting the +data set, the 40 simulations with their respective cosmolog- +ical parameters and Lmax values, into k equal-sized groups. +Afterwards, each of the k = 8 groups is used as a test set +after training the GP on the remaining data. This allows us +to empirically asses the predictive power of different kernels. +Of all the kernels, we find the Mat´ern 5/2 to give the best +performance. +After determining the best kernel, we train the GP on the +entire set of 40 simulations and use the interpolated Lmax as +our proxy for the cosmological posterior. We note that, con- +trary to the default analysis, this procedure does not account +for the impact of uncertainties in the simulation predictions +on the final cosmology posterior. However, this effect was +found in Lange et al. (2022) to be negligible when analysing +the mocks with Aemulus simulations. In Fig. A1, we com- +pare the inferred posteriors on S8 against the results obtained +from the default analysis procedure involving skew-normals. +Overall, we find both approaches to give highly consistent +results, providing further evidence for the robustness of our +analysis method. +7 https://github.com/SheffieldML/GPy/ +APPENDIX B: GALAXY–HALO CONNECTION +Here, we present and discuss posterior constraints on the +galaxy–halo connection G. Since we fit each of the 40 sim- +ulations to data, we naturally get 40 posterior constraints +P(G|Ci) for 40 different cosmologies Ci. We combine these 40 +by taking the average weighted by the profile likelihood each +a simulation, +P(G) = +� P(G|Ci)Lp(Ci) +� Lp(Ci) +. +(B1) +As an example, we show in Fig. B1 all one and two- +dimensional posteriors on the galaxy–halo connection param- +eters for the 0.18 ⩽ z < 0.30 sample in the SGC. Similar to +the results for cosmology as well as central velocity and as- +sembly bias discussed below, this figure indicates good agree- +ment between the constraints derived from redshift-space +clustering, the combination of projected clustering and lens- +ing and RSDs combined with lensing. This figure also demon- +strates that the addition of gravitational lensing as a con- +straint does not add significant constraining power on galaxy– +halo connection parameters compared to redshift-space clus- +tering alone. This is expected since we only include gravi- +tational lensing in the two-halo regime. At fixed cosmology, +gravitational lensing in this regime primarily contains infor- +mation on the large-scale bias, something already contained +within clustering measurements. +Our posterior constraints on the HOD parameters, par- +ticularly Mmin, M1 and σlog M, present significant variation +amongst the different galaxy samples studied in this work. +This is expected since the different samples represent galaxy +populations at different redshifts, stellar masses etc. In this +appendix we focus on velocity bias and assembly bias, as we +find that the conclusions drawn below apply equally well to +each of the galaxy samples we consider. +In Fig. B2, we present the posterior constraints on αc, the +central velocity bias parameter. As expected, redshift-space +clustering is able to put some constraints on αc, whereas we +do not show the combination of projected galaxy clustering +and galaxy–galaxy lensing since it is insensitive to this pa- +rameter. Overall, our results are consistent with little to no +central velocity bias, αc = 0, and agree with the findings in +Lange et al. (2022). Similarly, our results here do not con- +tradict the findings in Guo et al. (2015a) where αc > 0. In +the present work, we define central velocity with respect to +the inner 10% of the halo particles (Behroozi et al. 2013) in- +stead of the inner 25% as in Guo et al. (2015a). The latter +definition is expected to imply larger values for αc (Ye et al. +2017). +Fig. B3 shows our constraints on the central assembly bias +parameter Acen. We find no strong constraints on assembly +bias and our results are consistent with no assembly bias, +Acen = 0. As discussed in Lange et al. (2019b, 2022), this is +in part due to the degeneracy between Acen and cosmology. +At the same time, even at fixed cosmology, we find neither +redshift-space clustering nor the combination of projected +clustering and galaxy–galaxy lensing to be very sensitive to +assembly bias. Similarly, our analysis also does not yield any +noteworthy constraints on the satellite assembly bias param- +eter Asat, either. Other summary statistics beyond two-point +correlation functions might be needed to robustly constrain +assembly bias (see e.g. Wang et al. 2019; Storey-Fisher et al. +2022). +MNRAS 000, 1–20 (2023) + +Full-scale and full-shape analysis of RSD and GGL +21 +wp + ∆Σ +RSD-only +RSD + ∆Σ +0.5 +1.0 +σlog M +0.75 +1.00 +fΓ +13 +14 +log M0 +13.6 +14.4 +log M1 +0.8 +1.6 +α +0 +1 +Acen +0 +1 +Asat +−0.4 +0.0 +0.4 +log η +0.2 +0.4 +αc +13.2 +13.6 +log Mmin +1.0 +1.2 +αs +0.5 +1.0 +σlog M +0.75 +1.00 +fΓ +13 +14 +log M0 +13.6 +14.4 +log M1 +0.8 +1.6 +α +0 +1 +Acen +0 +1 +Asat +−0.4 +0.0 +0.4 +log η +0.2 +0.4 +αc +1.0 +1.2 +αs +Figure B1. Posterior constraints on galaxy–halo connection parameters for the 0.18 ⩽ z < 0.30 sample in the SGC after marginalisation +over cosmology. We show constraints coming from redshift-space clustering (blue), the combination of projected clustering and lensing +(purple) and RSDs combined with lensing (red). Contours denote 68 and 95% confidence regions. +MNRAS 000, 1–20 (2023) + +22 +J. U. Lange et al. +0.0 +0.1 +0.2 +0.3 +αc +Posterior +RSD-only +0.18 ≤ z < 0.30 +0.30 ≤ z < 0.36 +0.36 ≤ z < 0.43 +0.0 +0.1 +0.2 +0.3 +αc +RSD + ∆Σ +NGC +SGC +Figure B2. Posterior constraints on central velocity bias after marginalisation over cosmology. Different panels indicate the observa- +tional constraints used: redshift-space clustering (left) and the combination of redshift-space clustering and galaxy–galaxy lensing (right). +Different lines indicate different samples. +−1.0 +−0.5 +0.0 +0.5 +Acen +Posterior +RSD-only +0.18 ≤ z < 0.30 +0.30 ≤ z < 0.36 +0.36 ≤ z < 0.43 +−1.0 +−0.5 +0.0 +0.5 +Acen +wp + ∆Σ +NGC +SGC +−1.0 +−0.5 +0.0 +0.5 +Acen +RSD + ∆Σ +Figure B3. Same as Fig. B2 except for focussing on the central assembly bias parameter Acen and also showing results for the combination +of projected clustering and galaxy–galaxy lensing (middle). +MNRAS 000, 1–20 (2023) + diff --git a/N9FAT4oBgHgl3EQfyR7D/content/tmp_files/load_file.txt b/N9FAT4oBgHgl3EQfyR7D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..575f0de718ca7480f44b9e6e447506034cf8c2e9 --- /dev/null +++ b/N9FAT4oBgHgl3EQfyR7D/content/tmp_files/load_file.txt @@ -0,0 +1,2307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf,len=2306 +page_content='MNRAS 000, 1–20 (2023) Preprint 23 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 Constraints on S8 from a full-scale and full-shape analysis of redshift-space clustering and galaxy–galaxy lensing in BOSS Johannes U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange1,2,3,4⋆, Andrew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Hearin5, Alexie Leauthaud2, Frank C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' van den Bosch6, Enia Xhakaj2, Hong Guo7, Risa H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wechsler1 and Joseph DeRose8 1Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' CA 94305,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA 2Department of Astronomy and Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' CA 95064,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MI 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA 4Leinweber Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MI 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA 5Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Argonne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' IL 60439,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA 6Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' CT 06511,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA 7Key Laboratory for Research in Galaxies and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Shanghai 200030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' China 8Berkeley Center for Cosmological Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' USA Accepted xxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Received xxx ABSTRACT We present a novel simulation-based cosmological analysis of galaxy–galaxy lensing and galaxy redshift-space clus- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Compared to analysis methods based on perturbation theory, our simulation-based approach allows us to probe a much wider range of scales, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 h−1 Mpc to 63 h−1 Mpc, including highly non-linear scales, and marginalises over astrophysical effects such as assembly bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We apply this framework to data from the Baryon Oscillation Spec- troscopic Survey LOWZ sample cross-correlated with state-of-the-art gravitational lensing catalogues from the Kilo Degree Survey and the Dark Energy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We show that gravitational lensing and redshift-space clustering when analysed over a large range of scales place tight constraints on the growth-of-structure parameter S8 = σ8 � Ωm/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, we infer S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='022 when analysing the combination of galaxy–galaxy lensing and projected galaxy clustering and S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='771±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='027 for galaxy redshift-space clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' These findings highlight the potential constrain- ing power of full-scale studies over studies analysing only large scales, and also showcase the benefits of analysing multiple large-scale structure surveys jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our inferred values for S8 fall below the value inferred from the CMB, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='834 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' While this difference is not statistically significant by itself, our results mirror other findings in the literature whereby low-redshift large scale structure probes infer lower values for S8 than the CMB, the so-called S8-tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Key words: cosmology: large-scale structure of Universe – cosmology: cosmological parameters – cosmology: dark energy – cosmology: dark matter 1 INTRODUCTION The large-scale structure (LSS) distribution in the low- redshift Universe has emerged as one of the primary probes of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' LSS surveys such as the Baryon Oscillation Spectroscopic Survey (BOSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ahumada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020), the Kilo Degree Survey (KiDS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Giblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) and the Dark Energy Survey (DES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Gatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) have provided some of the most stringent constraints on the parameters of the cosmological standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the com- ing decade, LSS surveys such as the Dark Energy Spectro- scopic Instrument (DESI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Abareshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) survey, the Legacy Survey of Space and Time (LSST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The LSST Dark ⋆ email: julange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='astro@pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='me Energy Science Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2018) on the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ru- bin Observatory, and the Nancy Grace Roman Space Tele- scope (Spergel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015) will build upon this success and provide even more powerful constraints on the cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' LSS surveys are particularly sensitive to Ωm, the frac- tion of the energy density in matter, and σ8, the amplitude of matter fluctuations on scales of 8 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' There are two promising avenues to constrain Ωm and σ8 from LSS observa- tions: the deflection of light, so-called gravitational lensing, by the LSS mass distribution and the clustering of matter in redshift-space which is sensitive to peculiar velocities via redshift-space distortions (RSDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Recently, several LSS probes of the low-redshift Universe have reported tensions with respect to cosmological param- eters preferred by the analysis of the high-redshift cosmic microwave background (CMB) under the canonical Λ Cold © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='08692v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='CO] 20 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Dark Matter (ΛCDM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Most often this tension is ex- pressed in constraints on the cosmological parameter S8 = σ8 � Ωm/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 and, hence, is called the “S8-tension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similar to the well-known Hubble tension (Knox & Millea 2020), the S8-tension could potentially point to new physics beyond the standard ΛCDM cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Under ΛCDM, observa- tions of the CMB by Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (Planck2020, 2020) infer S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='834 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='016 (TT,TE,EE+lowE), a value that is 5 − 10% higher than the value preferred by LSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Currently, the tension is 2 − 4σ significant with respect to several LSS studies utilising gravitational lensing, while the significance is lower for redshift-space clustering studies (see Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' While no study by itself can claim a > 5σ tension between the low-redshift LSS and the high-redshift CMB, the similarity of the findings of multiple, independent LSS surveys using different techniques suggest that the S8-tension might be a genuine cosmological tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Most often, cosmology studies of the LSS distribution fo- cus on large scales where the statistical properties of the matter distribution can be calculated analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Further- more, on large scales, the relationship between the observed galaxy distribution and the underlying dark matter density field can be characterised by a small number of bias factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, methods applicable to large, linear scales typically break down on highly non-linear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In one approach to extending LSS predictions to small scales, an analytical halo model is used to make predictions for basic summary statis- tics of the density field such as the matter power spectrum, and the abundance and clustering of dark matter halos (Sel- jak 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When augmented with additional ingredients for the halo occupation statistics of galaxies, the halo model be- comes a prediction pipeline for the galaxy distribution on non-linear scales (Cooray & Sheth 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Krause & Eifler 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' One of the biggest challenges faced by this approach is meeting the stringent demands for percent-level accuracy with an analytic model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=', Tinker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Hayashi & White 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Fedeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Miy- atake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Mahony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' however, steady im- provements have been made over the last decade (Mead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Garc´ıa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Mead et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021), and by now nu- merous analyses have used such analytical methods to de- rive constraints on cosmology from LSS measurements in the non-linear regime (Cacciato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Reddick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Tr¨oster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We will generically refer to this type of analysis as a “full-scale” study, in contrast to analyses that restrict attention to large scales only (see also Brieden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021, for the term “full-shape”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In recent years, there has been tremendous progress in com- putational power and statistical methods (Parejko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Kwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' DeRose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Nishimichi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022b) such that one can now directly compare predictions from simulations against LSS observations for multiple cosmological models and perform a rigorous Bayesian quantification of cosmolog- ical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' These simulation-based methods differ from conventional analytical halo models in that cosmological sim- ulations are directly populated with synthetic galaxies, and summary statistics are predicted using statistical estimators of the resulting synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' These simulation-based ap- proaches simplify the task of incorporating systematic effects such as galaxy assembly bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=', Hearin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016) and velocity bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=', Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015a), and at the same time enable a full-scale analysis to obtain stringent cosmological constraints (Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Salcedo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, simulation-based full-scale studies have potential to exhaust the information content of LSS two-point correlation functions using analyses with a de- gree of complexity that is difficult to achieve with an analyt- ical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Simulation-based full-scale studies have matured in recent years to the point that several have been applied to actual LSS data, demonstrating that the long-forecasted constrain- ing power of full-scale studies can be realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For exam- ple, Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) use a full-scale approach to infer S8 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='712 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='031 from a combination of galaxy clustering and galaxy–galaxy lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In a similar study using updated lensing data from the Hyper Suprime-Cam (HSC) survey, Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022b) infer S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='795+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='049 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Closely re- lated to these works are studies of the so-called “lensing is low” effect: when assuming the best-fit Planck2020 cosmol- ogy and fitting a model for galaxy clustering, the measured galaxy–galaxy lensing amplitude is overpredicted (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' It was shown that the measured lensing am- plitude under Planck CMB parameters is around 15 − 35% lower than predicted, particularly on small scales(Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2023) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since the predicted lensing amplitude at fixed clustering is corre- lated with S8 (Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006), these findings can also be seen as evidence of an S8-tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, several re- cent studies (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) have applied a simulation-based modelling framework to the analysis of redshift-space clus- tering of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' All four studies, using largely independent galaxy samples, find a preference for lower values for cosmic structure growth than the best-fit Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) ΛCDM model predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For completeness, we point out that already in 2013 a com- bined full-scale analysis of projected galaxy clustering and galaxy-galaxy lensing based on SDSS data by Cacciato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2013) yielded constraints on Ωm and σ8 in excellent agree- ment with these more recent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, at that point in time, those constraints where consistent with the then best-fit CMB constraints provided by the WMAP7 data (Ko- matsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2011), and thus did not signal any tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ad- ditionally, this study was based on an approximate analytical halo model and did not capture the effects of assembly bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2014) performed a simulation-based full-scale analysis of redshift-space distortions in BOSS and also found a preference for a low structure growth amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, this study relied on re-scaling velocities in a single cosmological simulation which has been argued to lead to inaccurate results (Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In this work, we built upon previous efforts by modelling the redshift-space clustering and galaxy–galaxy lensing of BOSS LOWZ galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This work extends previous studies in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Compared to the lensing studies of Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) and Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022b), we incorporate a model for galaxy assembly bias which has been shown to be important for modelling lensing on non-linear scales (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, we measure and analyse updated high-precision galaxy–galaxy lensing mea- surements from the state-of-the-art DES Y3 and KiDS-1000 data sets with reduced systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Compared MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 3 to the redshift-space clustering-only analysis of Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), the present work doubles the number of BOSS LOWZ galaxies, analysing nearly the entire BOSS LOWZ sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, we introduce and apply a new blinding (mask- ing) methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Most importantly, the present study is the first simulation-based full-scale and full-shape joint cosmolog- ical analysis of galaxy-galaxy lensing and redshift-space clus- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We start by introduc- tion the observational data set and observabeles in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our modelling approach is described in section 3 and veri- fied on mock catalogues in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The masking strategy is described and tested in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In section 6, we apply our analysis technique to observations and present the cosmolog- ical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, we discuss our results in section 7 and section 8 presents our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2 OBSERVATIONS Here, we describe the observational data and the calculation of the summary statistics used to derive cosmological con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 Observational data sets Our primary data set is the BOSS LOWZ galaxy sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The clustering of BOSS LOWZ galaxies will be used as a summary statistic in our modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, we measure the galaxy–galaxy lensing effect around LOWZ galaxies using gravitational lensing catalogues from KiDS and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 Baryon Oscillation Spectroscopic Survey Galaxies in BOSS LOWZ are targeted for spectroscopic ob- servations if they fulfil a series of magnitude cuts aimed at selecting luminous red galaxies (LRGs) in the redshift range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='15 ⩽ z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the following, cuts on apparent magnitudes use cmodel magnitudes whereas colours are calculated using model magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The cuts are as follows: rcmod < 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 + c∥ /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 (1) |c⊥| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 (2) 16 < rcmod < 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 , (3) where c∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7(gmod − rmod) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2(rmod − imod − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18) (4) and c⊥ = rmod − imod − (gmod − rmod)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (5) The above colour cuts, which are based on apparent pho- tometric magnitudes, select galaxies that primarily fall in the redshift range, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='15 ⩽ z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, because the LOWZ target selection is based on apparent magnitudes, LOWZ galaxies are not uniformly selected in redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Par- ticularly, galaxies of similar intrinsic luminosities and colours will have different apparent magnitudes depending on the redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, such a selection would contradict our mod- elling approach, which implicitly assumes a volume-limited sample of red galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, additional selection cuts are needed to arrive at approximately volume-limited samples of Property Sample A Sample B Sample C zref 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='40 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Mr −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='412 −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='558 −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='208 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' c0 ⊥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='216 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='154 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='166 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' c0 ⊥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='126 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Mr − c0 ∥/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='874 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='729 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='901 volume V [Gpc3 h−3] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='26/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='35/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='15 ngal [10−4 Mpc−3 h3] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='11/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='13/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='38/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='35 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Definitions and properties of the samples analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' All three samples are designed to be subsets of the BOSS LOWZ sample that are roughly volume-limited in their respective redshift ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the above table, Mr is the absolute r-band magnitude and the superscript 0 indicates rest-frame colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Both absolute magnitudes and colours are k-corrected to zref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The final two rows indicate the volumes and galaxy number densities, split by NGC and SGC areas of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We follow Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) and construct three subsamples of the BOSS LOWZ galaxy sample in the redshift ranges, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 < z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 < z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 < z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43, whose cuts are based on absolute magnitudes and rest-frame colours, k-corrected to a reference redshift zref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The samples and basic properties are described in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Sample A in the Northern Galactic Cap (NGC) area is almost identical to the low-redshift sample in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Conversely, sam- ples B and C correspond to the single high-redshift sample in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) but combined they have roughly 60% more galaxies per sky area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, compared to Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), we also analyse galaxies from the Southern Galactic Cap (SGC) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Taken together, these changes roughly double the sample size compared to Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) to roughly 3×105 galaxies altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Note that due to the slightly different photometric zero-points in the NGC and SGC regions, we opt to model and analyse galaxy samples from these two hemispheres separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, we point out that we do not use the so-called LOWZE2 and LOWZE3 sam- ples in the NGC that have relied on an incorrect star–galaxy separation criterion (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Kilo Degree Survey We use galaxies imaged by KiDS as so-called source galaxies when measuring the galaxy–galaxy lensing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Specifically, we use the KiDS-1000 data set covering roughly 1000 deg2 of the extra-galactic sky, roughly half of which overlaps with the BOSS survey footprint, with a source density ∼ 6 arcmin−2 (Giblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Galaxies in KiDS-1000 are grouped into 5 broad tomographic redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As described in Hilde- brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2021), the source redshift distribution n(z) of each of the tomographic redshift bins has been derived using self-organizing maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use the tabulated n(z) to convert gravitational tangential shears into estimates of the excess surface density ∆Σ, as described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 Dark Energy Survey In addition to KiDS-1000, we also use the DES Y3 weak lensing shape catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' DES Y3 covers ∼ 4000 deg2, out of which roughly 800 deg2 overlap with BOSS (Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2023), with an effective source density of 6 arcmin−2 (Gatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similar to KiDS-1000, DES Y3 source galaxies are grouped into 4 tomographic redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Source red- shift distributions n(z) are derived from a combination of self-organising maps, small-scale shear ratios, clustering red- shifts, (Myles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) and take into account blending effects in the photometry (MacCrann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Galaxy clustering measurements Galaxy clustering is characterised by the two-point correla- tion function ξ(s, µ), which measures the excess probabil- ity of having a pair of galaxies separated by s, the three- dimensional separation and µ, the cosine of the angle between s and the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use the Landy—Szalay estimator (Landy & Szalay 1993) and correct for fibre collisions using the methodology presented in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Most of the information contained in the two-point corre- lation function can be described by its multipole moments, ξℓ(s) = 2ℓ + 1 2 1 � −1 Lℓ(µ)ξ(s, µ)dµ , (6) where Lℓ represents the Legendre polynomial of order ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use ℓ = 0, 2 and 4, the so-called monopole, quadrupole, and hexadecapole moments of the redshift-space correlation func- tion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The multipole moments contain informa- tion about galaxy peculiar velocities due to redshift-space distortions that change the apparent positions of galaxies along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, the redshift-space clustering of galaxies itself, even without gravitational lensing, contains information about cosmic structure growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We also use the projected correlation function wp wp(rp) = +πmax � −πmax ξ (s, π/s) dπ , (7) where πmax = 80 h−1 Mpc and s = � π2 + r2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The projected correlation function is nearly independent of galaxy pecu- liar velocities (van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' By combining the projected correlation function with the galaxy–galaxy lens- ing amplitude we can probe cosmological constraints that are practically independent of peculiar velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Both ξ(s) and wp(rp) are measured in 14 comoving loga- rithmic bins going from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 h−1 Mpc to 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 ≈ 63 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Uncertainties on the correlation function were derived from jackknife-resampling of 100 roughly equal-area patches of the BOSS LOWZ sample, separately for the NGC and SGC ar- eas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The cross-covariances between different clustering mea- sures at different radial bins as well as different multipole moments of the redshift-space correlation function are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We apply the smoothing procedure described and tested in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) in order to suppress noise in the resulting covariance matrix estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 Gravitational lensing measurements In addition to the clustering properties of BOSS LOWZ galaxies, we also measure the galaxy–galaxy lensing effect around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is done by analysing the mean tangential ellipticities et of source galaxies from KiDS and DES around BOSS LOWZ lens galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The mean tangential ellipticity is related to the so-called excess surface density ∆Σ around lens galaxies, defined via ∆Σ(rp) = ⟨Σ(< rp)⟩ − Σ(rp) , (8) where Σ denotes the surface mass density, rp the projected distance in the frame of the lens galaxy, and the ⟨Σ(< rp)⟩ is the mean surface density inside a circle of radius rp centred on the lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The induced tangential ellipticity depends on ∆Σ and Σcrit defined as Σcrit(zl, zs) = c2 4πG 1 (1 + zl)2 DA(zs) DA(zl)DA(zl, zs) , (9) where zl and zs are the redshifts of the lens and source galaxy, respectively, and DA denotes the angular diameter distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the weak lensing regime, ∆Σ ≪ Σcrit, gravitational lensing induces a tangential shear component, γt = ∆Σ Σcrit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (10) Galaxies have intrinsic ellipticities, such that et is a stochastic measure of γt and one needs to stack a large number of lens– source pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use the following estimator for the mean excess surface density of galaxies: � ∆Σ = M −1 [∆Σl − ∆Σr] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (11) In the above equation, M is an estimate of the mean mul- tiplicative bias of the tangential ellipticity, ∆Σl is the un- corrected estimate for ∆Σ around lens galaxies and ∆Σr the analogous quantity around random points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the absence of systematics, the excess surface density around random points is zero on average but subtracting this estimate also reduces the variance of the ∆Σ estimator (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The mean multiplicative shear bias differs slightly between KiDS and DES due to different shape measurement algorithms em- ployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For KiDS, our multiplicative shear bias estimate is MKiDS = 1 + � ls wlsms � wls , (12) where m is the multiplicative shear bias that depends on the source tomographic redshift bin (Giblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021), the sum � ls goes over all suitable lens–source pairs separated by rp and wls is a weight associated with each galaxy pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For DES, the shear bias correction factor is given by MDES = � 1 + � ls wlsms � wls � � 1 + � ls wls[RT + Rsel] � wls � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (13) In the above equation, m is the multiplicative shear bias in- duced by blending (MacCrann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, RT and Rsel are the tangential component of the metacalibra- tion shear response and the selection response, respectively (Huff & Mandelbaum 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, the raw ∆Σ estimator for lens galaxies is defined as ∆Σl = � ls wlset�Σcrit,ls � ls wls , (14) where �Σcrit is an estimator of the intrinsic critical surface density of each lens–source pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We do not know the pre- cise redshift for each individual source galaxy and, instead, MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 5 have estimates of the normalised source redshift distribution nˆb(z)1 in each tomographic source bin ˆb (Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Myles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In order to obtain unbiased lensing amplitudes, we use �Σcrit,ls = � � ∞ � 0 Σ−1 crit(zl, zs)nˆb(zs)dzs � � −1 , (15) where nˆb(zs) are the normalised intrinsic redshift distribu- tions in each tomographic photometric redshift bin (Hilde- brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Myles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, the lens–source weights are designed to minimise shape noise, wls = ws �Σ−2 crit,ls , (16) where ws is the source weight provided in the KiDS and DES shape catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' All lensing calculations are performed with the dsigma galaxy–galaxy lensing package (Lange & Huang 2022) version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Note that we do not apply a lens weight wl and instead correct for fibre collisions using a nearest- neighbour correction (Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We find that the alternative weighting by wl = wnoz + wcp − 1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017) results in a ∼ 1% lower lensing signal than our default choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, the choice of fibre correction scheme does not significantly affect our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We measure ∆Σ(rp) in 14 logarithmic bins going from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 to ∼ 63 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Covariance matrices for the lensing measurements are derived from jackknife re-sampling of 50 roughly equal-area patches of overlap regions of BOSS LOWZ with KiDS and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For the BOSS NGC area, we only use the KiDS shape catalogue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' ignoring the small overlap of BOSS and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For the SGC area, we only use the DES catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We employ the same smoothing procedure as for the clustering measurements to suppress noise in our covari- ance matrix estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We do not take into account the cross- covariance between clustering and lensing measuremens since those are expected to be negligible (Taylor & Markoviˇc 2022), especially when considering that the overlap regions of BOSS with KiDS and DES constitute only a small part of the to- tal BOSS footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, we neglect systematic uncertain- ties in the lensing measurements stemming from photometric redshift calibration and multiplicative shear bias corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' These uncertainties are at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5% (Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2023) and would translate into a ∼ 1% systematic uncertainty on S8, significantly below statistical uncertainties presented in sec- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3 MODELLING In this section, we describe our simulation-based modelling framework which closely follows the one presented in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, we only describe the most salient points here and refer the reader to Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) for a more in-depth discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 1 By default, the DES Y3 redshift distributions are not normalised in order to incorporate blending effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In this work, we absorb the normalisation into the multiplicative shear bias m, instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 Cosmological simulations We use the Aemulus cosmological simulations (DeRose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019) to make predictions for galaxy clustering and galaxy– galaxy lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Aemulus is a suite of 40 simulations with dif- ferent cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Each simulation traces struc- ture formation in a wCDM Universe using (1400)3 particles in a (1050 h−1 Mpc)3 cubic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As discussed in DeRose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2019), the resolution of these simulations is sufficient to be used in the study of non-linear clustering of LRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Dark matter haloes in the simulations are identified with the Rockstar halo finder (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the follow- ing, we will only use parent haloes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' no subhaloes, with a mass M at or above 100 times the particle mass mp, where mp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='51 × 1010(Ωm/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3)h−1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Galaxy–halo connection model We use a Halo Occupation Distribution (HOD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Berlind & Weinberg 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Bullock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2007) model as the basis for our galaxy–halo connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In particular, we parameterise the average number of central and satellite galaxies expected in a halo as a function of its mass M and maximum circular velocity Vmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The average number of cen- tral galaxies as a function of halo mass M is given by ⟨Ncen|M⟩ = fΓ 2 � 1 + erf �log M − log Mmin σlog M �� , (17) where fΓ, log Mmin and σlog M are free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The pa- rameter fΓ models incompleteness in the selection of LRGs (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The average number of satellites is given by ⟨Nsat|M⟩ = �M − M0 M1 �α (18) with M0, M1, and α being free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The depen- dence on Vmax is modelled via the decorated HOD frame- work (Hearin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016), a natural extension to the standard, mass-only HOD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is necessary to model the im- pact of galaxy assembly bias (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wechsler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zentner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2014) and its degeneracy with cosmo- logical parameters (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yuan & Eisenstein 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In short, we perturb the average number of expected galaxies in a halo of mass M based on whether Vmax is above or below average of haloes at that mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The perturbation is described by ⟨Ncen|M, Vmax⟩ = ⟨Ncen|M⟩ ± Acen �1 2 − ���� 1 2 − ⟨Ncen|M⟩ ���� � , (19) for centrals and ⟨Nsat|M, Vmax⟩ = (1 ± Asat)⟨Nsat|M⟩, (20) for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In both cases, ± indicates + when Vmax is above the median at that mass and − otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This adds two more free parameters, Acen and Asat, varying in the range [−1, +1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Once average galaxy numbers are specified, the dis- tribution of galaxy numbers follow Bernoulli and Poisson dis- tributions for centrals and satellites, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We note that several recent theoretical and observational studies in- dicate the possibility of non-Poisson satellite number distri- butions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=', Dvornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Chaves-Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Parameter Description Range log Mmin low-mass cut-off for ⟨Ncen⟩ [12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] σlog M low-mass transition for ⟨Ncen⟩ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] fΓ incompleteness for ⟨Ncen⟩ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] log M0 low-mass cut-off for ⟨Nsat⟩ [12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] log M1 characteristic halo mass for ⟨Nsat⟩ [13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] α power-law index for ⟨Nsat⟩ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] Acen central galaxy assembly bias [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] Asat satellite galaxy assembly bias [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0] log η satellite spatial bias [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5] αc central velocity bias [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4] αs satellite velocity bias [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2] Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' All galaxy–halo connection parameters modelled in this work together with a short description and flat prior ranges used for fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), we showed that the assumption of a non-Poisson satellite distribution did not significantly af- fect cosmological constraints from redshift-space clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, non-Poisson numbers are not expected to have a substantial impact on the galaxy–galaxy lensing predictions in the two-halo regime that we are modelling (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=', Zu 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Chaves-Montero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Centrals and satellites have different phase-space coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Central galaxies coincide spatially with the halo centre but have additional random Gaussian velocities along the line of sight (Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015a,b) with scatter σ, σ = αcVvir √ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (21) This simulates the velocities of central galaxies with respect to the halo centre due to the unrelaxed state of the halo (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The free parameter αc is commonly known as the central velocity bias parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Satellite galaxies follow a Navarro–Frenk–White (NFW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 1997) profile with respect to the halo centre, n(r) ∝ 1 r/rs (1 + r/rs)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (22) The concentration parameter csat = rs/rh, where rh is the host halo radius, is allowed to be different than that of the dark matter distribution in the same halo, cdm, csat = ηcdm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (23) Similar to central galaxies, satellites follow the halo velocity on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We add additional velocities with respect to the halo by drawing from a Gaussian distribution with scatter de- rived from the spherically symmetric, anisotropy-free Jeans equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The derived satellite velocities are likely an approx- imation since satellite populations will not be perfectly spher- ically symmetric, without velocity anisotropy or dynamically relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, we apply an additional free scaling factor αs (the satellite velocity bias parameter) to the velocity scatter derived from the Jeans equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, the phase-space co- ordinates of galaxies add an additional three free parameters, αc, αs, and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ultimately, our model for galaxies has 11 free parameters, which we list in Table 2 for reference, that we allow to vary when fitting for cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 Data likelihood We use Halotools (Hearin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017) to compute galaxy clustering and lensing observables for a given simulation with cosmological parameters C and galaxy–halo connection pa- rameters G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For sample A, we model the observables using simulation outputs at redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='25 whereas for samples B and C, we use z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='40 snapshots2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We make use of the dis- tant observer approximation and in all cases project galaxy populations onto each of the three simulation axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When making predictions for galaxy clustering, we take into ac- count the Alcock–Paczynski effect (Alcock & Paczynski 1979) by correcting simulation coordinates for the assumed cosmol- ogy when obtaining the clustering measurements in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, we correct the ∆Σ predictions for the assumed cos- mology following the methodology described in More (2013) and approximating ∆Σ ∝ r−1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' To speed up the calculation of the clustering and lensing predictions for a given Aemulus simulation, we make use of a tabulation method for galaxy correlation functions (Zheng & Guo 2016) as implemented in TabCorr3 version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For a given clustering and lensing prediction, the likelihood L of the observed data vector D is computed as ln L(D|C, G) = 1 2 � (ngal − ˆngal)2σ−2 ngal + (ξ − ˆξ)TΣ−1 ξ (ξ − ˆξ) +(∆Σ − � ∆Σ)TΣ−1 ∆Σ(∆Σ − � ∆Σ) � , (24) where ξ denotes the combination of all galaxy clustering mea- surements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' ξ0, ξ2 and ξ4 or just wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use three different choices of data sets D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The first set, which we call “RSD-only” consists of the three redshift-space clustering multipole moments, ξ0, ξ2, and ξ4, and no lensing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The second set labelled “wp + ∆Σ” combines the projected correlation function wp with galaxy–galaxy lensing ∆Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, the set called “RSD + ∆Σ” consists of the redshift-space clustering multipole moments and the lens- ing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' All clustering measurements, ξ0, ξ2, ξ4, and wp are fitted on scales larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4h−1 Mpc, as discussed and motivated in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Contrary, for lensing, we only consider scales 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc < rp < 25h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The lower limit is chosen to avoid biases associated with baryonic feedback (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amodeo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) which we do not model in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The up- per limit is chosen in order to avoid biases in the covariance matrix estimate related to the size of the jackknife fields (Shi- rasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We note that for all three choices of two-point correlation functions, we use the number density of galaxies, ngal, as a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This observable tightly limits one dimension of 2 For sample B, there is an apparent mismatch between the mean redshift of the sample, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='33, and the redshift of the simulation output used to analyse it, z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As shown in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), this should have a negligible impact on studies using RSDs since these observations are primarily sensitive to f(z)σ8(z), which evolves very slowly with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the predicted lensing signal at fixed clustering roughly scales as σ8(z) (Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, to account for the redshift mismatch, we multiply lensing predictions for sample B by σ8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='33)/σ8(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='40) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='com/johannesulf/TabCorr MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 7 the HOD parameter space due to the requirement that to- tal number density of galaxies matched the observed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At the same time, the galaxy–halo connection model has sig- nificant flexibility to change the predicted number density without affecting the clustering predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For example, the predictions for the two-point correlation functions are unaf- fected if the number of galaxies in all halos was changed by constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Consequently, if we replace fΓ → x fΓ and M1 → x−1/α M1, the number density prediction changes to ˆngal → x ˆngal while leaving the clustering and lensing pre- dictions unaffected4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ultimately, we do not expect that the number density constraint significantly affects the cosmology result since neither fΓ nor log M1 are tightly constrained by the observational data or pile up against the prior ranges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' fΓ ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Equation (24) describes the likelihood of any simulation and its underlying cosmology C as a function of galaxy pa- rameters G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, ultimately, we want to obtain a con- straint on C alone for which we need L(D|C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Therefore, we first have to marginalise the data likelihood over the galaxy parameters G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2019b) and Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), L(D|C) is the evidence Z(D|C), Z(D|C) = � L(D|C, G)P(G)dG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (25) As an alternative, one can also use the profile likelihood, Lp(D|C) = max G L(D|C, G) , (26) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the maximum likelihood obtained over all galaxy–halo connection parameters, as a replacement for the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The advantage of this approach is that L(D|C) becomes inde- pendent of poorly motivated priors on the galaxy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We will compare the performance of both approaches in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' To calculate the evidence, maximum likelihood and poste- riors on galaxy–halo connection parameters for each simula- tion, we employ the nautilus5 sampler version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 (Lange, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use 3000 live points, discard points during the exploration phase and run the sampler until an effective sam- ple size of 100, 000 is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 Cosmological inference Once we have computed the summary statistic, either the ev- idence Z or the profile likelihood Lp, we have to characterise the dependence of those summary statistics on the cosmol- ogy of each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Full-scale redshift-space clustering is sensitive to f(z)σ8(z) (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022), where f is the lin- ear growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Conversely, projected galaxy clustering and galaxy–galaxy lensing are sensitive to both Ωm and σ8(z) (Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since our analysis exhibits joint depen- dence upon these three cosmological parameters, we elect to model the summary statistic also as a function of three pa- rameters that fully specify f(z), σ8(z), and Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Here, we choose these three cosmological parameters to be S8, Ωm, and w, the Dark Energy equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We assume that the dependence of Z and Lp on S8, Ωm, and w can be mod- elled as a multi-variate skew-normal distribution (Azzalini & 4 This is strictly true only for Acen = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, observations do not tightly constrain Acen and it is always consistent with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='com/johannesulf/Nautilus Valle 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For reference, in one dimension, the probability distribution function (PDF) of a skew-normal distribution is given by, f(x) = 2 σ φ �x − µ σ � Φ � λx − µ σ � , (27) where φ and Φ are the PDF and cumulative distribution func- tion (CDF) of the standard normal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This func- tional form is motivated empirically (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019b) and is a natural extension of the assumption of a Gaussian pos- terior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' A three-dimensional skew-normal has 12 free param- eters: three means µ, three standard deviations ˆσ = σ/(1 + σ) ∈ [0, 1], three skew parameters δ = λ/ √ 1 + λ2 ∈ [−1, +1] as well as three parameters for the off-diagonals of the co- variance matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' rS8,Ωm, rS8,w, and rΩm,w ∈ [−1, +1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We obtain our full posterior constraints on S8, Ωm, and w as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Beginning with the collection of 406 values for Z(D|C) or Lp(D|C) (depending on which we use for the summary statistic), we approximate the distribution of these values with a skew-normal distribution and use an MCMC to derive full posterior distributions on the 12 skew-normal hyper-parameters plus one additional free parameter for the likelihood scatter of each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Each point in this 13- dimensional hyper-parameter space represents a particular skew-normal approximation to the cosmology-dependence of our summary statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We draw N samples of our hyper- parameters based on the posteriors on these quantities, and we derive our constraints on S8, Ωm and w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' P(S8, Ωm, w) based on the superposition of the resulting collection of nor- malised skew-normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), we place ex- plicit flat priors on cosmological parameters, in this case S8, Ωm and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is done to take into account that we cannot reliably extrapolate the dependence of Z and Lp on parts of the cosmological parameter space not probed by the Aemu- lus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The prior is determined by placing a three- dimensional minimum-volume enclosing ellipsoid around the parameter combinations of the Aemulus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Com- binations of S8, Ωm, and w outside this ellipse are assigned 0 prior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We refer the reader to Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) for a detailed discussion and motivation of the fitting proce- dure described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In order to verify that our conclusions are robust to our assumption that the cosmology-dependence of Z and Lp is well-described by a skew-normal form, in appendix A we conduct an alternate analysis in which we alternatively use a Gaussian Process regression to approxi- mate these distributions, finding very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, we note that our final constraints on cosmology should not be regarded as being model-independent or valid outside the wCDM cosmological parameter space probed by the Aemu- lus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 4 VERIFICATION ON MOCK CATALOGUES Before applying our analysis method to the observational data described in section 2, we test our methods on mock 6 In practice, we always exclude the simulation Box023 since it has an unusually low S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='595, well below the second lowest value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The motivation is that this simulation is almost always confidently excluded by observations and we do not want its very low likelihood to influence the fit to L(D|C) in regions of cosmological parameter space allowed by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' catalogues to ensure that our cosmological inferences do not suffer from significant biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 Mock observations The mock catalogues used here are described in more de- tail in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' They are created from the UNIT simulations (Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019) and the associated Rock- star halo catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We populate haloes in the z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='25 outputs of each of the four UNIT simulations with galaxies using the subhalo abundance matching framework (SHAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Vale & Ostriker 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In essence, we place galaxies at the centres of the haloes with the highest α log Vpeak + (1 − α) log Vvir, where Vvir is the halo virial ve- locity at the epoch of peak halo mass, Vpeak the correspond- ing value of Vmax at that epoch and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='73 (Lehmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We populate haloes until the number density of galaxies matches that of the LOWZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ⩽ z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Haloes in this procedure include both parent haloes as well as subhaloes such that satellite galaxies are naturally accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As described in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), small modifications to the SHAM procedure are applied to account for scatter between galaxy and halo properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Once mock galaxy cat- alogues are created, we compute mock observables assuming a distant observer approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since we have four simu- lations of 1 Gpc3 h−3 volume each, and we consider all three simulation axes as the line of sight, the mock measurements have an effective volume of ∼ 12 Gpc3 h−3 (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The mock observations for galaxy clustering and galaxy– galaxy lensing are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Error bars represent the observational uncertainties of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ⩽ z ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 NGC (SGC) sample for galaxy clustering (galaxy–galaxy lensing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the following, we will use these same observational uncer- tainties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' covariance matrix, to fit our model to the mock data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The best-fit model predictions, marginalised over all galaxy parameters and all 40 simulations, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 1 by the solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, our HOD-based galaxy model, which is an empirical model that is founded upon different as- sumptions than the SHAM model used to produce the mock catalogues, can produce good fits to the mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Cosmology results Cosmological parameter constraints for the simulated dataset are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In each of the figures, red lines indicate posterior constraints when fitting both RSDs and lensing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' ξ0, ξ2, ξ4 and ∆Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Blue lines indicate posterior constraints when only redshift-space clustering is considered in the fit, and purple lines indicate results obtained without redshift-space clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since galaxy–galaxy lensing is only sensitive to cosmology in combination with galaxy clustering (Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006), we also include the projected two-point correlation function, wp, in that particular fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since wp is obtained by projecting the redshift-space two-point correla- tion function along the line of sight, it is mostly insensitive to redshift-space distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As shown in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), wp by itself has little cosmological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The difference between Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2 and 3 is that the former uses the evidence Z as the summary statistic, whereas the latter uses the profile likelihood Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We see that our analysis places strong constraints on S8, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Both redshift- space clustering and galaxy–galaxy lensing obtain roughly 0 20 40 r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 ξ [h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 Mpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5] ξ0 ξ2 ξ4 Mock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 s [h−1 Mpc] 0 2 δξ 8 10 12 rp ∆Σ [106M⊙/pc] Mock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 rp [h−1 Mpc] 3 0 +3 δ∆Σ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Mock observations of galaxy clustering in redshift space (top) and galaxy–galaxy lensing (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Dots indicate the mock measurements themselves, error bars the assumed observational uncertainty, and the solid line is the best-fit model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The lower panels show the difference between mock observations and best-fit model predictions in units of the observational uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Grey backgrounds indicate the ranges of the small-scale data that are not included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' comparable constraints on S8 with slightly different degen- eracies with respect to the other cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' On the other hand, neither RSDs nor lensing lead to strong constraints on Ωm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the posterior PDF is very similar to the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, we nominally get noteworthy constraints on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, a significant fraction of this constraint may originate indirectly from the constraint on S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Specifically, our assumed prior implies a strong correlation between S8 and w such that any constraint on S8 also leads to a strong constraint on w, as is evident from the lower left panels in each of the two figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We find that both approaches, either employing the evi- dence or the profile likelihood, are successful in recovering the input cosmology of the UNIT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, the fit using the evidence as the summary statistic produces quantitatively similar results to the analysis using the pro- file likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As described before, the effective volume of the mock observations is roughly ∼ 12 Gpc3 h−3, a factor of ∼ 40 larger than the observational volume from which MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 9 wp + ∆Σ RSD-only RSD + ∆Σ truth prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='90 S8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 Ωm −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 w Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Posterior constraints on cosmological parameters when analysing the mock data set derived from the UNIT simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We show the results from analysis of projected clustering and galaxy–galaxy lensing (purple), the study of redshift-space mul- tipoles (blue), and the combination of the redshift-space cluster- ing and galaxy–galaxy lensing (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In all panels we highlight the cosmological parameters of the UNIT simulations (yellow) we seek to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, the grey lines indicate our prior: in the off- diagonal panels it shows the range and in the diagonal elements the implicit one-dimensional prior implied by projecting the vol- ume of a three-dimensional ellipsoid onto one axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For this figure, the evidence Z was used as the summary statistic for each of the 40 Aemulus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' wp + ∆Σ RSD-only RSD + ∆Σ truth prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='90 S8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 Ωm −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 w Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2 but with the profile likelihood Lp instead of the evidence used as the summary statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the assumed observational uncertainties come from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, we expect our results to reproduce the input to much bet- ter than the assumed observational uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' With this in mind, it is noteworthy that the posterior constraints on S8 are slightly off-centred when fitting the evidence for either the case of redshift-space clustering or galaxy–galaxy lens- ing in isolation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' this is not the case for the fit involving the profile likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Moreover, the profile likelihood has a com- paratively stronger theoretical motivation since it is indepen- dent of the arbitrary priors on the galaxy–halo connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For these reasons, we will choose the profile likelihood as the default summary statistic when analysing the real data in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 5 MASKING STRATEGY For scientific studies, we want to protect the experiment and analysis design from confirmation bias of the scientists lead- ing the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' A common strategy to avoid this issue is to “blind” the analysis with respect to the key scientific re- sults while still being able to perform critical null tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In our example, we wish to make analysis choices insensitive to the cosmological constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' S8, while still being able to judge, for example, the goodness of fit of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Through- out this work, we will use the term “masking” instead of the more commonly used term “blinding”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 Proposed method A masking procedure can in principle happen at various stages of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the case of posterior masking, one would simply hide or randomly perturb the final cosmological posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Data vector masking involves perturbing the sum- mary statistics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' ξ and ∆Σ, in ways that also randomly perturb the final cosmological result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, masking can be implemented at the catalogue level such that both summary statistics and cosmological posterior are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Data vec- tor and catalogue level masking are harder to implement than posterior masking but are more robust in the sense that they are less prone to accidental unmasking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We refer the reader to Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) for a detailed discussion and motiva- tion behind different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In this work, we will apply a data vector masking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The method employed here is a variation of the data vector masking procedure described in Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Let us assume that we have an unperturbed data vector D and a model for that data vector �D(θ) where θ represents the model parameters we wish to constrain with the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) propose perturbing D by ∆D(∆θ) = �D(θfid + ∆θ) − �D(θfid) , (28) where ∆θ is an offset in the model and θfid are suitably chosen set of fiducial model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Under idealised conditions, adding ∆D(∆θ) to the unperturbed data vector will shift the posterior of θ by ∆θ while leaving the goodness of fit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the minimum χ2, unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In practice, these idealised condi- tions are not perfectly met, such that the above statements are only approximately true and the masking procedure needs to be validated with simulations first (Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In our analysis, we wish to mask the final constraints on S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Constraints on the galaxy–halo connection are of less interest MNRAS 000, 1–20 (2023) 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 10−1 100 101 Distance s [h−1 Mpc] −20 −10 0 10 20 30 (d �D/dS8)/σD ξ0 ξ2 ξ4 ∆Σ wp Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Estimates of the derivatives of the best-fit predictions as a function of S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The derivatives are derived from the mock data RSD fits presented in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' in this study and could be described as nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The original method proposed in Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) would calculate ∆D(∆θ) by looking at the difference in predictions for two cosmologies with different S8 values while keeping the galaxy–halo connection parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We may choose to perturb S8 by up to ∆S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1, which, as we will show later, would correspond to up to 4σ shifts in the final S8 posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the impact of cosmology and galaxy–halo connec- tion parameters on the data vector is often degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For example, the large-scale clustering amplitude is sensitive to bσ8 where b is the galaxy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As a result, implementing the method described in Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) could result in data vector shifts that are large with respect to the observational uncertainties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' significantly larger than 4σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Here, we propose a slight variation of the method described in Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When calculating ∆D, instead of only changing S8 while keeping all other model parameters fixed we instead change S8 and then vary all other model parame- ters such that the difference between �D(θfid+∆θ) and �D(θfid) is minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Here, we define the difference between the data vectors as their χ2 difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This ensures that the shift in the data vector is as small as possible while ensuring the de- sired shift in S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For example, a 4σ shift in S8 would result in a roughly 4σ significant shift of the data vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, this procedure seems more likely than the original Muir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) method to preserve the goodness of fit after masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Specific implementation We now want to apply the above mentioned method to our application by masking the value of S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, in simulation-based modelling, we can only make predictions for a handful of cosmologies and the predictions are inher- ently noisy due to finite simulation sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' To overcome these problems, we slightly modify the method introduced above while keeping the main idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the mock analysis in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), we fit our model to a mock RSD vector and get best-fit model predictions �D for all 40 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We then use the 40 predictions to fit for a linear relation between �D and the input S8 value to estimate d �D/dS8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 S8 Posterior ∆S8 χ2 Change 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 → 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 → 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 wp + ∆Σ RSD-only RSD + ∆Σ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Change in the S8 posterior constraints from the mock data set when applying the masking procedure outlined in sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We show the original input S8 value (yellow dashed verti- cal line) and the expected S8 shift ∆S8 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 (black arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As expected, the posterior constraints obtained from the masked data (solid lines) are shifted by roughly −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 compared to the same con- straints obtained from the unmasked data (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the upper right corner, we also indicate the change in the goodness of fit due to applying the masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The resulting derivatives are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 4 and discussed further in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As expected, the figure indicates that S8 and ∆Σ predictions are positively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' To mask S8, we add the following offset to the data vector D: ∆D(∆S8) = d �D dS8 × ∆S8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (29) We test the above masking procedure on the mocks in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We choose a target shift of ∆S8 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 5, we show the shift in the S8 posterior induced by the this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As expected, irrespective of whether we analyse wp and ∆Σ, the redshift-space correlation function or the combination of lensing and redshift-space clustering, the S8 posterior shifts by roughly −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 in S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, as shown in the same figure, the goodness of fit is close to unaffected by the masking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' These findings on mock catalogues indicate that the masking procedure is expected to give a good performance in practical applications such as the one in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We then proceed to apply the masking procedure to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For each of the three redshift bins, we choose a random ∆S8 that is drawn from a uniform distribution in the range [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='075, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='075].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In principle, larger ranges would be ideal to erase any meaningful correlation between the masked and unmasked result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, for our simulation-based analy- sis, we need to ensure that the value of S8 that the masked data prefers is covered by the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, we cannot make the range for ∆S8 arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The range chosen here was selected as a compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Note that we apply the same S8 masking shift for all analyses within each redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Therefore, even with the masking, we were able to judge the consistency between constraints coming from RSDs and lensing as well as between results from the NGC and the SGC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The entire analysis in the next section was first MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 11 performed and checked with the masked data and only un- masked after all authors agreed on all analysis choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 6 RESULTS We now proceed to apply the modelling framework to the ob- servational galaxy clustering and galaxy–galaxy lensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Here, we concentrate on posterior constraints on cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Constraints on the galaxy–halo connection are presented and discussed in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 Model fits In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 6, we show the best-fit model predictions for each of the six samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Each model was fitted to the redshift- space clustering and galaxy–galaxy lensing data jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The fits were marginalised over both the galaxy–halo connection parameters as well as over the 40 Aemulus simulations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The minimum χ2 values are 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 and 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0) for the three bins from low to high red- shift in the NGC (SGC) with 39 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When fitting all data with a single simulation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 6, the best-fit χ2 is 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Although our HOD model has 11 free parameters, the number of effective degrees of freedom of the galaxy–halo connection is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We numerically determine this number by taking model predictions, randomly perturbing them ac- cording to the observational uncertainty and minimising χ2 over HOD parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We find that the number of ef- fective degrees of freedom of the HOD model with respect to fitting redshift-space clustering and lensing jointly is ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, while we have seven cosmological parameters that are varied in the Aemulus simulations, the likelihood seems to be only a function of around two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, we estimate to have around ∼ 2 effective degrees of freedom in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, when fitting redshift-space clustering and lensing, the number of effective degrees of freedom Ndof is approxi- mately 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 when fitting a single sample of galaxies and 6 × (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 = 181 when fitting all six galaxy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 7, we show the distribution of best-fit χ2 values for the mocks when fitting all six sam- ples to redshift-space clustering and galaxy–galaxy lensing together with the best-fit χ2 from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, we find χ2 ν = χ2/Ndof ≈ 1 for all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The largest χ2 ν value, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2/28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5, has a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, overall, our model provides a good fit to all the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Cosmology Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 8 shows our constraints on the cosmological parame- ter S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Results are presented for all six samples individu- ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We also distinguish between redshift-space clustering- only fits, the combination of projected clustering and galaxy– galaxy lensing, and joint redshift-space clustering and lens- ing fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, the different samples show good agreement for the value of S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, the RSD-only fits and the joint projected clustering plus lensing fits are also in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When combining all six galaxy samples, we obtain S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='022 from gravitational lensing and S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='027 from redshift-space clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We also investigate how the redshift-space clustering result depends on the scales considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' By default, we consider all scales sample wp + ∆Σ RSD-only RSD + ∆Σ A NGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='815 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='813 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='807 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='038 A SGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='818 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='783 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='810 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='035 B NGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='735 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='774 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='754 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='035 B SGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='776 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='750 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='761 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='045 C NGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='746 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='789 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='763 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='043 C SGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='802 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='727 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='798 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='045 combined 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='779 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='020 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Posterior constraints on the cosmological parameter S8 as a function of the galaxy sample analysed (different rows) and the observational constraints (different columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' larger than 400 h−1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When only analysing scales larger than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 h−1 Mpc, we obtain S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='806 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, when looking at the combination of reshift-space clustering and lensing for all six samples, we obtain S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='779 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The derived constraints on S8 are also listed in Table 3, for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 9, we show the full cosmology constraints on S8, Ωm and w when considering all six samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In ad- dition to constraints on S8, we infer w = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='915 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='113, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='967 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='076 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='963 + / − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='069 for observations of wp + ∆Σ, RSD-only and RSD +∆Σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' On the other hand, we do not obtain strong constraints on Ωm and are instead dominated by the Aemulus prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 Lensing amplitudes derived from different lensing data sets Here, we compare the measured galaxy–galaxy lensing am- plitudes between KiDS and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, we contrast them with galaxy–galaxy lensing measurements obtained with SDSS lensing catalogues (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In this sec- tion, we use lensing measurements that extend to smaller radial scales than used in the fiducial cosmology analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The lensing amplitude ∆Σ is a physical quantity that should only depend on lens properties and be independent of the shape catalogue used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In a recent study, Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) used this insight to compare the results of different lensing catalogues, including SDSS, KiDS and DES, finding good agreement between all of them within the statistical and systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the findings of Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) are based on older DES Y1 and KiDS-450 data whereas our new DES and KiDS lensing measurements have substantially more statistical constraining power and reduced systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We compare the SDSS, DES and KiDS lensing measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The lensing measurements used for our cosmology analysis are limited to scales where boost fac- tors corrections due to physical lens–source associations are unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Here, we extend the measurements to smaller radial scales and, thus, include boost factor corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Over- all, we see that the DES Y3 lensing measurements tend to be higher than the SDSS and KiDS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' To estimate the statistical significance of the difference in the lensing am- plitudes, we follow the approach of Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In particular, we first determine an overall lensing normal- isation A by fitting the observed lensing amplitude ∆Σobs with a template ∆Σtemplate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Afterwards, we compare the nor- malisations between the different lensing amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For the template, we choose the best-fit lensing prediction when ana- lyzing the combination of galaxy redshift-space clustering and MNRAS 000, 1–20 (2023) 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 0 20 40 r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 ξ [h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 Mpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5] A-N ξ0 ξ2 ξ4 B-N ξ0 ξ2 ξ4 C-N ξ0 ξ2 ξ4 3 0 +3 δξ 0 20 40 r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 ξ [h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 Mpc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5] A-S ξ0 ξ2 ξ4 B-S ξ0 ξ2 ξ4 C-S ξ0 ξ2 ξ4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 s [h−1 Mpc] 3 0 +3 δξ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 s [h−1 Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 s [h−1 Mpc] 0 5 10 15 rp ∆Σ [106M⊙/pc] A-N B-N C-N 3 0 +3 δ ∆Σ 0 5 10 15 rp ∆Σ [106M⊙/pc] A-S B-S C-S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 rp [h−1 Mpc] 3 0 +3 δ ∆Σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 rp [h−1 Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 10 rp [h−1 Mpc] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Data measurements and best-fit model predictions when analysing galaxy clustering and galaxy–galaxy lensing jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Upper panels show the clustering measurements and fits whereas the lower panels display the lensing measurements and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Each panel indicates the sample displayed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' “A-S” denotes sample A in the SGC area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For each set, the upper panels show the absolute measure- ments (data points) and predictions (solid lines) and the lower panels show the difference between the best-fit model predictions and the data in units of the observational uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 1, grey backgrounds indicate the ranges of the data that are not included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' All fits use the Aemulus simulation box B04 which provides the best fit to the combination of all observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 13 125 150 175 200 225 250 275 χ2 min Distribution Mocks χ2-fit Data Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The distribution of best-fit χ2-values in perturbed mocks when fitting all data (solid blue), an analytic χ2-distribution with the same mean as the mocks (dashed blue) and the best-fit χ2 from the actual data (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For each of the perturbed mocks, we fully marginalised over all 11 galaxy–halo connection parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' galaxy–galaxy lensing with the simulation box B04, though the exact choice of template does not strongly affect the re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The results are shown in Fig 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When comparing DES and SDSS on scales rp > 1h−1 Mpc, we find that SDSS lens- ing amplitudes are (20 ± 6)%, (14 ± 8)% and (23 ± 10)% lower on average than the DES amplitudes for samples A, B and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When doing the same comparison for KiDS, we find (−13 ± 9)%, (14 ± 16)% and (13 ± 19)%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' KiDS and SDSS agree well on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We note that the quoted uncertainties are the statistical uncertainties only and do not account for systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The latter should be dominated by the SDSS photometric redshift calibration and is of order 6% (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Taking into account this systematic uncertainty, none of the offsets at large scales are statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' On the other hand, differences on smaller scales have a stronger significance but are subject to uncertainties regarding boost factor estimates (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We leave an in-depth comparison of lensing am- plitudes measured with different lensing data sets to a future study with new lenses from the DESI survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 7 DISCUSSION To our knowledge, the current work represents the first full-scale combined analysis of redshift-space clustering and galaxy–galaxy lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Using this approach and combining all LOWZ samples, we obtain a ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5% constraint on S8, one of the most stringent constraints on S8 to date (Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 S8-tension In this work, we present two roughly independent constraints on the growth of structure amplitude S8 based on the analysis of redshift-space clustering and the combination of projected clustering and galaxy–galaxy lensing, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='027 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' By contrast, the most re- cent Planck2020 CMB analysis prefers S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='834 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This represents a ∼ 2σ discrepancy in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' While this offset is not statistically significant, our results follow a long-standing trend whereby low-redshift probes of cosmic structure growth infer a lower value for S8 than studies of the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 12, we compare our results against CMB results (Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Aiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020) and other low-redshift large-scale structure studies, including cosmic shear (Hikage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022), the combination of projected galaxy clustering and lensing cross-correlation (Krolewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022b), so-called 3 × 2pt stud- ies (Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022), redshift-space clustering (Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Philcox & Ivanov 2022), the combination of redshift-space clustering and lensing cross- correlation (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) as well as cluster counts (Mantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Bocquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, our results are in excel- lent agreement with other low-redshift probes which also pre- fer S8 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our analysis is the first to analyse BOSS LOWZ galaxies cross-correlated with KiDS-1000 and DES Y3 to fit for cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, our constraints from redshift-space clustering are primarily derived from scales not analysed in conventional large scale-only RSD studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=', Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Philcox & Ivanov 2022), which we have achieved through a simulation-based model that marginalises over uncertainties in galaxy assembly bias and other astro- physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We note that two other recent full-scale RSD studies of the BOSS CMASS (Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) and the eBOSS LRG samples (Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) also find a lower growth of structure amplitude than predicted by Planck2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, our results add further evidence for the existence of an S8-tension between low redshift and CMB data and to the evidence that this discrepancy is not limited to studies involving gravitational lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 Lensing is low For a given set of cosmological parameters, the observed galaxy clustering amplitude makes precise predictions for the galaxy–galaxy lensing amplitude, even after marginal- ising over uncertainties in the galaxy–halo connection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Cacciato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Recently, several studies have shown that the lensing ampli- tude is significantly over-predicted when assuming the best- fit Planck2020 cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This discrepancy is most significant on small scales, rp < 5 h−1 Mpc where the difference is around 30% (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the matter distribution on such small scales is also affected by baryonic feedback effects which are often not explicitly modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This may contribute to the apparent lensing-is-low effect on small scales (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amodeo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021), which we do not analyse here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' On large scales, the lensing measurements have increased uncertainties and therefore it is less clear to what extent a similar lensing-is-low tension exists at those scales, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Recently, Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2021), using BOSS LOWZ cluster- ing and SDSS galaxy–galaxy lensing measurements, also find a statistically significant difference of ∼ 30 − 35% on large MNRAS 000, 1–20 (2023) 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 S8 0 10 20 Posterior dp/dS8 RSD-only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 S8 wp + ∆Σ NGC SGC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 S8 RSD + ∆Σ combined Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Posterior constraints on S8 from the combination of wp and ∆Σ (left), redshift-space distortions (middle) and the combination of redshift-space clustering and lensing (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In each panel, we show constraints from different redshift bins (different coloured lines) from the NGC (solid) and SGC (dashed) area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, we show constraints when combining all six LOWZ data samples (black solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' wp + ∆Σ RSD-only RSD + ∆Σ prior 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 Ωm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='90 S8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 Ωm −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 w Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Cosmological constraints from BOSS LOWZ when analysing all six samples in this work jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We show con- straints derived from a combination of projected galaxy cluster- ing and galaxy–galaxy lensing (purple), redshift-space clustering (blue) and the combination of redshift-space clustering and lensing (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We also show the prior imposed by the Aemulus simulations (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Recall that the lensing predictions at fixed projected clustering roughly scale as S8 (Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, studies cross-correlating BOSS LOWZ galaxies with SDSS shape cat- alogues and inferring S8 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='03 (Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020) and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='04 (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020), considerably below the Planck CMB prediction, also corroborate the existence of a lensing-is-low problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' More recently, Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2023) re- evaluated the lensing-is-low tension using updated DES and KiDS lensing measurements, finding a difference at the level of 15% for LOWZ on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, their results are offset with respect to with the Planck CMB prediction at the ∼ 2σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Finally, our results also imply a mild, 2σ tension with the Planck2020 results but not at the level reported in earlier studies relying on SDSS lensing measurements (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As shown in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3, we find that for LOWZ galaxies the SDSS lensing catalogues imply lower lensing amplitudes for the same lens samples than the DES Y3 catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We leave a detailed investigation of the statistical significance of this finding to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Nonetheless, this difference helps explain why earlier studies based on SDSS lensing measure- ments (Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) find stronger levels of tension with Planck CMB pre- dictions than Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2023) and the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 Full-scale studies Using a full-scale approach, growth of structure constraints from redshift-space distortions become competitive with lead- ing gravitational lensing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Recent large scale-only, full- shape studies using BOSS LOWZ and CMASS achieve a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='045 uncertainty on S8 (Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Philcox & Ivanov 2022), whereas here we obtain a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='027 constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Fur- thermore, in this work we only use the BOSS LOWZ sam- ple and do not analyse the two times larger BOSS CMASS sample, unlike the aforementioned large scale-only studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 4, we show the derivative of the RSD multipoles with respect to changes in S8 after fully marginalising over galaxy– halo connection parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, this figure indicates where cosmological constraints from full-scale studies are coming from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' It indeed suggests that significant information on S8 de- rives from scales smaller than 10 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At the same time, the figure also confirms the finding in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) that little to no cosmological information is contained for RSD multipoles below s ∼ 1 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We also repeat the RSD analysis using only scales above 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 h−1 Mpc, finding that our constraints on S8 degrade by 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Additionally, we find that excluding the smallest scales from our analysis leads to a constraint on S8 that, while being higher (also see Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022), is in good agreement with the result from the default analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, our results suggest that full-scale studies have the potential of MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 15 10−1 100 101 Projected Radius rp [h−1 Mpc] 0 5 10 15 Lensing rp∆Σ [106M⊙/pc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ≤ zl <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 DES KiDS SDSS 10−1 100 101 Projected Radius rp [h−1 Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 ≤ zl <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 10−1 100 101 Projected Radius rp [h−1 Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 ≤ zl <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Galaxy–galaxy lensing measurements from cross-correlating BOSS LOWZ targets with lensing catalogues from DES (red), KiDS (purple), and SDSS (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Different panels correspond to the three different redshift-binned samples analysed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Measurements are slightly offset in the x-direction for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' DES KiDS SDSS < 1 Mpc/h > 1 Mpc/h all scales 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 A/ADES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ≤ zl <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 < 1 Mpc/h > 1 Mpc/h all scales 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 A/ADES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 ≤ zl <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 < 1 Mpc/h > 1 Mpc/h all scales 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 A/ADES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 ≤ zl <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lensing amplitudes averaged over different scales as a function of the lensing data set, the lens galaxy sample and the scale range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Three different redshift bins are shown from low redshift (top) to higher redshift (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' improving cosmological constraints from redshift-space clus- tering by a factor of 2−3 over traditional large scale-only full- shape studies, even after marginalising over complex galaxy– halo connection models (also see Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In ad- dition to placing independent, high-precision constraints on the S8-tension, analyses of the non-linear regime could also enable precise tests of modifications to gravity in the future (Blake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Regarding the combination of projected clustering and galaxy–galaxy lensing, we model the lensing signal down to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Nominally, this is not a significant improvement to, for example, the recent analysis by Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), where scales down to 6 h−1 Mpc are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the analysis of Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) marginalises over a free point-mass term, which scales as ∆Σ ∝ r−2 p and is designed to remove systematics in the modelling of non-linear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As described in Prat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022), this reduces the signal-to- noise ratio of the lensing measurements analysed in Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) from 67 to 32 and primarily affects small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) and similar analyses marginal- ising over a point-mass term, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020) and Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2020), are by design less sensitive to the pre- dicted lensing amplitude on small scales relative to studies without point-mass marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In this work, we do not marginalise over a point-mass term since we argue that our simulation-based modelling framework and complex galaxy– halo connection model should already accurately model scales down to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc, thereby allowing us to leverage this ad- ditional information without the sacrifice in signal-to-noise that results from point-mass marginalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Currently we do not explore even smaller scales in order to avoid the need to model baryonic feedback processes (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amodeo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the lensing mea- surements on scales down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 h−1 Mpc have a signal-to- noise ratio of 57 compared to 26 when limiting the anal- ysis to scales > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Thus, if we would be able to empirically constrain and marginalise over baryonic feed- back processes, we could potentially get even more stringent growth-of-structure constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Unfortunately, galaxy clus- tering alone is unlikely to place strong constraints on baryonic feedback, thus marginalising over flexible baryonic feedback models with the present data is unlikely to result in more stringent S8 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, combining galaxy cluster- ing and lensing with data on the baryon distribution around galaxies, such as measurements of the Sunyaev–Zel’dovich (Schaan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amodeo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021), may break that de- generacy, and could unlock the constraining power of small- scale lensing measurements for cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The constraining power of full-scale studies might be fur- MNRAS 000, 1–20 (2023) 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 S8 = σ8 � Ωm,0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 Cosmic Microwave Background Planck Planck20 ACT+WMAP Aiola+20 Cosmic Shear DES Amon+22 HSC Hikage+19 KiDS Asgari+21 Projected Clustering and Lensing DES×DES Porredon+22 BOSS×HSC Miyatake+21 unWISE×Planck-κ Krolewski+21 LOWZ×DES/KiDS this work Shear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Clustering and Lensing DES×DES Abbott+22 BOSS/2dFLenS×KiDS Heymans+21 Redshift-Space Clustering BOSS Ivanov+20 BOSS Philcox+21 LOWZ this work Redshift-Space Clustering and Lensing BOSS×Planck-κ Chen+22 LOWZ×DES/KiDS this work Cluster Counts ROSAT Mantz+15 Planck-SZ Planck16 SPT Bocquet+19 DES Abbott+20 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Comparison of different literature constraints on S8 against the results derived in this work (results from Mantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Hikage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Bocquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Aiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Krolewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Philcox & Ivanov 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Blue band indicates the prediction of Planck2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' ther improved by using larger simulations with reduced sam- ple variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The current Aemulus simulations have a vol- ume of (∼ 1 h−1 Gpc3) each, roughly the same as the total LOWZ volume analysed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since our analysis approach incorporates uncertainties in simulation predictions, con- straints would likely become more stringent with larger sim- ulations such as the AbacusSummit simulation suite (Mak- simova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' On the observational side, most cur- rent spectroscopic large-scale structure surveys are designed to be cosmic variance limited on large scales, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' up to k ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 h Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Due to this design choice, highly non-linear scales are currently mostly dominated by shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Future galaxy surveys with a higher sampling density might improve the constraining power of non-linear scales further for the same observational volume (Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Such high- density surveys would also be ideal targets for multi-tracer studies, another method to reduce the impact of cosmic vari- ance (McDonald & Seljak 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At the same time, simulations and modelling frameworks need to be developed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For example, current gravity solvers predict an up to 1% different halo (matter) clustering amplitude at k ∼ 1(10) h Mpc−1 respectively at low redshifts (Grove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' With increasing precision of large-scale structure measurements, these differences might soon domi- nate uncertainties in the cosmological interpretation of non- linear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, more work needs to be done to en- sure that the galaxy–halo connection models used to analyse the data are sufficiently general and do not bias cosmology results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' While several cosmology recovery tests (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) have been performed on SHAM mock galaxy catalogues, including in this work, tests on additional, more complex galaxy models, including semi-analytic mod- els and hydrodynamic simulations (see Wechsler & Tinker 2018, for a review), are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At the same time, marginal- ising over more physically-motivated galaxy–halo connection models than HODs such as the UniverseMachine (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019) or Emerge (Moster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2018) might reduce cosmological uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We leave such investigations to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 8 CONCLUSION In this work, we present a novel simulation-based joint cosmological analysis of galaxy redshift-space clustering and galaxy–galaxy lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our work extends previous simulation-based full-scale studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wibking et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Miyatake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022) in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' For example, this is the first full- scale cosmological study involving gravitational lensing that also models galaxy assembly bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, this is the first simulation-based cosmological full-scale study that uses the most recent DES Y3 and KiDS-1000 gravitational lens- ing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Most importantly, we also present the first joint full-scale analysis of redshift-space clustering and galaxy–galaxy lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our analysis incorporates a complex galaxy–halo connec- tion model, including the effects of galaxy assembly bias as well as central and satellite velocity bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our best-fit model is able to provide a good fit to the observed galaxy clustering and lensing amplitudes over a wide range of scales, down to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc for lensing and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 h−1 Mpc for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our main result are new, highly competitive constraints on the cosmic growth of structure, particularly S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' These findings can be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When analysing the combination of projected clustering and galaxy–galaxy lensing, we infer S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='022 while for redshift-space clusetering we infer S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='771 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Fi- nally, combining redshift-space clustering and galaxy–galaxy lensing, we find S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='779 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We find good agreement regarding S8 between multiple independent galaxy samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, constraints derived only from redshift-space clustering are consistent with those relying on gravitational lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' When repeating our analysis of redshift-space clustering using only larger scales above s > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 h−1 Mpc instead of MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 17 400 h−1 kpc, we achieve statistically consistent results, but our constraining power on S8 degrades by 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our results favour a value for S8 below the best-fit value inferred by the Planck2020 CMB analysis, S8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is in agreement with results from other analyses of the low- redshift Universe, the so-called S8-tension (see Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similar to other recent full-scale studies of galaxy redshift-space clustering (Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022), we find evidence for an S8-tension with predictions from Planck2020, even without incorporat- ing gravitational lensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our analysis also highlights the strong constraining power of full-scale studies over similar analyses targeting only large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Particularly, our constraints on S8 using only redshift- space clustering are a factor of two more stringent than recent large scale-only studies of BOSS galaxy redshift-space clus- tering (Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Philcox & Ivanov 2022), despite only using a small fraction of the data compared to these other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, our constraints derived from the com- bination of projected galaxy clustering and galaxy–galaxy lensing are significantly more stringent than, for example, a recent DES Y3 analysis using those observables (Porredon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We attribute part of this improvement to con- sidering scales down to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc for the lensing signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Furthermore, we argue that our complex modelling frame- work alleviates the need to marginalise over a point mass, further increasing sensitivity to high-S/N non-linear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The current study represents a new advance in full-scale cosmological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We anticipate building upon this in the future in several respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the present study, we do not use scales below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 h−1 Mpc since our galaxy model model does not include baryonic feedback (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, the signal-to-noise ratio of the lensing amplitude more than doubles when considering scales down to rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 h−1 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Observations of the Sunyaev– Zel’dovich effect can place independent constraints on the strength of baryonic feedback (Schaan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Amodeo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' thus, a combined analysis of clustering, lensing and the Sunyaev–Zel’dovich effect could improve constraining power even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We aim to apply the full-scale approach to upcoming data from the DESI survey using the high- resolution, large-volume AbacusSummit simulations (Maksi- mova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) instead of Aemulus for the modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is also expected to improve cosmological constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At the same time, in light of this increased constraining power, more tests on complex, highly-realistic galaxy mock catalogues are needed to verify the robustness of full-scale constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' An- other avenue for full-scale studies is to provide tight priors on galaxy bias models to be used with large-scale hybrid effec- tive field theory cosmology studies (see Kokron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022, for a recent example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the Aemulus collaboration and the UnitSims team for making their simulations publicly available as well as Sandy Yuan, Jeremy Tinker, and Zhongxu Zhai for in- teresting discussions on several aspects of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We also thank Sukhdeep Singh for providing the SDSS lensing measurements discussed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We acknowledge use of the lux supercomputer at UC Santa Cruz, funded by NSF MRI grant AST 1828315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This work was partially supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of High Energy Physics under Award Num- ber DE-SC0019301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' JUL received support from a fellowship from the Leinweber Center for Theoretical Physics and from a Stanford-Santa Cruz Fellowship including support from the Kavli Institute for Particle Astrophysics and Cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' AL acknowledges support from the David and Lucille Packard foundation, and from the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Sloan foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' HG ac- knowledges the support from the National Natural Science Foundation of China (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 11833005, 11922305).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Work done by APH was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Nuclear Physics, under contract DE-AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' FvdB is supported by the National Aeronautics and Space Administration through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 19-ATP19-0059 issued as part of the Astrophysics Theory Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This work made use of the following software packages: matplotlib (Hunter 2007), SciPy, NumPy (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2011), Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013), Colossus (Diemer 2015), halotools (Hearin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017), MultiNest (Feroz & Hobson 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Feroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2009, 2019), PyMultiNest (Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2014), scikit-learn (Pe- dregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2011), emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013), UltraNest (Buchner 2021), Spyder and Setzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This project used public archival data from the Dark En- ergy Survey (DES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Funding for the DES Projects has been provided by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Department of Energy, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='S.' metadata={'source': 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+page_content=' the Institut de Ci`encies de l’Espai (IEEC/CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the Institut de F´ısica d’Altes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lawrence Berkeley National Lab- oratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the Ludwig-Maximilians Universit¨at M¨unchen and the associated Excellence Cluster Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the National Optical Astronomy Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the OzDES Membership Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the University of Pennsyl- vania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' SLAC National Acceler- ator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the University of Sus- sex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' and Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Based in part on observations at Cerro Tololo Inter- American Observatory, National Optical Astronomy Obser- vatory, which is operated by the Association of Universi- ties for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='A-3016, 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='A-3017, 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='A-3018 and 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='A-2004, and on data prod- ucts produced by the KiDS consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The KiDS produc- tion team acknowledges support from: Deutsche Forschungs- gemeinschaft, ERC, NOVA and NWO-M grants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the University of Padova, and the University Federico II (Naples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We use the gold sample of weak lensing and photomet- ric redshift measurements from the fourth data release of the Kilo-Degree Survey (Kuijken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Hildebrandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Giblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021) (Kuijken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019), hereafter referred to as KiDS-1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Cosmological parameter constraints from KiDS-1000 have been presented in (Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021, cosmic shear), (Heymans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021, 3 × 2pt) and (Tr¨oster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2021, beyond ΛCDM), with the methodology presented in Joachimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Funding for SDSS-III has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Sloan Foundation, the Participating Institutions, the Na- tional Science Foundation, and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='S.' metadata={'source': 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Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Max Planck Institute for Extraterrestrial Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' New Mexico State Uni- versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Pennsyl- vania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' the Spanish Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' DATA AVAILABILITY The Aemulus and UNIT simulations used in this ar- ticle are publicly available at https://aemulusproject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='io/ and http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='unitsims.' 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+page_content=', 2011, Computing in Science and Engineering, 13, 22 APPENDIX A: GAUSSIAN PROCESS MODELLING Our analysis method requires generalising the dependence of the maximum likelihood or evidence as a function of cosmol- ogy for the 40 Aemulus simulations to arbitrary cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' By default, we utilise the method described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 that fits the results for the 40 simulations with a multi-dimensional skew-normal distribution in the (S8, Ωm, w)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Here, we test an alternative approach using Gaussian Process (GP) emulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This follows similar applications of GP emulation in the literature (Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The main difference to the aforementioned works is that we are emulating only a single summary statistic, Lmax, as a func- MNRAS 000, 1–20 (2023) 20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='00 S8 Posterior wp + ∆Σ RSD-only RSD + ∆Σ Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Inferred posterior constraints on S8 from the un- masked mock catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We compare the results from the de- fault procedure to model the likelihood as a function of cosmology (solid) using skew-normals to an alternative approach employing Gaussian Process fitting (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' tion of cosmology instead of all observables as a function of galaxy and cosmology parameters (Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We apply the alternative GP emulation technique to the mock analyses described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The cosmological pos- terior is based on Lmax, similar to the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 3 and the cosmological parameters considered are S8, Ωm and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' GP interpolation is performed using the publicly available GPy package7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' To train a GP one needs to first determine the kernel that best constrains the covariance matrix of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We first perform a k-fold cross validation over a set of kernels such as polynomial, exponential, RBF, Mat´ern 3/2 and Mat´ern 5/2 to determine the kernel with maximum predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' k-fold cross validation involves splitting the data set, the 40 simulations with their respective cosmolog- ical parameters and Lmax values, into k equal-sized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Afterwards, each of the k = 8 groups is used as a test set after training the GP on the remaining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This allows us to empirically asses the predictive power of different kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Of all the kernels, we find the Mat´ern 5/2 to give the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' After determining the best kernel, we train the GP on the entire set of 40 simulations and use the interpolated Lmax as our proxy for the cosmological posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We note that, con- trary to the default analysis, this procedure does not account for the impact of uncertainties in the simulation predictions on the final cosmology posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' However, this effect was found in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022) to be negligible when analysing the mocks with Aemulus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' A1, we com- pare the inferred posteriors on S8 against the results obtained from the default analysis procedure involving skew-normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, we find both approaches to give highly consistent results, providing further evidence for the robustness of our analysis method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='com/SheffieldML/GPy/ APPENDIX B: GALAXY–HALO CONNECTION Here, we present and discuss posterior constraints on the galaxy–halo connection G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Since we fit each of the 40 sim- ulations to data, we naturally get 40 posterior constraints P(G|Ci) for 40 different cosmologies Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We combine these 40 by taking the average weighted by the profile likelihood each a simulation, P(G) = � P(G|Ci)Lp(Ci) � Lp(Ci) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (B1) As an example, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' B1 all one and two- dimensional posteriors on the galaxy–halo connection param- eters for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ⩽ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 sample in the SGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similar to the results for cosmology as well as central velocity and as- sembly bias discussed below, this figure indicates good agree- ment between the constraints derived from redshift-space clustering, the combination of projected clustering and lens- ing and RSDs combined with lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This figure also demon- strates that the addition of gravitational lensing as a con- straint does not add significant constraining power on galaxy– halo connection parameters compared to redshift-space clus- tering alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is expected since we only include gravi- tational lensing in the two-halo regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At fixed cosmology, gravitational lensing in this regime primarily contains infor- mation on the large-scale bias, something already contained within clustering measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Our posterior constraints on the HOD parameters, par- ticularly Mmin, M1 and σlog M, present significant variation amongst the different galaxy samples studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' This is expected since the different samples represent galaxy populations at different redshifts, stellar masses etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In this appendix we focus on velocity bias and assembly bias, as we find that the conclusions drawn below apply equally well to each of the galaxy samples we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' B2, we present the posterior constraints on αc, the central velocity bias parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As expected, redshift-space clustering is able to put some constraints on αc, whereas we do not show the combination of projected galaxy clustering and galaxy–galaxy lensing since it is insensitive to this pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Overall, our results are consistent with little to no central velocity bias, αc = 0, and agree with the findings in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, our results here do not con- tradict the findings in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2015a) where αc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' In the present work, we define central velocity with respect to the inner 10% of the halo particles (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2013) in- stead of the inner 25% as in Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' The latter definition is expected to imply larger values for αc (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' B3 shows our constraints on the central assembly bias parameter Acen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We find no strong constraints on assembly bias and our results are consistent with no assembly bias, Acen = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' As discussed in Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' (2019b, 2022), this is in part due to the degeneracy between Acen and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' At the same time, even at fixed cosmology, we find neither redshift-space clustering nor the combination of projected clustering and galaxy–galaxy lensing to be very sensitive to assembly bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Similarly, our analysis also does not yield any noteworthy constraints on the satellite assembly bias param- eter Asat, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Other summary statistics beyond two-point correlation functions might be needed to robustly constrain assembly bias (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Storey-Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) Full-scale and full-shape analysis of RSD and GGL 21 wp + ∆Σ RSD-only RSD + ∆Σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 σlog M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='00 fΓ 13 14 log M0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 log M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 α 0 1 Acen 0 1 Asat −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 log η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 αc 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 log Mmin 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 αs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 σlog M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='00 fΓ 13 14 log M0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 log M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='6 α 0 1 Acen 0 1 Asat −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 log η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='4 αc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 αs Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Posterior constraints on galaxy–halo connection parameters for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ⩽ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 sample in the SGC after marginalisation over cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' We show constraints coming from redshift-space clustering (blue), the combination of projected clustering and lensing (purple) and RSDs combined with lensing (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Contours denote 68 and 95% confidence regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 αc Posterior RSD-only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='3 αc RSD + ∆Σ NGC SGC Figure B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Posterior constraints on central velocity bias after marginalisation over cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Different panels indicate the observa- tional constraints used: redshift-space clustering (left) and the combination of redshift-space clustering and galaxy–galaxy lensing (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Different lines indicate different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 Acen Posterior RSD-only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='18 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='30 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='36 ≤ z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='43 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 Acen wp + ∆Σ NGC SGC −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content='5 Acen RSD + ∆Σ Figure B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' B2 except for focussing on the central assembly bias parameter Acen and also showing results for the combination of projected clustering and galaxy–galaxy lensing (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FAT4oBgHgl3EQfyR7D/content/2301.08692v1.pdf'} diff --git a/NtE0T4oBgHgl3EQfjgEx/content/2301.02459v1.pdf b/NtE0T4oBgHgl3EQfjgEx/content/2301.02459v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb714cd3a4e1e176406ce6bf00d8dbd1e8a184bb --- /dev/null +++ b/NtE0T4oBgHgl3EQfjgEx/content/2301.02459v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eb2cc32e3742c772e75a2fa7dcbbb2975c36ac1d9a9fd38ee76ff1c22645e466 +size 1270738 diff 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Henann†† +School of Engineering, Brown University, Providence, RI 02912, USA +Abstract +Dense granular systems that consist of particles of disparate sizes segregate based on size during flow, resulting in +complex, coupled segregation and flow patterns. The ability to predict how granular mixtures segregate is important +in the design of industrial processes and the understanding of geophysical phenomena. The two primary drivers +of size-segregation are pressure gradients and shear-strain-rate gradients. In this work, we isolate size-segregation +driven by shear-strain-rate gradients by studying two dense granular flow geometries with constant pressure fields: +gravity-driven flow down a long vertical chute with rough parallel walls and annular shear flow with rough inner and +outer walls. We perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems +in both flow geometries, while varying system parameters, such as the flow rate, flow configuration size, fraction of +large/small grains, and grain-size ratio, and use DEM data to inform continuum constitutive equations for the relative +flux of large and small particles. When the resulting continuum model for the dynamics of size-segregation is coupled +with the nonlocal granular fluidity model–a nonlocal continuum model for dense granular flow rheology–we show +that both flow fields and segregation dynamics may be simultaneously captured using this coupled, continuum system +of equations. +1 +Introduction +Dense granular systems in nature and industry are often non-monodisperse–i.e., consisting of particles of disparate +sizes. In non-monodisperse granular systems, the constituent grains segregate based on size during flow, forming +complex patterns (e.g., Shinbrot and Muzzio, 2000; Gray and Thornton, 2005; Hill and Fan, 2008; Fan and Hill, 2010; +Schlick et al., 2015; Gray, 2018; Umbanhowar et al., 2019). The ability to predict the dynamics of segregation is +important across applications. For example, in geophysics, granular size segregation can manifest in landslides and +debris flows (Johnson et al., 2012), in which larger grains segregate to the top of the flow, potentially causing more +damage, while in industry, size-segregation can be an undesirable effect when blending granular constituents of various +sizes. +The current understanding is that there are two driving forces for size-segregation in dense granular flows. The +first is pressure-gradients, which are typically induced by gravity. In pressure-gradient-driven size-segregation, small +particles move more readily through the interstitial spaces that open and close during flow through a process referred +to in the literature as “kinetic sieving,” leading to a system stratified along the direction of pressure gradients (Savage +and Lun, 1988; Gray and Thornton, 2005; Gray and Chugunov, 2006; Thornton et al., 2012; Fan et al., 2014). While +pressure-gradient-driven segregation has been the focus of significant study, Hill and coworkers (Hill and Fan, 2008; Fan +and Hill, 2010, 2011b; Hill and Tan, 2014) demonstrated that grains can also segregate in inhomogeneous flows along +directions orthogonal to gravitationally-induced pressure gradients, driven instead by gradients in the shear-strain-rate. +As an example, this mechanism has been observed in the split-bottom cell experiments of Hill and Fan (2008). In these +experiments, not only do the larger particles segregate to the top of the cell, but they also segregate perpendicular to the +direction of pressure gradients towards more rapidly shearing regions. Shear-strain-rate-gradient-driven segregation +has received comparatively less attention in modeling efforts. +†These authors contributed equally to this work. +††Email address for correspondence: david_henann@brown.edu +1 +arXiv:2301.11453v1 [cond-mat.soft] 26 Jan 2023 + +Due to the complexity of flow and segregation patterns, developing a general, predictive, continuum model for +coupled size-segregation and flow in dense granular materials remains an open challenge. Although much progress +has been made over recent decades (e.g., Savage and Lun, 1988; Gray and Thornton, 2005; Gray and Chugunov, 2006; +Fan and Hill, 2011b; Fan et al., 2014; Tunuguntla et al., 2017; Gray, 2018; Umbanhowar et al., 2019), the development +of continuum models that are capable of simultaneously predicting the evolution of both segregation and flow fields, +based solely on the geometry of the flow configuration, applied loads, and boundary/initial conditions is still in its +infancy. Instead, most continuum models for size-segregation require some flow field quantity, such as the velocity or +stress fluctuation field, to be measured first from experiments or DEM simulations and then used as model input. A +crucial reason for the incompleteness of current models is the lack of a dense granular flow rheology theory that may +be coupled to segregation models. A widely-used class of viscoplastic models for steady, dense granular flow is based +on the 𝜇(𝐼) rheology (MiDi, 2004; Jop et al., 2005; da Cruz et al., 2005; Srivastava et al., 2021). One recent work that +couples rheology and segregation in dense granular flows is that of Barker et al. (2021), which combines a regularized +version of the 𝜇(𝐼) rheology (Barker and Gray, 2017) with a model for gravity-driven segregation. However, it has +been well-documented in the literature (e.g., Kamrin, 2019) that the 𝜇(𝐼) rheology, even in its regularized form, can +break down in the presence of spatial flow inhomogeneity, which can be attributed to nonlocal effects not accounted +for in the 𝜇(𝐼) rheology. +To address this point, significant effort has gone into the development of size-dependent, nonlocal continuum +constitutive theories for dense granular flow rheology, and coupling a nonlocal rheological model with a segregation +model provides a route to robust, simultaneous prediction of flow and segregation fields. In this paper, we focus on +the nonlocal granular fluidity (NGF) model of Kamrin and coworkers (Kamrin and Koval, 2012; Henann and Kamrin, +2013; Kamrin, 2019), which has been successfully applied to predicting dense flows of monodisperse grains in a +wide variety of flow geometries. Then, the overarching aim of this paper is to formulate a predictive continuum +theory for simultaneous flow and size-segregation in dense granular systems by integrating the NGF model with a +phenomenological size-segregation model. This is a broad goal, and in this paper, we narrow our focus to several simpler, +quasi-one-dimensional flow configurations. In most real-world flows, both the pressure-gradient-driven and shear- +strain-rate-gradient-driven segregation mechanisms are present, making it difficult to disentangle them. Therefore, +our plan for this paper is to isolate and examine the shear-strain-rate-gradient-driven mechanism. Specifically, we +study the shear-strain-rate-gradient-driven segregation mechanism by considering flows of dense, bidisperse systems +of both disks and spheres in flow geometries in which the pressure field is spatially uniform–namely, vertical chute +flow and annular shear flow. Therefore shear-strain-rate-gradients are the sole drivers of segregation. In order to +inform continuum model development, we perform discrete element method (DEM) simulations using the open source +software LAMMPS (Plimpton, 1995), which function as “numerical experiments.” The coupled continuum model +that we develop is then validated by comparing its predictions of the transient evolution of segregation and flow fields +against additional DEM simulation results. +This paper is organized as follows. In Section 2, we discuss the continuum model that we use to describe flow and +size-segregation in bidisperse, dense granular materials. Specifically, Sections 2.1 and 2.2 introduce the mixture theory +framework used to describe dense, bidisperse granular mixtures, and in Section 2.3, we briefly revisit the 𝜇(𝐼) rheology +and the NGF model for monodisperse granular systems and discuss their extension to bidisperse systems. Then in +Section 2.4, we propose a model for shear-strain-rate-gradient-driven size-segregation. In Sections 3 and 4, we consider +granular diffusion and shear-strain-rate-gradient-driven segregation, respectively, and independently determine the two +dimensionless material parameters that appear in the size-segregation model for both disks and spheres. Then, in +Section 5, the proposed segregation model is coupled with the NGF model and applied to both vertical chute flow and +annular shear flow to predict the transient evolution of the segregation dynamics, and the predicted continuum fields +are compared against DEM measurements. In the end, our model demonstrates a level of fidelity in simultaneously +predicting flow and segregation dynamics that has not been previously achieved. We close with a discussion of the +segregation model and some concluding remarks in Section 6. +2 +Continuum framework +In this section, we discuss the continuum framework used to describe dense, bidisperse granular systems and propose +constitutive equations for rheology and size-segregation. Throughout, we utilize a mixture-theory-based approach, +which is common in continuum modeling of dense, bidisperse mixtures (e.g., Gray and Thornton, 2005; Gray and +Chugunov, 2006; Fan and Hill, 2011b; Gray, 2018; Umbanhowar et al., 2019; Barker et al., 2021; Bancroft and Johnson, +2 + +AB8HicbVBNSwMxEJ2tX7V+V +T16CVbBU9mVUj0WvHisYD+kXUs2m21Dk+ySZIWy9Fd48aCIV3+O +N/+NabsHbX0w8Hhvhpl5QcKZNq7RTW1jc2t4rbpZ3dvf2D8u +FRW8epIrRFYh6rboA15UzSlmG026iKBYBp51gfDPzO09UaRbLe +zNJqC/wULKIEWys9BA+Zn0lEJ8OyhW36s6BVomXkwrkaA7KX/0w +Jqmg0hCOte5bmL8DCvDCKfTUj/VNMFkjIe0Z6nEgmo/mx8Red +WCVEUK1vSoLn6eyLDQuJCGynwGakl72Z+J/XS0107WdMJqmhki +wWRSlHJkaz71HIFCWGTyzBRDF7KyIjrDAxNqOSDcFbfnmVtC+rX +r1au6tVGmd5HEU4gVO4A+uoAG30IQWEBDwDK/w5ijnxXl3Phat +BSefOY/cD5/AK+SkEA= +dl +AB8HicbVBNSwMxEJ2tX7V+V +T16CVbBU9mVUj0WvHisYD+kXUs2m21Dk+ySZIWy9Fd48aCIV3+O +N/+NabsHbX0w8Hhvhpl5QcKZNq7RTW1jc2t4rbpZ3dvf2D8u +FRW8epIrRFYh6rboA15UzSlmG026iKBYBp51gfDPzO09UaRbLe +zNJqC/wULKIEWys9BA+Zn0lkJ4OyhW36s6BVomXkwrkaA7KX/0w +Jqmg0hCOte5bmL8DCvDCKfTUj/VNMFkjIe0Z6nEgmo/mx8Red +WCVEUK1vSoLn6eyLDQuJCGynwGakl72Z+J/XS0107WdMJqmhki +wWRSlHJkaz71HIFCWGTyzBRDF7KyIjrDAxNqOSDcFbfnmVtC+rX +r1au6tVGmd5HEU4gVO4A+uoAG30IQWEBDwDK/w5ijnxXl3Phat +BSefOY/cD5/ALo1kEc=ds +Figure 1: A representative schematic of a dense, bidisperse granular system consisting of two-dimensional disks. +2021; Duan et al., 2021), and we use standard component notation, which supposes an underlying set of Cartesian +basis vectors {e𝑖|𝑖 = 1, 2, 3}, and in which the components of vectors, v, and tensors, 𝝈, are denoted by 𝑣𝑖 and 𝜎𝑖 𝑗, +respectively. The Einstein summation convention is employed, and the Kronecker delta, 𝛿𝑖 𝑗, is utilized to denote the +components of the identity tensor. +2.1 +Bidisperse systems +We consider granular mixtures consisting of particles with two sizes–large grains with an average diameter of 𝑑l and +small grains with an average diameter of 𝑑s. We consider both two-dimensional systems of disks, as illustrated in +Fig. 1, and three-dimensional systems of spheres. To eliminate the effect of density-based segregation (e.g., Tripathi +and Khakhar, 2013) and isolate size-based segregation, all particles are made of the same material with density 𝜌s, +which represents the area-density for disks and the volume-density for spheres. Throughout, we utilize the notational +convention in which we denote large-grain quantities using a superscript l and small-grain quantities using a superscript +s. The species-specific solid fractions–i.e., the areas occupied by each species per unit total area for disks and the +volumes occupied by each species per unit total volume for spheres–are 𝜙l and 𝜙s, respectively, and the total solid +fraction is 𝜙 = 𝜙l + 𝜙s. The concentration of each species then follows as 𝑐l = 𝜙l/𝜙 and 𝑐s = 𝜙s/𝜙, so that 𝑐l + 𝑐s = 1. +The average mixture grain size is defined as the sizes of both species weighted by their concentrations, ¯𝑑 = 𝑐l𝑑l + 𝑐s𝑑s. +We make the common idealization that the total area for dense systems of disks or total volume for dense systems of +spheres does not change (Savage, 1998; Gray and Thornton, 2005; Gray and Chugunov, 2006; Fan and Hill, 2011b), +and therefore 𝜙 is idealized as constant at each point in space and at each instant in time during the segregation process. +We have verified in our DEM simulations that area (or volume) dilatation at flow initiation occurs over a much shorter +time scale than the process of segregation, so that this idealization is reasonable. Throughout this study, we use 𝜙 = 0.8 +for disks, and 𝜙 = 0.6 for spheres. +Regarding the kinematics of flow, each species has an associated partial velocity, 𝑣l +𝑖 and 𝑣s +𝑖, and the mixture velocity +is given by 𝑣𝑖 = 𝑐l𝑣l +𝑖 + 𝑐s𝑣s +𝑖. The mixture strain rate tensor is then defined using the mixture velocity in the standard +way: 𝐷𝑖 𝑗 = (1/2)(𝜕𝑣𝑖/𝜕𝑥 𝑗 + 𝜕𝑣 𝑗/𝜕𝑥𝑖), where 𝐷𝑘𝑘 = 0 since we have assumed that the mixture area (or volume) does +not change. The equivalent shear strain-rate is defined as �𝛾 = (2𝐷𝑖 𝑗𝐷𝑖 𝑗)1/2. +Then, the relative area (or volume) flux for each grain type 𝛼 = l, s is defined through the difference between its +partial velocity and the mixture velocity as 𝑤𝛼 +𝑖 = 𝑐𝛼 �𝑣𝛼 +𝑖 − 𝑣𝑖 +� , so that 𝑤l +𝑖 + 𝑤s +𝑖 = 0. Conservation of mass for each +species requires that 𝐷𝑐𝛼/𝐷𝑡 + 𝜕𝑤𝛼 +𝑖 /𝜕𝑥𝑖 = 0, where 𝐷(•)/𝐷𝑡 is the material time derivative. Due to the fact that +𝑐l + 𝑐s = 1, only one of 𝑐l and 𝑐s is independent. Therefore, we will utilize 𝑐l as the field variable that describes the +dynamics of size-segregation in the following discussion, and the evolution of 𝑐l is governed by its conservation of +mass equation: +𝐷𝑐l +𝐷𝑡 + 𝜕𝑤l +𝑖 +𝜕𝑥𝑖 += 0. +(2.1) +3 + +2.2 +Stress and the equations of motion +We recognize that the symmetric Cauchy stress tensor 𝜎𝑖 𝑗 = 𝜎𝑗𝑖 represents the Cauchy stress of the mixture, rather +than the partial stress of either species. Regarding stress-related quantities for the granular mixture, we define the +pressure 𝑃 = −(1/3)𝜎𝑘𝑘, the stress deviator 𝜎′ +𝑖 𝑗 = 𝜎𝑖 𝑗 + 𝑃𝛿𝑖 𝑗, the equivalent shear stress 𝜏 = (𝜎′ +𝑖 𝑗𝜎′ +𝑖 𝑗/2)1/2, and the +stress ratio 𝜇 = 𝜏/𝑃. The Cauchy stress is then governed by the standard equations of motion +𝜙𝜌s +𝐷𝑣𝑖 +𝐷𝑡 = 𝜕𝜎𝑖 𝑗 +𝜕𝑥 𝑗 ++ 𝑏𝑖, +(2.2) +where 𝜙 is the constant total solid fraction, and 𝑏𝑖 is the non-inertial body force per unit volume (typically gravitational). +In order to close the system of equations, we require (1) rheological constitutive equations for the Cauchy stress 𝜎𝑖 𝑗 +and (2) a constitutive equation for the flux 𝑤l +𝑖, each of which are discussed in the following subsections. +2.3 +Rheological constitutive equations for bidisperse mixtures +In this section, we discuss the rheology of dense, bidisperse granular mixtures. Our strategy for formulating rheological +constitutive equations for bidisperse mixtures is to relate mixture-related quantities, such as the Cauchy stress 𝜎𝑖 𝑗 and +the strain-rate tensor 𝐷𝑖 𝑗, instead of specifying constitutive equations for species-specific partial stresses and then +combining them to obtain the mixture stress. +The starting point of this discussion is the local inertial, or 𝜇(𝐼), rheology (MiDi, 2004; Jop et al., 2005; da Cruz +et al., 2005), which follows from dimensional arguments. For a dense, monodisperse system of dry, stiff grains with +mean grain diameter 𝑑 subjected to homogeneous shearing, the local inertial rheology asserts that the stress ratio 𝜇 is +given through the equivalent shear strain-rate �𝛾 and the pressure 𝑃 through the dimensionless relationship 𝜇 = 𝜇loc(𝐼), +where 𝐼 = �𝛾 +√︁ +𝑑2𝜌s/𝑃 is the inertial number, representing the ratio of the microscopic time scale associated with +particle motion +√︁ +𝑑2𝜌s/𝑃 to the macroscopic time scale of applied deformation 1/ �𝛾. As shown by Rognon et al. (2007) +and Tripathi and Khakhar (2011), the inertial rheology function 𝜇loc(𝐼) may be straightforwardly generalized from +monodisperse to bidisperse systems by defining the inertial number for a bidisperse system as 𝐼 = �𝛾 +√︁ ¯𝑑2𝜌s/𝑃, where +the average mixture grain size for a bidisperse system ¯𝑑 has been used in place of 𝑑 for a monodisperse system. Then, +the same local rheology function 𝜇loc(𝐼) utilized for the monodisperse system may be used for bidisperse systems +without any changes to the parameters appearing in the fitting function. This approach neglects potential effects of +new dimensionless quantities that arise in a bidisperse granular system, such as the grain size ratio 𝑑l/𝑑s, but has been +shown to capture DEM data well (Rognon et al., 2007; Tripathi and Khakhar, 2011). +To demonstrate this point, consider DEM simulations of homogeneous, simple shearing of a dense, bidisperse +system of disks, illustrated in Fig. 2(a) for the case of 𝑑l/𝑑s = 1.5 and a system-wide large-grain concentration of +𝑐l = 0.5. Details of the simulated granular systems, including grain interaction properties, for both two-dimensional +disks and three-dimensional spheres are given in Appendix A.1. The large particles are dark gray, and the small +particles are light gray. With the system-wide mean grain size denoted by ¯𝑑0 = 𝑐l𝑑l + (1−𝑐l)𝑑s, the rectangular domain +has a length of 𝐿 = 60 ¯𝑑0 in the 𝑥-direction and a height of 𝐻 = 60 ¯𝑑0 in the 𝑧-direction, which is filled with ∼ 5000 +flowing grains. Shearing is driven through the relative motion of two parallel, rough walls, which each consist of a +thin layer of touching glued grains, which are denoted as black in Fig. 2(a). The bottom wall is fixed, and the top wall +moves with a velocity 𝑣w along the 𝑥-direction. Following previous works in the literature (da Cruz et al., 2005; Koval +et al., 2009; Kamrin and Koval, 2012; Zhang and Kamrin, 2017; Liu and Henann, 2018; Kim and Kamrin, 2020), the +𝑧-position of the top wall is not fixed but continuously adjusted using a feedback scheme so that the normal stress +applied by the top wall is maintained at a target value of 𝜎𝑧𝑧(𝑧 = 0) = −𝑃w. Periodic boundary conditions are applied +along the 𝑥-direction. For homogeneous simple shearing, no segregation will occur since the flow is homogeneous +and no pressure or strain-rate gradients are present. We utilize the DEM procedures described in detail in Liu and +Henann (2018) in order to extract the relationship between 𝜇 and 𝐼 for bidisperse mixtures with grain size ratios +of 𝑑l/𝑑s = 1.5, 2.0, 2.5, and 3.0 and 𝑐l = 0.5. The simulated relationships are plotted in Fig. 2(b) using triangular +symbols of different colors, along with the monodisperse data from Liu and Henann (2018) plotted as gray circles. The +relationship between 𝜇 and 𝐼 for dense systems of disks is observed to be approximately independent of 𝑑l/𝑑s. As for +the monodisperse case, the DEM data for bidisperse mixtures of disks may be fit by a linear, Bingham-like functional +form: +𝜇loc(𝐼) = 𝜇s + 𝑏𝐼, +(2.3) +4 + +0 +0.05 +0.1 +0.15 +0.2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.05 +0.1 +0.15 +0.2 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AgfONMw=(a) +ACAXicbVDL +SgMxFL3js9bXqBvBTbAIrspME +XVZcOygn1AZyiZNG1Dk8yQZCp +lqBt/xY0LRdz6F+78G9N2Ftp6 +IORwzr03uSdKONPG876dldW19Y +3NwlZxe2d3b989OGzoOFWE1kn +MY9WKsKacSVo3zHDaShTFIuK0G +Q1vpn5zRJVmsbw34SGAvcl6z 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Upper and lower +layers of black grains denote rough walls. Dark gray grains indicate large flowing grains, and light gray grains indicate +small flowing grains. A 10% polydispersity is utilized for each species to prevent crystallization. (b) The local inertial +rheology (𝜇 versus 𝐼 = �𝛾 +√︁ ¯𝑑2𝜌s/𝑃) for monodisperse as well as bidisperse mixtures of disks for grain-size ratios of +𝑑l/𝑑s = 1.5, 2.0, 2.5, and 3.0 and 𝑐l = 0.5. The solid black line represents the best fit to the monodisperse DEM data +using (2.3) with 𝜇s = 0.272 and 𝑏 = 1.168. (c) The local inertial rheology for monodisperse and bidisperse mixtures +of spheres for grain size ratios of 𝑑l/𝑑s = 1.5 and 2.0 and 𝑐l = 0.5 along with the DEM data of Tripathi and Khakhar +(2011). The solid black curve represents the best fit to the monodisperse DEM data using (2.4) with 𝜇s = 0.37, +𝜇2 = 0.95, and 𝐼0 = 0.58. +as shown by the solid line in Fig. 2(b), where 𝜇s = 0.272 and 𝑏 = 1.168 are the dimensionless material parameters for +monodisperse disks (Liu and Henann, 2018). +Similarly, we consider DEM simulations of homogeneous, simple shearing of dense, bidisperse systems of spheres. +The simulation domain consists of a rectangular box of length 𝐿 = 20 ¯𝑑0 in the 𝑥-direction (i.e., the shearing direction), +width 𝑊 = 10 ¯𝑑0 in the 𝑦-direction (i.e., the direction perpendicular to the plane of shearing), and height 𝐻 = 40 ¯𝑑0 in +the 𝑧-direction. The domain is filled with ∼10000 flowing grains, and periodic boundary conditions are applied along +both the 𝑥- and 𝑦-directions. The simulation domain is bounded along the 𝑧-direction by two parallel, rough walls, +consisted of touching glued grains, and as for the case of disks, shearing along the 𝑥-direction and normal stress along +the 𝑧-direction are applied by the walls. We perform DEM simulations of steady simple shearing for size ratios of +𝑑l/𝑑s = 1.5 and 2.0 for a system-wide large-grain concentration of 𝑐l = 0.5 as well as for the monodisperse case over +a range of top wall velocities. The 𝜇 versus 𝐼 relationship extracted from DEM simulations for these cases along with +data from the prior DEM study of Tripathi and Khakhar (2011) collapse quite well as shown in Fig. 2(c), showing +minimal dependence on 𝑑l/𝑑s. This relationship for dense systems of spheres may be fitted using a nonlinear functional +form of Jop et al. (2005) for 𝜇loc(𝐼): +𝜇loc(𝐼) = 𝜇s + 𝜇2 − 𝜇s +𝐼0/𝐼 + 1, +(2.4) +as shown by the solid curve in Fig. 2(c), where {𝜇s = 0.37, 𝜇2 = 0.95, 𝐼0 = 0.58} are the dimensionless parameters +for frictional spheres. (We note that these parameters are nearly the same as those determined by Zhang and Kamrin +(2017) for monodisperse frictional spheres.) In this way, one may capture the rheology of bidisperse mixtures of +5 + +both disks and spheres in homogeneous simple shearing without introducing additional fitting functions or adjustable +parameters beyond those used for the monodisperse case. +Despite the successes of the local inertial rheology in capturing steady, homogeneous shear flow, it has been +well established in the literature that a local rheological modeling approach cannot be applied to a broad set of +inhomogeneous flows, such as annular shear flow (Tang et al., 2018), split-bottom flow (Fenistein and van Hecke, +2003), and gravity-driven heap flow (Komatsu et al., 2001). Therefore, to consider inhomogeneous flows of bidisperse +granular systems, it is necessary to generalize a nonlocal rheological modeling approach to the case of dense, bidisperse +mixtures. In the present work, we focus attention on the nonlocal granular fluidity (NGF) model of Kamrin and Koval +(2012), which has been shown to robustly capture a variety of inhomogeneous, steady flows of monodisperse granular +systems (Kamrin, 2019). As is standard in the NGF model, we introduce the granular fluidity 𝑔, which is a positive, +scalar field quantity, and recognize that 𝑔 represents the fluidity of the mixture. (See Zhang and Kamrin (2017) and Kim +and Kamrin (2020) for further discussion of the kinematic description of the granular fluidity field for monodisperse +granular systems.) Then, we utilize the steady-state form of the NGF model, which relates the stress state, the strain-rate, +and the granular fluidity through two constitutive equations: (1) the flow rule and (2) the nonlocal rheology. +First, invoking the common idealization that the Cauchy stress deviator and the strain-rate tensor are co-directional +(Rycroft et al., 2009), the flow rule relates the Cauchy stress tensor 𝜎𝑖 𝑗, the strain-rate tensor 𝐷𝑖 𝑗, and the granular +fluidity through +𝜎𝑖 𝑗 = −𝑃𝛿𝑖 𝑗 + 2𝑃 +𝑔 𝐷𝑖 𝑗. +(2.5) +Taking the magnitude of the deviatoric part of (2.5) and rearranging leads to the following scalar form of the flow rule: +�𝛾 = 𝑔𝜇. +(2.6) +Second, the granular fluidity of the bidisperse mixture is governed by the following differential relation: +𝑔 = 𝑔loc(𝜇, 𝑃) + 𝜉2(𝜇) 𝜕2𝑔 +𝜕𝑥𝑖𝜕𝑥𝑖 +, +(2.7) +where 𝑔loc(𝜇, 𝑃) is the local fluidity function and 𝜉(𝜇) is the stress-dependent cooperativity length. The local fluidity +function gives the granular fluidity during steady, homogeneous shear flow at a given state of stress and is related to +the local inertial rheology function 𝜇loc(𝐼). Denote the inverted form of the local inertial rheology function 𝜇loc(𝐼) as +𝐼loc(𝜇) = +� +𝜇−1 +loc(𝜇) +if 𝜇 > 𝜇s, +0 +if 𝜇 ≤ 𝜇s, +(2.8) +which is a function of the stress ratio 𝜇. Then, consistent with the new definition of the inertial number involving ¯𝑑 for +a bidisperse mixture, the local fluidity function is 𝑔loc(𝜇, 𝑃) = +√︁ +𝑃/ ¯𝑑2𝜌s 𝐼loc(𝜇)/𝜇. For the case of bidisperse disks, +using (2.3), the local fluidity function is +𝑔loc(𝜇, 𝑃) = +���� +���� +√︄ +𝑃 +𝜌s ¯𝑑2 +(𝜇 − 𝜇s) +𝑏𝜇 +if 𝜇 > 𝜇s, +0 +if 𝜇 ≤ 𝜇s, +(2.9) +with {𝜇s = 0.272, 𝑏 = 1.168}, and for the case of bidisperse spheres, using (2.4), the local fluidity function is +𝑔loc(𝜇, 𝑃) = +���� +���� +𝐼0 +√︄ +𝑃 +𝜌s ¯𝑑2 +(𝜇 − 𝜇s) +𝜇(𝜇2 − 𝜇) +if 𝜇 > 𝜇s, +0 +if 𝜇 ≤ 𝜇s, +(2.10) +with {𝜇s = 0.37, 𝜇2 = 0.95, 𝐼0 = 0.58}. No additional adjustable parameters beyond those used to describe the local +inertial rheology for the monodisperse case are introduced in the local fluidity function. +As discussed in several of our previous works (Henann and Kamrin, 2014; Kamrin and Henann, 2015; Liu and +Henann, 2017), the functional form for the cooperativity length 𝜉(𝜇) is also connected to the choice of the 𝜇loc(𝐼) +6 + +function. Without going into details here, the functional forms for the cooperativity length corresponding to (2.3) and +(2.4) are +𝜉(𝜇) = +𝐴 ¯𝑑 +√︁ +|𝜇 − 𝜇s| +and +𝜉(𝜇) = 𝐴 ¯𝑑 +√︄ +(𝜇2 − 𝜇) +(𝜇2 − 𝜇s)|𝜇 − 𝜇s| , +(2.11) +respectively. In the monodisperse case, the cooperativity length is directly proportional to the grain size 𝑑, and in +(2.11) for the bidisperse case, it is taken to be proportional to ¯𝑑. This is analogous to the process undertaken above +for generalizing the local inertial rheology, in which 𝑑 for monodisperse grains is replaced by ¯𝑑 for bidisperse grains. +Moreover, in (2.11), the parameter 𝐴 is a dimensionless material constant, referred to as the non-local amplitude, +which quantifies the spatial extent of cooperative effects. Again, following an approach analogous to the generalization +of the local inertial rheology, we utilize values of 𝐴 previously determined for monodisperse frictional disks and +spheres–namely, 𝐴 = 0.9 as determined by Liu and Henann (2018) for monodisperse disks and 𝐴 = 0.43 as determined +by Zhang and Kamrin (2017) for monodisperse spheres. These choices of 𝐴 for bisdisperse mixtures will be tested in +later sections by comparing flow fields predicted by the NGF model to measured flow fields in DEM simulations of +bidisperse, inhomogeneous flows. +2.4 +Segregation model +The segregation model consists of the constitutive equation for the large-grain flux 𝑤l +𝑖. In the present work, we focus +on dense flows in the absence of pressure gradients, and we take the large-grain flux 𝑤l +𝑖 to be comprised of two +contributions: (1) a diffusion flux 𝑤diff +𝑖 +and (2) a shear-strain-rate-gradient-driven segregation flux 𝑤seg +𝑖 , so that +𝑤l +𝑖 = 𝑤diff +𝑖 ++ 𝑤seg +𝑖 +. +(2.12) +First, the diffusion flux acts counter to segregation to mix the species and is taken to be given in the standard form, +in which the diffusion flux is driven by concentration gradients: 𝑤diff +𝑖 += −𝐷 �𝜕𝑐l/𝜕𝑥𝑖 +�, where 𝐷 is the binary diffusion +coefficient (Utter and Behringer, 2004; Artoni et al., 2021; Bancroft and Johnson, 2021). Based on dimensional +arguments, we expect that +𝐷 = 𝐶diff ¯𝑑2 �𝛾, +(2.13) +where 𝐶diff is a dimensionless material parameter which remains to be calibrated (Fan et al., 2014; Tripathi and +Khakhar, 2013). Therefore, we have that the diffusion flux is +𝑤diff +𝑖 += −𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑥𝑖 +. +(2.14) +Second, regarding segregation, a major question is what field quantity drives the segregation flux in the absence of +pressure gradients. Gradients of a number of kinematic quantities are possible–e.g., strain-rate, velocity fluctuations, +or fluidity. For perspective, we note that recent works (Fan and Hill, 2011b; Hill and Tan, 2014; Tunuguntla et al., 2016, +2017) have shown that gradients in kinetic stress, which are defined through the velocity fluctuations, correlate well +with segregation flux. In the present work, we adopt the simplest approach and hypothesize that the segregation flux is +driven by gradients in the shear strain-rate �𝛾 and take the segregation flux to be given in the following phenomenological +form: +𝑤seg +𝑖 += 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕 �𝛾 +𝜕𝑥𝑖 +. +(2.15) +The factor 𝑐l(1 − 𝑐l) ensures that segregation ceases when the bidisperse mixture becomes either all large (𝑐l = 1) or +all small (𝑐l = 0) grains, and the factor ¯𝑑2 is present for dimensional consistency. The quantity 𝐶S +seg is a dimensionless +material property. While it is possible for 𝐶S +seg to depend on the size ratio 𝑑l/𝑑s, we will demonstrate that this effect is +negligible in the DEM simulations of Section 4 and therefore treat 𝐶S +seg as a constant, dimensionless material parameter, +which will be determined by fitting to DEM simulation results for disks and spheres, respectively. +Combining (2.14), (2.15), and (2.12) with conservation of mass (2.1), we obtain the following differential relation +governing the dynamics of 𝑐l: +𝐷𝑐l +𝐷𝑡 + 𝜕 +𝜕𝑥𝑖 +� +−𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑥𝑖 ++ 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕 �𝛾 +𝜕𝑥𝑖 +� += 0, +(2.16) +7 + +where {𝐶diff, 𝐶S +seg} represent two constant dimensionless material parameters that remain to be determined. +We close this section by noting that the incompressibility constraint, the equations of motion (2.2), the nonlocal +rheology (2.7), and the segregation dynamics equation (2.16) represent a closed system of equations for the velocity +field 𝑣𝑖, the pressure field 𝑃, the fluidity field 𝑔, and the large-grain concentration field 𝑐l, which may be used to +simultaneously predict flow fields and segregation dynamics in the absence of pressure gradients. +3 +Diffusion flux +In this section, we determine values of 𝐶diff for dense, bidisperse systems of frictional disks and spheres. Consider +homogeneous simple shear flow of such a bidisperse mixture, as shown in Fig. 2(a) for disks. Again, no segregation +occurs in this setting, since neither of the segregation driving forces (pressure gradients or shear-strain-rate gradients) +are present (Tripathi and Khakhar, 2011). During steady, simple shearing, the motion of individual grains in the +direction transverse to flow (the 𝑧-direction in Fig. 2(a)) approximates a random walk for both two-dimensional systems +of disks and three-dimensional systems of spheres. Therefore, by measuring the mean square displacement (MSD) of +the system of 𝑁 particles as a function of time, we may determine the binary diffusion coefficient 𝐷 (Natarajan et al., +1995; Dufty et al., 2002; Utter and Behringer, 2004; Fan et al., 2015; Kharel and Rognon, 2017; Bancroft and Johnson, +2021) through +MSD(𝑡) = 1 +𝑁 +𝑁 +∑︁ +𝑛=1 +(𝑧𝑛(𝑡) − 𝑧𝑛(0))2 = 2𝐷𝑡, +(3.1) +where 𝑧𝑛(𝑡) is the 𝑧-coordinate of the 𝑛th grain at time 𝑡. We simulate homogeneous, steady simple shear flows of disks +for grain-size ratios of 𝑑l/𝑑s = 1.5, 2.0, 2.5, 3.0, and 4.0 and at various shearing rates. We also simulate homogeneous, +steady simple shear flows of spheres for the monodisperse case as well as for grain-size ratios of 𝑑l/𝑑s = 1.5, 2.0, and +2.5 over a range of shearing rates. To avoid wall effects in the calculation of the MSD (3.1), grains that are initially +within 15 ¯𝑑0 of either the top or bottom wall in Fig. 2(a) are excluded from the system of particles used to calculate the +MSD for disks, leaving a set of 𝑁 ≈ 2400 grains. For spheres, particles initially within 5 ¯𝑑0 of the top and bottom walls +are excluded, so that a set of 𝑁 ≈ 9000 grains are used to calculate the MSD. Both large and small grains are included +in the calculation of the MSD for the mixture. After a sufficiently long time, the calculated MSD is linear in time in all +cases for both disks and spheres, allowing one to extract the diffusion coefficient 𝐷 for each case. +The diffusion coefficient 𝐷 is plotted against �𝛾 ¯𝑑2 (with both quantities normalized by 𝑑s√︁ +𝑃w/𝜌s) for disks in +Fig. 3(a) and for spheres in Fig. 3(b). The DEM data for the binary diffusion coefficient collapses to a nearly linear +relation with 𝐷 ∼ �𝛾 ¯𝑑2 across the range of size ratios and shearing rates considered. A best fit of the slopes of the linear +relations–the solid black lines in Figs. 3(a) and (b)–yields: +𝐶diff = +𝐷 +�𝛾 ¯𝑑2 = 0.20 for disks and 𝐶diff = +𝐷 +�𝛾 ¯𝑑2 = 0.045 for spheres. +(3.2) +We note that our results are consistent with previous results in the literature. For example, the recent work of Bancroft +and Johnson (2021) found a value of 𝐶diff ≈ 0.05 with a weak dependence on the inertial number for dense spheres. +In order to further assess the fitted value of 𝐶diff in a diffusion-dominated setting, we have performed a consistency +test for disks by considering simple shearing of an initially fully-segregated cell, which is described in Appendix B. In +this case, diffusion drives remixing of the two species. Using the fitted value of 𝐶diff for disks in (3.2), we are able to +quantitatively capture the diffusive remixing process, which provides confidence in our fitted value of 𝐶diff. +4 +Shear-strain-rate-gradient-driven segregation flux +Having independently determined the material parameter 𝐶diff for both frictional disks and spheres, we next turn to +testing the constitutive equation for the shear-strain-rate-gradient-driven segregation flux (2.15) and determining the +material parameter 𝐶S +seg by studying two representative flow configurations in the absence of pressure gradients: (1) +vertical chute flow and (2) annular shear flow. +8 + +10-2 +10-1 +10-4 +10-3 +10-2 +10-3 +10-2 +10-1 +100 +10-4 +10-3 +10-2 +10-1 +AB6nicbVDLSgNBEOyNrxhfU +Y9eBqMQL2E3iHoMePEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf +5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxc +Ojpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+m +nGCfkQHkoecUWOlh3Jw0SuW3Io7B1klXkZKkKHeK351+zFLI5SG +Cap1x3MT40+oMpwJnBa6qcaEshEdYMdSPU/mR+6pScW6VPwlj +ZkobM1d8TExpPY4C2xlRM9TL3kz8z+ukJrzxJ1wmqUHJFovCVB +ATk9nfpM8VMiPGlCmuL2VsCFVlBmbTsG4C2/vEqa1Yp3Vbm8r +5ZqZ1kceTiBUyiDB9dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZz +DH/gfP4Ag3iNA=(b) +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AgfONMw=(a) +Figure 3: The binary diffusion coefficient 𝐷, calculated using the mean square displacement (3.1), versus �𝛾 ¯𝑑2 in +homogeneous, steady simple shear DEM simulations. (a) Simple shearing of bidisperse mixtures of disks for grain- +size ratios of 𝑑l/𝑑s = 1.5, 2.0, 2.5, 3.0, and 4.0. Both axes are normalized by 𝑑s√︁ +𝑃w/𝜌s. Each symbol represents 𝐷 +calculated from one DEM simulation of a specified size ratio at one shearing rate. The solid line represents the best +fit of a linear relation with 𝐶diff = 0.20. (b) Simple shearing of bidisperse mixtures of spheres for the monodisperse +case and for grain-size ratios of 𝑑l/𝑑s = 1.5, 2.0, and 2.5. The solid line represents the best fit of a linear relation with +𝐶diff = 0.045. +4.1 +Vertical chute flow +Consider a dense, bidisperse granular mixture flowing down a long vertical chute with parallel, rough walls separated +by a distance 𝑊 under the action of gravity 𝐺. This flow geometry has been utilized extensively in the literature to +study dense flows of monodisperse, frictional disks (Kamrin and Koval, 2012; Liu and Henann, 2018) and spheres +(Zhang and Kamrin, 2017; Kim and Kamrin, 2020) as well as flows of bidisperse, frictional spheres (Fan and Hill, +2011a,b). Beginning with the case of bidisperse disks, the DEM setup is shown in Fig. 4(a) for 𝑊 = 60 ¯𝑑0, where ¯𝑑0 is +the system-wide average grain size. In all cases for disks, we take the chute length to be 𝐿 = 60 ¯𝑑0 and apply periodic +boundary conditions along the 𝑧-direction. The parallel, rough walls consist of touching glued large grains, denoted +as black in Fig. 4(a). The left vertical wall is fixed, and the right wall is fixed in the 𝑧-direction but can move slightly +in the 𝑥-direction so as to maintain a constant compressive normal stress 𝑃w on the granular material, utilizing the +same wall-position control method described in Liu and Henann (2018). We have verified that the chute length 𝐿 is +sufficiently large, so that it does not affect the resulting flow and segregation fields and all fields are invariant along +the 𝑧-direction. In the resulting flow fields, the only non-zero component of the velocity is 𝑣𝑧, which only depends on +the cross-channel coordinate 𝑥. A typical steady velocity field is qualitatively sketched in Fig. 4(a), illustrating that the +shear-strain-rate is greatest at the walls (𝑥 = ±𝑊/2). +In all of our DEM simulations of vertical chute flow of bidisperse disks, we observe that the stress field quickly +becomes independent of time, so that macroscopic inertia (i.e., the left-hand-side of (2.2)) may be neglected. Moreover, +we observe that the normal stresses are approximately equal, i.e., 𝜎𝑧𝑧 ≈ 𝜎𝑥𝑥. Therefore, due to the force balance along +the 𝑧-direction, the equivalent shear stress field is 𝜏(𝑥) = |𝜎𝑥𝑧(𝑥)| = |𝜎𝑧𝑥(𝑥)| = 𝜙𝜌s𝐺|𝑥|, where 𝑥 is measured from the +centerline of the chute, and due to the force balance along the 𝑥-direction, the pressure field is 𝑃(𝑥) = −𝜎𝑥𝑥(𝑥) = 𝑃w. +The stress ratio field then follows as +𝜇(𝑥) = 𝜇w +� |𝑥| +𝑊/2 +� +, +(4.1) +where 𝜇w = 𝜙𝜌s𝐺𝑊/2𝑃w is the maximum value of 𝜇, occurring at the walls (𝑥 = ±𝑊/2). We note that while flow is +driven by gravity, the pressure field is constant throughout the chute, and no pressure gradients are present. Therefore, +segregation occurs only due to shear-strain-rate-gradients, enabling us to consider this effect in isolation. +Apart from the grain interaction properties that are held constant throughout this work (Appendix A.1), there +are four important dimensionless parameters that fully describe each case of vertical chute flow of dense, bidisperse +granular mixtures: (1) 𝑊/ ¯𝑑0, the dimensionless chute width; (2) 𝜇w, the maximum stress ratio, which occurs at the +walls and controls the total flow rate; (3) 𝑐l +0(𝑥), the initial large-grain concentration, which is not necessarily constant +but can be a spatially-varying field; and (4) 𝑑l/𝑑s, the bidisperse grain-size ratio. This list of system parameters +9 + +-20 +-10 +0 +10 +20 +0 +1 +2 +3 +4 +105 +0 +0.2 +0.4 +0.6 +0.8 +1 +-30 +-20 +-10 +0 +10 +20 +30 +0 +0.2 +0.4 +0.6 +0.8 +1 +-30 +-20 +-10 +0 +10 +20 +30 +10-2 +10-1 +100 +-20 +-10 +0 +10 +20 +0 +1 +2 +3 +4 +105 +0 +0.2 +0.4 +0.6 +0.8 +1 +AB+HicbVBN +S8NAEJ34WetHox69LFbBU0mkV +C9CwYvHCvYD2hA2m027dLMJuxu +hv4SLx4U8epP8ea/cdvmoK0P +Bh7vzTAzL0g5U9pxvq219Y3Nre +3STnl3b/+gYh8edVSULbJOG +J7AVYUc4EbWumOe2lkuI4LQbj +G9nfveRSsUS8aAnKfViPBQsYg +RrI/l2pYtuUMZBFjm4dR3fLv +q1Jw50CpxC1KFAi3f/hqECcliK +jThWKm+6Tay7HUjHA6LQ8yRV +NMxnhI+4YKHFPl5fPDp+jcKCGK +EmlKaDRXf0/kOFZqEgemM8Z6p +Ja9mfif1890dO3lTKSZpoIsFkU +ZRzpBsxRQyCQlmk8MwUQycysi +Iywx0SarsgnBX5lXQua26jV +r+vV5tnRwlOIFTuAXrqAJd9C +CNhDI4Ble4c16sl6sd+tj0bpm +FTPH8AfW5w/obJHi +W = 60 ¯d0 +AB6nicbVDL +SgNBEOyNrxhfUY9eBqMQL2E3i +HoMePEY0TwgWcLspDcZMju7zMw +KIeQTvHhQxKtf5M2/cZLsQRML +Goqbrq7gkRwbVz328mtrW9sbu +W3Czu7e/sHxcOjpo5TxbDBYhG +rdkA1Ci6xYbgR2E4U0igQ2ApGt +zO/9YRK81g+mnGCfkQHkoecUW +OlhzK96BVLbsWdg6wSLyMlyFD 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+eFCUdGounrqMcUJYaPLcFEMXsrIgOsMDE2oLwNwVt8eZk0LsreZ +blyVylWT7M4cnAMJ1ACD6gCrdQgzoQeIBneIU3RzovzrvzMW9d +cbKZI/gD5/MHFWOuQ=vz(x) +AB+HicbVBN +S8NAEJ3Ur1o/GvXoZbEKnkoip +XoRCl48eKhgP6ANYbPZtks3m7C +7EWroL/HiQRGv/hRv/hu3bQ7a ++mDg8d4M/OChDOlHefbKqytb2 +xuFbdLO7t7+2X74LCt4lQS2iI +xj2U3wIpyJmhLM81pN5EURwGn +WB8M/M7j1QqFosHPUmoF+GhYA +NGsDaSb5fv0DWqO/0Ayc+o5 +vV5yqMwdaJW5OKpCj6dtf/TAma +USFJhwr1XOdRHsZlpoRTqelfq +pogskYD2nPUIEjqrxsfvgUnRkl +RINYmhIazdXfExmOlJpEgemMs +B6pZW8m/uf1Uj248jImklRTQRa +LBilHOkazFDIJCWaTwzBRDJz +KyIjLDHRJquSCcFdfnmVtC+qb +r1au69VGqd5HEU4hM4BxcuoQG +30IQWEjhGV7hzXqyXqx362PR +WrDymSP4A+vzB9cmkdc= +L = 60 ¯d0 +ACAHicbVC7 +TsMwFHXKq5RXgIGBxaJCYqSC +gFjJRbGItGH1ESV49y0Vh0nsh2 +girwKywMIMTKZ7DxN7htBmg5 +0pWOzrnXvcEKWdKO863VpZXV +vfKG9WtrZ3dvfs/YO2SjJoU +TnshuQBRwJqClmebQTSWQODQC +UbXU79zD1KxRNzpcQp+TAaCRY +wSbaS+feREBokEwMcsUcIPQ8 +/EM7dtWpOTPgZeIWpIoKNPv2l +xcmNIvNa5QTpXquk2o/J1Izym +FS8TIFKaEjMoCeoYLEoPx8dsAE +nxolxFEiTQmNZ+rviZzESo3jw +HTGRA/VojcV/N6mY6u/JyJNM 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+6XaWRZHk7gFMrgwTXU4A7q0AGA3iGV3hzhPivDsfi9ack80c +wx84nz+IB403(e) +Figure 4: (a) Initial well-mixed configuration for two-dimensional DEM simulation of bidisperse vertical chute flow +with 4327 flowing grains. The chute width is 𝑊 = 60 ¯𝑑0. As in Fig. 2, black grains on both sides represent rough walls +(only large particles are used as wall grains here). (b) Segregated configuration after flowing for a total simulation time +of ˜𝑡 = 𝑡/ +� +𝑑s√︁ +𝜌s/𝑃w +� += 4.3 × 105. (c) Spatiotemporal evolution of the large grain concentration field. Spatial profiles +of (d) the concentration field 𝑐l and (e) the normalized velocity field (𝑣cen − 𝑣𝑧) +√︁ +𝜌s/𝑃w at three times (˜𝑡 = 4 × 103, +4 × 104 and 4 × 105) as indicated by the horizontal lines in (c). +{𝑊/ ¯𝑑0, 𝜇w, 𝑐l +0, 𝑑l/𝑑s} specifies the geometry, loads, and initial conditions of a given case of vertical chute flow. As +a representative base case, we consider the parameter group {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.45, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}. We +then run the corresponding DEM simulation starting from the well-mixed initial configuration shown in Fig. 4(a) +and observe that after a simulation time of ˜𝑡 = 𝑡/(𝑑s√︁ +𝜌s/𝑃w) = 4.3 × 105, the large, dark-gray grains segregate +towards the regions near the walls where the shear-strain-rate is greatest, while the small, light-gray grains gather in +bands just inside these regions, as shown in Fig. 4(b). A well-mixed core persists along the center of the vertical +chute where the shear-strain-rate is nearly zero. To obtain a more quantitative picture of the segregation process, we +10 + +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-3 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +10-4 +-1 +0 +1 +10-3 +-1.5 +-1 +-0.5 +0 +0.5 +1 +10-4 +-3 +-2 +-1 +0 +1 +2 +3 +10-3 +-5 +0 +5 +10-4 +-3 +-2 +-1 +0 +1 +2 +3 +10-3 +-5 +0 +5 +10-4 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AgfONMw=(a) +AB6nicbVDLSgNBEOyNrxhfU +Y9eBqMQL2E3iHoMePEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf +5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxc +Ojpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+m +nGCfkQHkoecUWOlh3Jw0SuW3Io7B1klXkZKkKHeK351+zFLI5SG +Cap1x3MT40+oMpwJnBa6qcaEshEdYMdSPU/mR+6pScW6VPwlj +ZkobM1d8TExpPY4C2xlRM9TL3kz8z+ukJrzxJ1wmqUHJFovCVB +ATk9nfpM8VMiPGlCmuL2VsCFVlBmbTsG4C2/vEqa1Yp3Vbm8r +5ZqZ1kceTiBUyiDB9dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZz +DH/gfP4Ag3iNA=(b) +Figure 5: Collapse of 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) versus ¯𝑑2𝑐l(1 − 𝑐l)(𝜕 �𝛾/𝜕𝑥) for several cases of vertical chute flow of (a) +bidisperse disks and (b) bidisperse spheres. Symbols represent coarse-grained, quasi-steady DEM field data, and the +solid lines are the best linear fits using (a) 𝐶S +seg = 0.23 for disks and (b) 𝐶S +seg = 0.08 for spheres. +coarse-grain the concentration field 𝑐l in both space and time and plot contours of the spatiotemporal evolution of +𝑐l in Fig. 4(c). The large-grain concentration field evolves quickly in time during flow initiation. Then, over longer +times, the evolution becomes slower. Spatial profiles of the concentration and velocity fields at three snapshots in +time–specifically, ˜𝑡 = 𝑡/(𝑑s√︁ +𝜌s/𝑃w) = 4 × 103, 4 × 104 and 4 × 105 as indicated by the horizontal lines in Fig. 4(c)–are +plotted in Figs. 4(d) and (e). These three snapshots correspond to early, medium, and late times with respect to the +segregation process. The spatial 𝑐l profiles shown in Fig. 4(d) demonstrate the transition from a well-mixed state +to a segregated state with large-grain-rich and small-grain-rich regions. In Fig. 4(e), the normalized velocity fields +(𝑣cen − 𝑣𝑧) +√︁ +𝜌s/𝑃w, relative to the velocity at the center of the chute, 𝑣cen = 𝑣𝑧(𝑥 = 0), show that the velocity field +rapidly develops into a steady flow field, even while the segregation process is still ongoing, and the 𝑐l field continues +to evolve. +At long times, near the end of the simulated time window (˜𝑡 = 𝑡/(𝑑s√︁ +𝜌s/𝑃w) ≳ 3 × 105), the concentration field +evolves very slowly, so that 𝐷𝑐l/𝐷𝑡 ≈ 0, and the state of segregation may be regarded as quasi-steady. Therefore, +according to equations (2.1) and (2.12) and the no-flux boundary condition at the walls, the total flux is approximately +zero (𝑤l +𝑖 = 𝑤diff +𝑖 ++ 𝑤seg +𝑖 +≈ 0𝑖) at each 𝑥-position, meaning that the segregation flux is approximately balanced by the +diffusion flux in this quasi-steady flow regime. Then, using the expressions for the two fluxes, equations (2.14) and +(2.15), this observation implies that +𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑥 ≈ 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕 �𝛾 +𝜕𝑥 . +(4.2) +The field quantities appearing in this expression may be obtained by coarse-graining the DEM data in the quasi-steady +flow regime. Therefore, since𝐶diff has been previously determined, (4.2) may be used to determine the parameter𝐶S +seg as +follows. First, we acquire the field quantities 𝑐l (and hence ¯𝑑) and 𝑣𝑧 by spatially coarse-graining 152 evenly-distributed +snapshots in time in the quasi-steady regime (˜𝑡 > 3 × 105).1 Then, we arithmetically average these fields in time, +yielding fields that only depend on the spatial coordinate 𝑥, and take spatial gradients to obtain 𝜕𝑐l/𝜕𝑥, �𝛾 = 𝜕𝑣𝑧/𝜕𝑥, +and 𝜕 �𝛾/𝜕𝑥 = 𝜕2𝑣𝑧/𝜕𝑥2.2 +Next, as suggested by (4.2), we plot 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) versus ¯𝑑2𝑐l(1 − 𝑐l)(𝜕 �𝛾/𝜕𝑥) in +Fig. 5(a), in which each point represents a unique 𝑥-position. A linear relation is observed, supporting our choice +for the form of the constitutive equation for the segregation flux (2.15). In order to obtain further evidence for this +choice, we consider four additional cases: (1) a lower flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.375, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; +1Results are not particularly sensitive to the quasi-steady regime criterion, and we only provide this value as a guideline. +2Detailed descriptions of our coarse-graining techniques may be found in Appendix A.2. +11 + +(2) a narrower channel case {𝑊/ ¯𝑑0 = 40, 𝜇w = 0.45, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (3) a more large grains case {𝑊/ ¯𝑑0 = +60, 𝜇w = 0.45, 𝑐l +0 = 0.75, 𝑑l/𝑑s = 1.5}; and (4) a larger size ratio case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.45, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 3.0}. +Coarse-graining the quasi-steady fields for each case and including the field data in Fig. 5(a), we observe a strong linear +collapse. Finally, the dimensionless material parameter 𝐶S +seg may be obtained from the slope of the linear relation in +Fig. 5(a) (indicated by the solid line). We determine the numerical value for disks to be 𝐶S +seg = 0.23 and note that this +value indeed appears to be independent of the grain-size ratio 𝑑l/𝑑s. +We carry out an analogous set of DEM simulations for dense, bidisperse mixtures of spheres to determine the +value of 𝐶S +seg for spheres. The DEM setup for spheres is similar to that shown in Fig 4(a) for disks with a domain +size of length 𝐿 = 20 ¯𝑑0 in the 𝑧-direction, width 𝑊 in the 𝑥-direction, which is varied in our DEM simulations, and +out-of-plane thickness 𝐻 = 10 ¯𝑑0 in the 𝑦-direction. Periodic boundary conditions are applied along both the 𝑦- and +𝑧-directions for spheres, and a constant compressive normal stress 𝜎𝑥𝑥 = −𝑃𝑤 is applied using the same feedback +scheme as utilized for disks. In DEM simulations of dense flows of spheres, we observe normal stress differences, +in which the normal stresses 𝜎𝑦𝑦 and 𝜎𝑧𝑧 are slightly different from the prescribed value of 𝜎𝑥𝑥 = −𝑃𝑤, which +is a widely-reported feature of dense flows of spheres in the literature (e.g., Srivastava et al., 2021). However, all +normal stresses, and hence the pressure field, are spatially uniform, and segregation occurs only due to shear-strain-rate +gradients. As for disks, the set of system parameters {𝑊/ ¯𝑑0, 𝜇w, 𝑐l +0, 𝑑l/𝑑s} specifies the geometry, loads, and initial +conditions for a given case of vertical chute flow. Here, the dimensionless parameter 𝜇w = 𝜙𝜌s𝐺𝑊/2𝑃w continues +to control the total flow rate down the chute. We consider five different cases: (1) a base case {𝑊/ ¯𝑑0 = 60, 𝜇w = +0.51, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (2) a lower flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.46, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (3) a narrower +channel and higher flow rate case {𝑊/ ¯𝑑0 = 50, 𝜇w = 0.59, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (4) a more large grains and higher +flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.58, 𝑐l +0 = 0.75, 𝑑l/𝑑s = 1.5}; and (5) a larger size ratio and higher flow rate case +{𝑊/ ¯𝑑0 = 60, 𝜇w = 0.58, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 2.0}. Each case involves ∼ 20000 flowing grains. After a sufficiently long +simulation time, a quasi-steady state is attained in each case, implying the flux balance (4.2). The quasi-steady field +quantities appearing in (4.2) are extracted for 1000 snapshots in time using the coarse-graining techniques described in +Appendix A.2 and then arithmetically averaged in time. The calculated quasi-steady diffusion flux 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) at +discrete 𝑥-positions for each of the five cases is plotted versus the calculated quantity ¯𝑑2𝑐l(1 − 𝑐l)(𝜕 �𝛾/𝜕𝑥) in Fig. 5(b), +and we observe a linear relation. Therefore, the form of the constitutive equation for the segregation flux (4.2) is also +applicable to dense, bidisperse systems of spheres. The numerical value of the dimensionless material parameter 𝐶S +seg +for spheres is determined from the slope of the linear relation to be 𝐶S +seg = 0.08. +4.2 +Annular shear flow +The constitutive equation for the segregation flux (2.15) and the fitted values of the material parameters should be +general across different flow geometries. To test this for the case of disks, we apply the same process described in the +preceding section for vertical chute flow to a different flow geometry–annular shear flow. In this flow geometry, flow is +driven through motion of the boundary rather than by gravity, but as in vertical chute flow, the pressure field is spatially +uniform, which eliminates hydrostatic pressure gradients so that only shear-strain-rate-gradient-driven size-segregation +occurs. +Our DEM simulations of annular shear flow of a dense, bidisperse granular mixture of disks follow the procedures +utilized in prior works in the literature for monodisperse, frictional disks (Koval et al., 2009; Kamrin and Koval, 2012, +2014; Liu and Henann, 2018). Consider a dense, bidisperse granular mixture in a two-dimensional annular shear cell +with rough circular walls of inner radius 𝑅 and outer radius 𝑅o, as shown in Fig. 6(a) for the case of 𝑅 = 60 ¯𝑑0. The +inner and outer walls consist of rings of glued large grains, denoted as black in Fig. 6(a). The circumferential velocity +of the inner wall is prescribed to be 𝑣w, and its radial position is fixed. The outer wall does not rotate, and its radius +𝑅o fluctuates slightly so as to maintain a constant imposed compressive normal stress 𝑃w, utilizing the wall-position +control method used throughout this work (Koval et al., 2009). As in Liu and Henann (2018), we simulate the full +shear cell, as shown in Fig. 6(a), instead of applying periodic boundary conditions along the circumferential direction +to a slice (Koval et al., 2009; Kamrin and Koval, 2012, 2014). In this flow geometry, all fields are axisymmetric (i.e., +invariant along the 𝜃-direction), and the only non-zero component of the velocity field is the circumferential component +𝑣 𝜃. Moreover, flow tends to localize near the inner wall with 𝑣 𝜃 rapidly decaying with radial position, as qualitatively +illustrated by the steady velocity field sketched in Fig. 6(a). We choose the outer radius 𝑅o to be sufficiently large +so that it does not affect the resulting flow and segregation fields, and based on our experience, we take 𝑅o = 2𝑅. +Therefore, the role of the outer wall is simply to apply a far-field pressure, and otherwise, it does not affect the flow +12 + +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +10 +20 +30 +10-3 +10-2 +10-1 +100 +0 +20 +40 +60 +0 +0.2 +0.4 +0.6 +0.8 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +0.2 +0.4 +0.6 +0.8 +1 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo 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flowing grains. The inner wall radius is 𝑅 = 60 ¯𝑑0, and the outer wall radius is 𝑅o = 2𝑅. The inner and +outer walls consist of rings of glued large grains, denoted as black. (b) Segregated configuration after flowing for +a total simulation time of ˜𝑡 = 𝑡/(𝑅/𝑣w) = 584. (c) Spatiotemporal evolution of the large grain concentration field. +Spatial profiles of (d) the concentration field 𝑐l and (e) the normalized circumferential velocity field 𝑣 𝜃/𝑣w at three +times (˜𝑡 = 5, 50 and 500) as indicated by the horizontal lines in (c). +and segregation fields. +Regarding the stress field, in all of our DEM simulations of annular shear flow of bidisperse disks, we observe that +the normal stresses are approximately equal, i.e., 𝜎𝑟𝑟 ≈ 𝜎𝜃 𝜃, and spatially uniform, so that the force balance along +the 𝑟-direction gives that the pressure field is 𝑃(𝑟) = −𝜎𝑟𝑟 (𝑟) = 𝑃w. Since the pressure field is spatially uniform, all +segregation in annular shear flow is due to shear-strain-rate-gradients. The moment balance gives that the equivalent +shear stress field is 𝜏(𝑟) = |𝜎𝑟 𝜃 (𝑟)| = |𝜎𝜃𝑟 (𝑟)| = 𝜏w(𝑅/𝑟)2, where 𝜏w is the inner-wall shear stress. It is important to +13 + +-8 +-6 +-4 +-2 +0 +10-3 +-2 +-1.5 +-1 +-0.5 +0 +10-3 +-8 +-6 +-4 +-2 +0 +10-3 +-2 +-1.5 +-1 +-0.5 +0 +10-3 +Figure 7: Collapse of 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑟) versus ¯𝑑2𝑐l(1−𝑐l)(𝜕 �𝛾/𝜕𝑟) for several cases of annular shear flow of bidisperse +disks. Symbols represent coarse-grained, quasi-steady DEM field data, and the solid line is the best linear fit using +𝐶S +seg = 0.23. +note that 𝜏w is not directly prescribed in our DEM simulations. Instead, the inner-wall velocity 𝑣w is prescribed, and +𝜏w arises as a result. The stress ratio field is then +𝜇(𝑟) = 𝜇w(𝑅/𝑟)2, +(4.3) +where 𝜇w = 𝜏w/𝑃w is the maximum value of 𝜇, occurring at the inner wall (𝑟 = 𝑅). +As for vertical chute flow, there are four important dimensionless parameters that specify the geometry, loads, +and initial conditions for a given case of annular shear flow of dense, bidisperse granular mixtures: (1) 𝑅/ ¯𝑑0, the +dimensionless inner-wall radius; (2) ˜𝑣w = (𝑣w/𝑅) +√︃ +𝜋𝜌s ¯𝑑0 +2/(4𝑃w), the dimensionless inner-wall velocity, which +determines 𝜏w and hence 𝜇w; (3) 𝑐l +0(𝑟), the initial large-grain concentration field; and (4) 𝑑l/𝑑s, the bidisperse grain- +size ratio. We choose a representative base case of annular shear flow identified by the parameter set {𝑅/ ¯𝑑0 = 60, ˜𝑣w = +0.01, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}. The well-mixed initial configuration for the base-case DEM simulation is shown in +Fig. 6(a), and the segregated configuration after driving flow for a total simulation time of ˜𝑡 = 𝑡/(𝑅/𝑣w) = 584 is shown +in Fig. 6(b). The large, dark-gray grains segregate into a ring near the inner wall, while the small, light-gray grains +form a band just outside this region. Outside of these bands, where the shear strain-rate is very small, the large and +small grains remain well-mixed. Contours of the spatiotemporal evolution of the coarse-grained concentration field 𝑐l +are plotted in Figs. 6(c), illustrating the time-evolution of 𝑐l field. Spatial profiles of the concentration and velocity +fields at three selected snapshots during the segregation dynamics (˜𝑡 = 𝑡/(𝑅/𝑣w) = 5, 50, and 500 as indicated by the +dashed lines in Fig. 6(c)) are shown in Figs. 6(d) and (e). The radial 𝑐l profiles in Fig. 6(d) demonstrate the formation +of large-grain-rich and small-grain-rich regions with a persistent well-mixed far-field. The normalized velocity fields +in Fig. 6(e) demonstrate that the flowing zone is localized near the inner wall with slow creeping flow observed far +from the wall. As in the case of vertical chute flow, the velocity field quickly develops into a steady flow field, while +the large grain concentration field 𝑐l evolves over a longer time-scale before approaching a quasi-steady state near the +end of our simulated time window. +Again, at long times, near the end of the simulated time window (˜𝑡 = 𝑡/(𝑅/𝑣w) ≳ 500), the concentration field +evolves very slowly, which we identify as the quasi-steady regime. In this regime, the segregation and diffusion fluxes +approximately balance at each 𝑟-position, implying that +𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑟 ≈ 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕 �𝛾 +𝜕𝑟 . +(4.4) +As for vertical chute flow, we spatially coarse-grain the DEM data to obtain the 𝑐l and 𝑣 𝜃 fields for 144 evenly- +distributed snapshots in time in the quasi-steady regime (˜𝑡 ≳ 500), which are arithmetically averaged in time to obtain +the quasi-steady 𝑐l(𝑟) and 𝑣 𝜃 (𝑟) fields and then spatially differentiated to obtain the remaining field quantities in +(4.4). Next, we plot 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑟) versus ¯𝑑2𝑐l(1 − 𝑐l)(𝜕 �𝛾/𝜕𝑟) in Fig. 7 with each point representing a unique +𝑟-position. Finally, this process is repeated for four additional cases of annular shear flow: (1) a lower inner-wall +14 + +velocity {𝑅/ ¯𝑑0 = 60, ˜𝑣w = 0.001, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (2) a smaller inner-wall radius {𝑅/ ¯𝑑0 = 40, ˜𝑣w = 0.01, 𝑐l +0 = +0.5, 𝑑l/𝑑s = 1.5}; (3) more large grains {𝑅/ ¯𝑑0 = 60, ˜𝑣w = 0.01, 𝑐l +0 = 0.75, 𝑑l/𝑑s = 1.5}; and (4) a larger size ratio +{𝑅/ ¯𝑑0 = 60, ˜𝑣w = 0.01, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 3.0}, and the coarse-grained, quasi-steady fields are included in the data +plotted in Fig. 7. Collectively, we observe a strong collapse to a linear relation. Crucially, the slope of the linear +relation in Fig. 7 (indicated by the solid line) gives the same value for the dimensionless material parameter 𝐶S +seg +obtained by fitting to vertical chute flow data, 𝐶S +seg = 0.23. This observation of agreement between the best fit values +of 𝐶S +seg obtained using two different flow geometries provides support for our choice of the constitutive equation for the +segregation flux (2.15) and the fitted value of 𝐶S +seg for disks. Having established that the parameter 𝐶S +seg is independent +of the flow geometry, driving conditions, and initial conditions, we henceforth regard 𝐶S +seg as a material parameter for +a given dense granular system, analogous to how rheological material parameters such as 𝜇s are regarded. Of course, +the values of 𝐶S +seg determined above for disks and spheres likely depend on grain interaction properties, such as the +inter-particle friction coefficient, but elucidating this dependence is beyond the scope of the present work. +5 +Validation of the continuum model in the transient regime +In the preceding section, we only used DEM data from the quasi-steady regime to test the constitutive equation for the +shear-strain-rate-gradient-driven segregation flux (2.15) and to determine the material parameter 𝐶S +seg. In this section, +we compare continuum model predictions of the transient evolution of segregation and flow fields to the DEM data for +both vertical chute flow and annular shear flow as a validation test of the model. To obtain continuum model predictions +in the transient regime, we couple the segregation dynamics equation (2.16) with the NGF model, (2.6) and (2.7), and +a use fixed sets of material parameters for disks, +{𝜇s = 0.272, 𝑏 = 1.168, 𝐴 = 0.9, 𝐶diff = 0.20, 𝐶S +seg = 0.23}, +(5.1) +and for spheres, +{𝜇s = 0.37, 𝜇2 = 0.95, 𝐼0 = 0.58, 𝐴 = 0.43, 𝐶diff = 0.045, 𝐶S +seg = 0.08}. +(5.2) +5.1 +Vertical chute flow +First, we describe in detail how the continuum model is solved to obtain predictions for the transient evolution of +segregation and flow fields for the case of vertical chute flow of disks. In vertical chute flow, the stress field may be +straightforwardly deduced from a static force balance, giving that the pressure field is uniform, 𝑃(𝑥) = 𝑃w, and that +the stress ratio field 𝜇(𝑥) is given through (4.1), and therefore, the balance of linear momentum (2.2) is satisfied and +does not further enter the solution procedure. Continuum model predictions are obtained by numerically solving the +remaining governing equations using finite-differences. Summarizing the coupled boundary/initial-value problem for +flow and segregation in the context of vertical chute flow, the unknown fields are the velocity field 𝑣𝑧(𝑥, 𝑡) and the +accompanying strain-rate field �𝛾(𝑥, 𝑡) = 𝜕𝑣𝑧/𝜕𝑥, the granular fluidity field 𝑔(𝑥, 𝑡), and the large-grain concentration +field 𝑐l(𝑥, 𝑡). The governing equations are (1) the flow rule (2.6) +�𝛾 = 𝑔𝜇, +(5.3) +(2) the nonlocal rheology (2.7) +𝑔 = 𝑔loc(𝜇, 𝑃w) + 𝜉2(𝜇) 𝜕2𝑔 +𝜕𝑥2 +(5.4) +with 𝑔loc and 𝜉 given through (2.9) and (2.11)1, respectively, and (3) the segregation dynamics equation (2.16) +𝜕𝑐l +𝜕𝑡 + 𝜕 +𝜕𝑥 +� +−𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑥 + 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕 �𝛾 +𝜕𝑥 +� += 0, +(5.5) +where ¯𝑑 = 𝑐l𝑑l + (1 − 𝑐l)𝑑s. +Regarding boundary conditions, we impose Dirichlet fluidity boundary conditions at the walls (i.e., 𝑔 = 𝑔loc(𝜇w, 𝑃w) +at 𝑥 = ±𝑊/2) as well as no flux boundary conditions at the walls (i.e., 𝑤l +𝑥 = −𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) + 𝐶S +seg ¯𝑑2𝑐l(1 − +𝑐l)(𝜕 �𝛾/𝜕𝑥) = 0 at 𝑥 = ±𝑊/2). Due to the time derivative in (5.5), an initial condition for the concentration field +15 + +𝑐l +0(𝑥) = 𝑐l(𝑥, 𝑡 = 0) is required. In order to account for the concentration fluctuations inherent in the initial state, we +obtain the coarse-grained 𝑐l-field from the initial DEM configuration for each case and utilize this field as the initial +condition field 𝑐l +0(𝑥) in each of the respective continuum simulations. +Then, for a given case identified through a set of input parameters {𝑊/ ¯𝑑0, 𝜇w, 𝑐l +0(𝑥), 𝑑l/𝑑s}, we obtain numerical +predictions of the continuum model utilizing finite differences as follows. First, at a given point in time, the concentration +field 𝑐l(𝑥) is known, allowing the average grain-size field to be calculated through ¯𝑑 = 𝑐l𝑑l + (1−𝑐l)𝑑s. Using the stress +ratio field 𝜇(𝑥) for vertical chute flow (4.1), the local fluidity 𝑔loc(𝜇, 𝑃) and the cooperativity length 𝜉(𝜇) (equations +(2.9) and (2.11)1) may be calculated at each spatial grid point. Then, the nonlocal rheology (5.4) may be used to solve +for the fluidity field 𝑔(𝑥) at the current step, using central differences in space. The strain-rate field follows using (5.3), +which may be integrated to obtain the velocity field 𝑣𝑧(𝑥). Next, (5.5) is used to determine the concentration field +at the next time step utilizing the forward Euler method and central differences in space with one modification–the +spatial derivatives of 𝑐l appearing in the diffusion flux term in (5.5) are treated implicitly in order to improve numerical +stability. This completes one time step, and this process is repeated to step forward in time and calculate the transient +evolution of the concentration and flow fields, 𝑐l(𝑥, 𝑡) and 𝑣𝑧(𝑥, 𝑡). In our finite-difference calculations, we utilize a +fine spatial resolution of Δ𝑥 ≪ ¯𝑑0, and we have verified that the time-step is sufficiently small in order to ensure stable, +accurate results. +We compare predictions of the continuum model against DEM data for all five cases of vertical chute flow considered +in Section 4.1 in order to test the generality of the model. Figures 8 and 9 summarize the comparisons for these five +cases. The first column of Figs. 8 and 9 shows the spatiotemporal contours of the evolution of the 𝑐l field measured in the +DEM simulations for each case, and the second column shows comparisons of the DEM simulations (solid black lines) +and the continuum predictions (dashed gray lines) for the 𝑐l field at four snapshots in time (˜𝑡 = 𝑡/ +� +𝑑s√︁ +𝜌s/𝑃w +� += 4×103, +2 × 104, 1 × 105 and 4 × 105) indicated by the horizontal lines in the first column of Figs. 8 and 9. Based on Figs. 8 +and 9, the coupled model generally does a good job capturing the salient features of the evolution of the 𝑐l field across +all cases. For instance, for the narrower chute case shown in Fig. 8(c), the segregation process nearly completes within +the simulated time window with the mixed core along the center of the chute nearly disappearing, and the continuum +model prediction captures this observation well. +Regarding flow fields, comparisons of the quasi-steady, normalized velocity fields at ˜𝑡 = 4 × 105 from the DEM +simulations and the continuum model predictions are shown in the third column of Fig. 8. Since the velocity field +evolves minimally during the segregation process, only the flow field at long time is shown. We note that the velocity +field is very well-predicted in all cases, including the creeping regions far from the wall. This favorable comparison +provides support of our generalization of the NGF model to bidisperse granular systems discussed in Section 2.3–in +particular, the choices to use the average grain size ¯𝑑 in the expression for the cooperativity length (2.11) and to +continue to use the numerical value for the nonlocal amplitude determined for monodisperse systems, 𝐴 = 0.9, without +refitting. +Next, we make comparisons between continuum model predictions and DEM data for vertical chute flow of +bidisperse spheres. The governing equations and boundary conditions are same as those used above for bidisperse +disks. That is, the fluidity field is governed by the nonlocal rheology (5.4) with Dirichlet fluidity boundary conditions at +the walls (𝑔 = 𝑔loc(𝜇w, 𝑃w) at 𝑥 = ±𝑊/2), and the concentration field is governed by the segregation dynamics equation +(5.5) with no flux boundary conditions at the walls (𝑤l +𝑥 = 0 at 𝑥 = ±𝑊/2). The only difference from the process +described above for bidisperse disks is that the values 𝑃w and 𝜇w used in the continuum simulations are obtained from +the coarse-grained stress fields in the DEM data for each case, rather than based on the nominal value of the compressive +wall stress 𝑃w applied in the DEM simulation. This is done to account for the normal stress differences that arise for +spheres, which slightly affect the predicted velocity fields and hence the consequent concentration fields. We consider +three different cases: (1) the base case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.51, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}, (2) the more large grains and +higher flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.58, 𝑐l +0 = 0.75, 𝑑l/𝑑s = 1.5}, and (3) the larger size ratio and higher flow +rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.58, 𝑐𝑙 +0 = 0.5, 𝑑𝑙/𝑑𝑠 = 2.0}, which are shown in Figs. 10(a), (b), and (c), respectively. +The leftmost column of Fig. 10 shows the spatiotemporal contours of the evolution of the 𝑐l field from the DEM +data. The middle column shows comparisons between the DEM simulations (solid black lines) and the continuum +model predictions (dashed gray lines) for the 𝑐l fields at four time snapshots in time (˜𝑡 = 𝑡/ +� +𝑑s√︁ +𝜌s/𝑃w +� += 5 × 102, +2 × 103, 1 × 104 and 4.5 × 104) indicated by the horizontal lines in the first column of Fig. 10. Lastly, comparisons +of the quasi-steady, normalized velocity fields at ˜𝑡 = 4.5 × 104 from the DEM simulations and the continuum model +predictions are shown in the third column of Fig. 10. The continuum model is able to capture the decaying velocity +16 + +-20 +0 +20 +0 +0.5 +1 -20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +0 +0.5 +1 +-10 +0 +10 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-10 +0 +10 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +1 +2 +3 +4 +105 +0 +1 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KIvWyEQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWl5ZXSulzY2t7 +Z37N29po5TRVmDxiJWbZ9oJrhkDeAgWDtRjES+YC1/eDOptx6Z0 +jyW9zBKmBeRvuQhpwSM1bMPusBFwDIYX1cNRky7zsN5zy47FWcq +vAhuDmWUq96zv7pBTNOISaCaN1xnQS8jCjgVLBxqZtqlhA6JH3 +WMSiJWeRl0/PH+MQ4AQ5jZ4EPHV/T2Qk0noU+aYzIjDQ87WJ+V ++tk0J45WVcJikwSWeLwlRgiPEkCxwxSiIkQFCFTe3YjogilAwi +ZVMCO78lxeheVZxLyrVu2q5dpjHURH6BidIhdohq6RXUQBRl +6Bm9ojfryXqx3q2PWvBymf20R9Znz+QupUZ˜t = 4 ⇥ 105 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KokXdCAU3LivYC7SxTCaTduhkEmZOhBIKvobF4q4 +9Tnc+TZO2y09YeBj/+cwznz+4ngGhzn2yosLa+srhXSxubW9 +s79u5eU8epoqxBYxGrtk80E1yBnAQrJ0oRiJfsJY/vJnUW49Ma +R7LexglzItIX/KQUwLG6tkHXeAiYBmMr6sGI6Zd5+G8Z5edijMV +XgQ3hzLKVe/ZX90gpmnEJFBtO64TgJeRhRwKti41E01Swgdkj7 +rGJTELPKy6fljfGKcAIexMk8Cnrq/JzISaT2KfNMZERjo+drE/K +/WSG8jIukxSYpLNFYSowxHiSBQ64YhTEyAChiptbMR0QRSiYx +EomBHf+y4vQPKu4F5XqXbVcO8zjKIjdIxOkYsuUQ3dojpqIoy +9Ixe0Zv1ZL1Y79bHrLVg5TP76I+szx+NspUX˜t = 4 ⇥ 103 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KUoq6EQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWV1b3yhulra2d3 +b37P2Dlo5TRVmTxiJWHZ9oJrhkTeAgWCdRjES+YG1/dDOtx+Z0 +jyW9zBOmBeRgeQhpwSM1bePesBFwDKYXFcNRky7zkOtb5edijMT +XgY3hzLK1ejbX70gpmnEJFBtO6TgJeRhRwKtik1Es1SwgdkQH +rGpTELPKy2fkTfGacAIexMk8Cnrm/JzISaT2OfNMZERjqxdrU/K +/WTSG8jIukxSYpPNFYSowxHiaBQ64YhTE2AChiptbMR0SRSiYx +EomBHfxy8vQqlbci0rtrlauH+dxFNEJOkXnyEWXqI5uUQM1EUZ +ekav6M16sl6sd+tj3lqw8plD9EfW5w+MGpUW˜t = 2 ⇥ 104 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KIvWyEQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWl5ZXSulzY2t7 +Z37N29po5TRVmDxiJWbZ9oJrhkDeAgWDtRjES+YC1/eDOptx6Z0 +jyW9zBKmBeRvuQhpwSM1bMPusBFwDIYX7sGI6Zd5+G8Z5edijMV +XgQ3hzLKVe/ZX90gpmnEJFBtO64TgJeRhRwKti41E01Swgdkj7 +rGJTELPKy6fljfGKcAIexMk8Cnrq/JzISaT2KfNMZERjo+drE/K +/WSG8jIukxSYpLNFYSowxHiSBQ64YhTEyAChiptbMR0QRSiYx +EomBHf+y4vQPKu4F5XqXbVcO8zjKIjdIxOkYsuUQ3dojpqIoy +9Ixe0Zv1ZL1Y79bHrLVg5TP76I+szx+MEJUW˜t = 1 ⇥ 105 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KIvWyEQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWl5ZXSulzY2t7 +Z37N29po5TRVmDxiJWbZ9oJrhkDeAgWDtRjES+YC1/eDOptx6Z0 +jyW9zBKmBeRvuQhpwSM1bMPusBFwDIYX1cNRky7zsN5zy47FWcq +vAhuDmWUq96zv7pBTNOISaCaN1xnQS8jCjgVLBxqZtqlhA6JH3 +WMSiJWeRl0/PH+MQ4AQ5jZ4EPHV/T2Qk0noU+aYzIjDQ87WJ+V ++tk0J45WVcJikwSWeLwlRgiPEkCxwxSiIkQFCFTe3YjogilAwi +ZVMCO78lxeheVZxLyrVu2q5dpjHURH6BidIhdohq6RXUQBRl +6Bm9ojfryXqx3q2PWvBymf20R9Znz+QupUZ˜t = 4 ⇥ 105 +AB8H +icbVBNSwMxEJ +3Ur1q/qh69BIv +gqexKUY8FLx4r +2A9p15JNs21ok +l2SrFCW/govH +hTx6s/x5r8xbf +egrQ8GHu/NMDM +vTAQ31vO+UWF +tfWNzq7hd2tnd +2z8oHx61TJxqy +po0FrHuhMQwR +VrWm4F6ySaER +kK1g7HNzO/cS +04bG6t5OEBZIM +FY84JdZJD/Qx +62mJxbRfrnhVb +w68SvycVCBHo1 +/+6g1imkqmLBX +EmK7vJTbIiLa +cCjYt9VLDEkLH +ZMi6jioimQmy+ +cFTfOaUAY5i7 +UpZPFd/T2REGj +ORoeuUxI7Msjc +T/O6qY2ug4yr +JLVM0cWiKBXY +xnj2PR5wzagVE +0cI1dzdiumIaE +Kty6jkQvCX1 +4lrYuqf1mt3dU +qdZzHUYQTOIVz +8OEK6nALDWgCB +QnP8ApvSKMX9 +I4+Fq0FlM8cwx ++gzx+s0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBNSwMxEJ3Ur1q/q +h69BIvgqexKUY8FLx4r2A9p15JNs21okl2SrFCW/govHhTx6s/x +5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx +61TJxqypo0FrHuhMQwRVrWm4F6ySaERkK1g7HNzO/cS04bG6t +5OEBZIMFY84JdZJD/Qx62mJxbRfrnhVbw68SvycVCBHo1/+6g1i 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+dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AhP2NQ=(c) +Figure 8: Comparisons of continuum model predictions with corresponding DEM simulation results for the transient +evolution of the segregation dynamics for three cases of vertical chute flow of disks: (a) Base case {𝑊/ ¯𝑑0 = 60, 𝜇w = +0.45, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (b) Lower flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.375, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; and (c) +Narrower chute width case {𝑊/ ¯𝑑0 = 40, 𝜇w = 0.45, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}. Additional cases are shown in Fig. 9. For +each case, the first column shows spatiotemporal contours of the evolution of 𝑐l measured in the DEM simulations. +The second column shows comparisons of the DEM simulations (solid black lines) and continuum model predictions +(dashed gray lines) of the 𝑐l field at four time snapshots representing different stages of the segregation process: +˜𝑡 = 4 × 103, 2 × 104, 1 × 105, and 4 × 105 in the sequence of top left, top right, bottom left, bottom right. The third +column shows comparisons of the quasi-steady, normalized velocity profiles at ˜𝑡 = 4 × 105 from DEM simulations and +continuum model predictions. +field quite well in all cases, and therefore, our choice to continue using the value of the nonlocal amplitude estimated for +monodisperse systems, 𝐴 = 0.43, without readjustment works well for spheres. Overall, the coupled continuum model +is capable of quantitatively predicting both the flow fields and the transient evolution of the segregation dynamics in +vertical chute flow of bidisperse spheres. +17 + +-20 +0 +20 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +1 +2 +3 +4 +105 +0 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +2 +4 +6 +105 +0 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +2 +4 +6 +105 +0 +1 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv 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+/WTSG8jIukxSYpPNFYSowxHiaBQ64YhTE2AChiptbMR0SRSiYx +EomBHfxy8vQqlbci0rtrlauH+dxFNEJOkXnyEWXqI5uUQM1EUZ +ekav6M16sl6sd+tj3lqw8plD9EfW5w+MGpUW˜t = 2 ⇥ 104 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KIvWyEQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWl5ZXSulzY2t7 +Z37N29po5TRVmDxiJWbZ9oJrhkDeAgWDtRjES+YC1/eDOptx6Z0 +jyW9zBKmBeRvuQhpwSM1bMPusBFwDIYX1cNRky7zsN5zy47FWcq +vAhuDmWUq96zv7pBTNOISaCaN1xnQS8jCjgVLBxqZtqlhA6JH3 +WMSiJWeRl0/PH+MQ4AQ5jZ4EPHV/T2Qk0noU+aYzIjDQ87WJ+V ++tk0J45WVcJikwSWeLwlRgiPEkCxwxSiIkQFCFTe3YjogilAwi +ZVMCO78lxeheVZxLyrVu2q5dpjHURH6BidIhdohq6RXUQBRl +6Bm9ojfryXqx3q2PWvBymf20R9Znz+QupUZ˜t = 4 ⇥ 105 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KIvWyEQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWl5ZXSulzY2t7 +Z37N29po5TRVmDxiJWbZ9oJrhkDeAgWDtRjES+YC1/eDOptx6Z0 +jyW9zBKmBeRvuQhpwSM1bMPusBFwDIYX7sGI6Zd5+G8Z5edijMV +XgQ3hzLKVe/ZX90gpmnEJFBtO64TgJeRhRwKti41E01Swgdkj7 +rGJTELPKy6fljfGKcAIexMk8Cnrq/JzISaT2KfNMZERjo+drE/K +/WSG8jIukxSYpLNFYSowxHiSBQ64YhTEyAChiptbMR0QRSiYx +EomBHf+y4vQPKu4F5XqXbVcO8zjKIjdIxOkYsuUQ3dojpqIoy +9Ixe0Zv1ZL1Y79bHrLVg5TP76I+szx+MEJUW˜t = 1 ⇥ 105 +AB/nicbZDLSsNAFIYn9VbrL +SqC4GawCK5KIvWyEQpuXFawF2hjmUwm7dDJMycCUfBU3LhRx +63O482ctlo6w8DH/85h3Pm9xPBNTjOt1VYWl5ZXSulzY2t7 +Z37N29po5TRVmDxiJWbZ9oJrhkDeAgWDtRjES+YC1/eDOptx6Z0 +jyW9zBKmBeRvuQhpwSM1bMPusBFwDIYX1cNRky7zsN5zy47FWcq +vAhuDmWUq96zv7pBTNOISaCaN1xnQS8jCjgVLBxqZtqlhA6JH3 +WMSiJWeRl0/PH+MQ4AQ5jZ4EPHV/T2Qk0noU+aYzIjDQ87WJ+V ++tk0J45WVcJikwSWeLwlRgiPEkCxwxSiIkQFCFTe3YjogilAwi +ZVMCO78lxeheVZxLyrVu2q5dpjHURH6BidIhdohq6RXUQBRl +6Bm9ojfryXqx3q2PWvBymf20R9Znz+QupUZ˜t = 4 ⇥ 105 +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8H +icbVBNSwMxEJ +3Ur1q/qh69BIv +gqexKUY8FLx4r +2A9p15JNs21ok +l2SrFCW/govH +hTx6s/x5r8xbf +egrQ8GHu/NMDM +vTAQ31vO+UWF +tfWNzq7hd2tnd +2z8oHx61TJxqy +po0FrHuhMQwR +VrWm4F6ySaER +kK1g7HNzO/cS +04bG6t5OEBZIM +FY84JdZJD/Qx +62mJxbRfrnhVb +w68SvycVCBHo1 +/+6g1imkqmLBX +EmK7vJTbIiLa +cCjYt9VLDEkLH +ZMi6jioimQmy+ +cFTfOaUAY5i7 +UpZPFd/T2REGj +ORoeuUxI7Msjc +T/O6qY2ug4yr 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+wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +Figure 9: Comparisons of continuum model predictions with corresponding DEM simulation results for the transient +evolution of the segregation dynamics for two cases of vertical chute flow of disks: (a) More large grains case {𝑊/ ¯𝑑0 = +60, 𝜇w = 0.45, 𝑐l +0 = 0.75, 𝑑l/𝑑s = 1.5} and (b) Larger size ratio case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.45, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 3.0}. +Additional cases are shown in Fig. 8. Results are organized as described in the caption of Fig. 8. +5.2 +Annular shear flow +We utilize an analogous process to obtain continuum model predictions for the transient evolution of concentration and +flow fields in annular shear flow of disks. In annular shear, based on the static force and moment balances, the pressure +field is uniform, 𝑃(𝑟) = 𝑃w, and the stress ratio field is given by (4.3). The governing equations (5.4) and (5.5) are +modified to appropriately account for the divergence and Laplacian operators in cylindrical coordinates. Also, since +𝑣w, not 𝜇w, is specified in our DEM simulations of annular shear, while 𝜇w is specified in our continuum simulations, +we iteratively adjust the value of 𝜇w input into our continuum simulations in order to achieve the target value of 𝑣w in +the predicted quasi-steady flow field. Otherwise, our process for obtaining numerical predictions from the continuum +model is the same. Dirichlet fluidity boundary conditions and no flux boundary conditions are imposed at the walls, +and the initial concentration field is extracted from the initial DEM configuration for each case. +Then, we compare continuum model predictions against DEM data for the five cases of annular shear flow discussed +in Section 4.2 in order to further validate the model. Figures 11 and 12 summarize the comparisons for these fives +cases and are organized in the same manner as Figs. 8 and 9. Again, the coupled, continuum model does a good job +capturing the segregation dynamics and its dependence on the input parameters. Moreover, the quasi-steady velocity +fields are well-predicted by the NGF model in all cases, including the creeping region far from the inner wall. We +reiterate that all continuum model predictions are obtained using the same set of material parameters for disks (5.1). +6 +Discussion and Conclusion +In this paper, we studied coupled size-segregation and flow in dense, bidisperse granular systems of disks and spheres +and developed a phenomenological continuum model that captures the simultaneous evolution of both segregation +and flow fields. We focused on the shear-strain-rate-gradient-driven size-segregation mechanism in two configurations +in which the pressure field is uniform–vertical chute flow and annular shear flow–and based on observations from +DEM simulations, we proposed a phenomenological constitutive equation for the shear-strain-rate-gradient-driven +18 + +-20 +0 +20 +0 +1 +2 +3 +4 +5 +104 +0 +1 +-20 +0 +20 +1 +2 +3 +4 +5 +104 +0 +1 +-20 +0 +20 +1 +2 +3 +4 +104 +0 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +10-2 +100 +-20 +0 +20 +10-2 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +0.5 +1 +-20 +0 +20 +0 +1 +2 +3 +4 +5 +104 +0 +1 +-20 +0 +20 +1 +2 +3 +4 +5 +104 +0 +1 +-20 +0 +20 +1 +2 +3 +4 +104 +0 +1 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ 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+yae/tNlWSsgZNRCLbPlFM8Jg1gINg7VQyEvmCtfzh9cRvPTCpe +BLfwShlXkT6MQ85JaClnlnpAhcBy2F85VpnuoiYwo597/bMqm3Z +U+BF4hSkigrUe+ZXN0hoFrEYqCBKdRw7BS8nEjgVbFzuZoqlhA5 +Jn3U0jYne5OXTC8b4WCsBDhOpXwx4qv6eyEmk1CjydWdEYKDmvY +n4n9fJILz0ch6nGbCYzhaFmcCQ4EkcOCSURAjTQiVXP8V0wGRh +IOraxDcOZPXiTNU8s5t9xbt1qrFHGU0CE6QifIQReohm5QHTUQ +RY/oGb2iN+PJeDHejY9Z65JRzBygPzA+fwDZpW5˜t = 4.5 ⇥ 104 +AB/3icbZDLSsNAFIYn9VbrL +SqI4GawCK5KUuplIxTcuKxgL9DGMplM26GTSZg5EUrswldx40IR +t76GO9/GaZuFtv4w8PGfczhnfj8WXIPjfFu5peWV1bX8emFjc2 +t7x97da+goUZTVaSQi1fKJZoJLVgcOgrVixUjoC9b0h9eTevOBK +c0jeQejmHkh6Uve45SAsbr2Qe4CFgK46szgyHT2HXuy1276JSc +qfAiuBkUaZa1/7qBFNQiaBCqJ123Vi8FKigFPBxoVOolM6JD +0WdugJGaTl07vH+MT4wS4FynzJOCp+3siJaHWo9A3nSGBgZ6vTc +z/au0EepdeymWcAJN0tqiXCAwRnoSBA64YBTEyQKji5lZMB0QRC +iaygnBnf/yIjTKJfe8VLmtFKuHWRx5dISO0Sly0QWqohtUQ3VE +0SN6Rq/ozXqyXqx362PWmrOymX30R9bnD+qxlUE=˜t = 5 ⇥ 102 +AB/3icbZDLSsNAFIYn9VbrL +SqI4GawCK5KUou6EQpuXFawF2hjmUym7dDJMycCV24au4caGI +W1/DnW/jtM1CW38Y+PjPOZwzvx8LrsFxvq3c0vLK6lp+vbCxub +W9Y+/uNXSUKMrqNBKRavlEM8ElqwMHwVqxYiT0BWv6w+tJvfnAl +OaRvINRzLyQ9CXvcUrAWF37oANcBCyF8VXZYMg0dp37s65dErO +VHgR3AyKFOta391gogmIZNABdG67ToxeClRwKlg40In0SwmdEj +6rG1QErPJS6f3j/GJcQLci5R5EvDU/T2RklDrUeibzpDAQM/XJu +Z/tXYCvUsv5TJOgEk6W9RLBIYIT8LAVeMghgZIFRxcyumA6IB +RNZwYTgzn95ERrlknteqtxWitXDLI48OkLH6BS56AJV0Q2qoTqi +6BE9o1f0Zj1ZL9a79TFrzVnZzD76I+vzB+eIlT8=˜t = 2 ⇥ 103 +AB/3icbZDLSsNAFIYn9VbrL +SqI4GawCK5KIkXdCAU3LivYC7SxTCaTduhkEmZOhBK78FXcuFDE +ra/hzrdx2mahrT8MfPznHM6Z308E1+A431ZhaXlda24XtrY3N +resXf3mjpOFWUNGotYtX2imeCSNYCDYO1EMRL5grX84fWk3npgS +vNY3sEoYV5E+pKHnBIwVs8+6AIXActgfOUajJjGrnNf7dlp+JM +hRfBzaGMctV79lc3iGkaMQlUEK07rpOAlxEFnAo2LnVTzRJCh6T +POgYlMZu8bHr/GJ8YJ8BhrMyTgKfu74mMRFqPIt90RgQGer42Mf ++rdVIL72MyQFJulsUZgKDGehIEDrhgFMTJAqOLmVkwHRBEKJ +rKSCcGd/IiNM8q7nmlelst1w7zOIroCB2jU+SiC1RDN6iOGoi +R/SMXtGb9WS9WO/Wx6y1YOUz+iPrM8f532VPw=˜t = 1 ⇥ 104 +ACAXicbVDLSsNAFJ34rPUVd +VNwM1gEVyGR+NgIBTcuK9gHtLFMJpN26OTBzI1Qt34K25cKOLW +v3Dn3zhts9DWAwPnMvd+7xU8EV2Pa3sbS8srq2Xtob25t7+ +yae/tNlWSsgZNRCLbPlFM8Jg1gINg7VQyEvmCtfzh9cRvPTCpe +BLfwShlXkT6MQ85JaClnlnpAhcBy2F85VpnuoiYwo597/bMqm3Z +U+BF4hSkigrUe+ZXN0hoFrEYqCBKdRw7BS8nEjgVbFzuZoqlhA5 +Jn3U0jYne5OXTC8b4WCsBDhOpXwx4qv6eyEmk1CjydWdEYKDmvY +n4n9fJILz0ch6nGbCYzhaFmcCQ4EkcOCSURAjTQiVXP8V0wGRh +IOraxDcOZPXiTNU8s5t9xbt1qrFHGU0CE6QifIQReohm5QHTUQ +RY/oGb2iN+PJeDHejY9Z65JRzBygPzA+fwDZpW5˜t = 4.5 ⇥ 104 +Figure 10: Comparisons of continuum model predictions with corresponding DEM simulation results for the transient +evolution of the segregation dynamics for three cases of vertical chute flow of spheres: (a) Base case {𝑊/ ¯𝑑0 = +60, 𝜇w = 0.51, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (b) More large grains and higher flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.58, 𝑐l +0 = +0.75, 𝑑l/𝑑s = 1.5}; and (c) Larger grain-size ratio and higher flow rate case {𝑊/ ¯𝑑0 = 60, 𝜇w = 0.58, 𝑐l +0 = 0.5, 𝑑l/𝑑s = +2.0}. Results are organized as described in the caption of Fig. 8. +flux. When combined with a standard model for granular diffusion, the segregation model involves two dimensionless +parameters {𝐶diff, 𝐶S +seg}, which multiply the two fluxes appearing in the model–the diffusion and shear-strain-rate- +gradient-driven fluxes, respectively. By coupling the segregation model with the NGF model adapted to bidisperse +systems, we may quantitatively predict both the flow fields and the segregation dynamics for dense flows of bidisperse +disks and spheres for two distinct flow geometries and under a number of different flow conditions. +Size-segregation in granular materials is a complex and rich problem, so there remain many avenues for model +improvement and unresolved research questions to be answered. One important question relates to the constitutive +equation for the shear-strain-rate-gradient-driven segregation flux (2.15). Although our use of a constitutive equation +driven by gradients in �𝛾 does a good job capturing the DEM data, there are other theories in the literature based on +gradients of other field quantities. In particular, Hill and coworkers (Fan and Hill, 2011b; Hill and Tan, 2014) have +proposed that gradients in the kinetic stress, which is related to velocity fluctuations and hence the granular temperature, +drive segregation. Since Zhang and Kamrin (2017) have established a connection between velocity fluctuations and +the granular fluidity 𝑔, it is possible to propose other forms for the constitutive equation for 𝑤seg +𝑖 +based on gradients in +19 + +0 +10 +20 +30 +10-2 +100 +0 +10 +20 +10-2 +100 +0 +50 +0 +0.5 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +0 +10 +20 +10-2 +100 +10 +20 +30 +0 +100 +200 +300 +400 +500 +0 +1 +0 +20 +40 +0 +0.5 +1 +0 +20 +40 +0 +0.5 +1 +0 +20 +40 +0 +0.5 +1 +0 +20 +40 +0 +0.5 +1 +10 +20 +30 +0 +100 +200 +300 +400 +500 +0 +1 +0 +20 +40 +0 +0.5 +1 +0 +20 +40 +0 +0.5 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq 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+ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK7 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AhP2NQ=(c) +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8H +icbVBNSwMxEJ +3Ur1q/qh69BIv +gqexKUY8FLx4r +2A9p15JNs21ok +l2SrFCW/govH +hTx6s/x5r8xbf +egrQ8GHu/NMDM +vTAQ31vO+UWF +tfWNzq7hd2tnd +2z8oHx61TJxqy +po0FrHuhMQwR +VrWm4F6ySaER +kK1g7HNzO/cS +04bG6t5OEBZIM +FY84JdZJD/Qx +62mJxbRfrnhVb +w68SvycVCBHo1 +/+6g1imkqmLBX +EmK7vJTbIiLa +cCjYt9VLDEkLH +ZMi6jioimQmy+ +cFTfOaUAY5i7 +UpZPFd/T2REGj +ORoeuUxI7Msjc +T/O6qY2ug4yr +JLVM0cWiKBXY +xnj2PR5wzagVE +0cI1dzdiumIaE +Kty6jkQvCX1 +4lrYuqf1mt3dU +qdZzHUYQTOIVz +8OEK6nALDWgCB +QnP8ApvSKMX9 +I4+Fq0FlM8cwx ++gzx+s0pA7 +cl +AB8nicbVDLSsNAFJ34rPVd +SO4GSyCq5JIfWyEghuXFewD0lAmk0k7dDITZm6EvoZblwo4tav +cefOG2z0NYDFw7n3Mu94Sp4AZc9tZWV1b39gsbZW3d3b39i +sHh2jMk1ZiyqhdDckhgkuWQs4CNZNSNJKFgnHN1N/c4T04Yr+ +QjlAUJGUgec0rASn4PuIhYDpPby36l6tbcGfAy8QpSRQWa/cpX +L1I0S5gEKogxvuemEOREA6eCTcq9zLCU0BEZMN9SRJmgnx28gS +fWSXCsdK2JOCZ+nsiJ4kx4yS0nQmBoVn0puJ/np9BfBPkXKYZME +ni+JMYFB4+j+OuGYUxNgSQjW3t2I6JpQsCmVbQje4svLpH1R8 +65q9Yd6tXFcxFCJ+gUnSMPXaMGukdN1EIUKfSMXtGbA86L8+58 +zFtXnGLmCP2B8/kDMkeREw=˜t = 5 +AB83icbVBNS8NAEJ34WetX1 +YvgJVgETyWR+nERCl48VrAf0ISy2WzapZtN2J0IJfRvePGgiFf/ +jDf/jds2B219MPB4b4aZeUEquEbH+bZWVtfWNzZLW+Xtnd29/c +rBYVsnmaKsRORqG5ANBNcshZyFKybKkbiQLBOMLqb+p0npjRP5 +COU+bHZCB5xClBI3kechGyHCe3l06/UnVqzgz2MnELUoUCzX7l +ywsTmsVMIhVE657rpOjnRCGngk3KXqZSuiIDFjPUElipv18dvP +EPjNKaEeJMiXRnqm/J3ISaz2OA9MZExzqRW8q/uf1Moxu/JzLNE +Mm6XxRlAkbE3sagB1yxSiKsSGEKm5utemQKELRxFQ2IbiLy+T9 +kXNvarVH+rVxnERwlO4BTOwYVraMA9NKEFJ4hld4szLrxXq3 +PuatK1YxcwR/YH3+AKRLkU0=˜t = 50 +AB9HicbVDLSgNBEJz1GeMr6 +kXwMhgET2E3BPUiBLx4jGAekCxhdrY3GTL7cKY3EJZ8hxcPinj1 +Y7z5N06SPWhiQUNR1U13l5dIodG2v6219Y3Nre3CTnF3b/gsH +R03NJxqjg0eSxj1fGYBikiaKJACZ1EAQs9CW1vdDfz2NQWsTRI +04ScEM2iEQgOEMjuT0U0ocMp7dV2+6XynbFnoOuEicnZKj0S9 +9fyYpyFEyCXTuvYCboZUyi4hGmxl2pIGB+xAXQNjVgI2s3mR0/ +phVF8GsTKVIR0rv6eyFio9ST0TGfIcKiXvZn4n9dNMbhxMxElKU +LEF4uCVFKM6SwB6gsFHOXEMaVMLdSPmSKcTQ5FU0IzvLq6RVr +ThXldpDrVw/zeMokDNyTi6JQ65JndyTBmkSTp7IM3klb9bYerHe +rY9F65qVz5yQP7A+fwAR/pGE˜t = 200 +AB9HicbVDLSgNBEOyNrxhfU +S+Cl8EgeAq7kqgXIeDFYwTzgGQJs7OTZMjsw5neQFjyHV48KOLV +j/Hm3zhJ9qCJBQ1FVTfdXV4shUb/rZya+sbm1v57cLO7t7+Qf +HwqKmjRDHeYJGMVNujmksR8gYKlLwdK04DT/KWN7qb+a0xV1pE4 +SNOYu4GdBCKvmAUjeR2Uifpzi9rVbtXrFkl+05yCpxMlKCDPVe +8avrRywJeIhMUq07jh2jm1KFgk+LXQTzWPKRnTAO4aGNODaTed +HT8m5UXzSj5SpEMlc/T2R0kDrSeCZzoDiUC97M/E/r5Ng/8ZNR +gnyEO2WNRPJMGIzBIgvlCcoZwYQpkS5lbChlRhiangnBWX5l +TQvy85VufJQKdVOsjycApncAEOXEMN7qEODWDwBM/wCm/W2Hqx +3q2PRWvOymaO4Q+szx8eKZGM˜t = 550 +AB9HicbVDLSgNBEOyNrxhfU +S+Cl8EgeAq7kqgXIeDFYwTzgGQJs7OTZMjsw5neQFjyHV48KOLV +j/Hm3zhJ9qCJBQ1FVTfdXV4shUb/rZya+sbm1v57cLO7t7+Qf +HwqKmjRDHeYJGMVNujmksR8gYKlLwdK04DT/KWN7qb+a0xV1pE4 +SNOYu4GdBCKvmAUjeR2Uifpzi9rVbtXrFkl+05yCpxMlKCDPVe +8avrRywJeIhMUq07jh2jm1KFgk+LXQTzWPKRnTAO4aGNODaTed +HT8m5UXzSj5SpEMlc/T2R0kDrSeCZzoDiUC97M/E/r5Ng/8ZNR +gnyEO2WNRPJMGIzBIgvlCcoZwYQpkS5lbChlRhiangnBWX5l +TQvy85VufJQKdVOsjycApncAEOXEMN7qEODWDwBM/wCm/W2Hqx +3q2PRWvOymaO4Q+szx8eKZGM˜t = 550 +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBNSwMxEJ3Ur1q/q +h69BIvgqexKUY8FLx4r2A9p15JNs21okl2SrFCW/govHhTx6s/x +5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx +61TJxqypo0FrHuhMQwRVrWm4F6ySaERkK1g7HNzO/cS04bG6t +5OEBZIMFY84JdZJD/Qx62mJxbRfrnhVbw68SvycVCBHo1/+6g1i +mkqmLBXEmK7vJTbIiLacCjYt9VLDEkLHZMi6jioimQmy+cFTfOa +UAY5i7UpZPFd/T2REGjORoeuUxI7MsjcT/O6qY2ug4yrJLVM0c +WiKBXYxnj2PR5wzagVE0cI1dzdiumIaEKty6jkQvCX14lrYuqf +1mt3dUqdZzHUYQTOIVz8OEK6nALDWgCBQnP8ApvSKMX9I4+Fq0F +lM8cwx+gzx+s0pA7 +cl +AB8HicbVBNSwMxEJ3Ur1q/q +h69BIvgqexKUY8FLx4r2A9p15JNs21okl2SrFCW/govHhTx6s/x +5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx +61TJxqypo0FrHuhMQwRVrWm4F6ySaERkK1g7HNzO/cS04bG6t +5OEBZIMFY84JdZJD/Qx62mJxbRfrnhVbw68SvycVCBHo1/+6g1i +mkqmLBXEmK7vJTbIiLacCjYt9VLDEkLHZMi6jioimQmy+cFTfOa +UAY5i7UpZPFd/T2REGjORoeuUxI7MsjcT/O6qY2ug4yrJLVM0c +WiKBXYxnj2PR5wzagVE0cI1dzdiumIaEKty6jkQvCX14lrYuqf +1mt3dUqdZzHUYQTOIVz8OEK6nALDWgCBQnP8ApvSKMX9I4+Fq0F +lM8cwx+gzx+s0pA7 +cl +AB8HicbVBNSwMxEJ3Ur1q/q +h69BIvgqexKUY8FLx4r2A9p15JNs21okl2SrFCW/govHhTx6s/x +5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx +61TJxqypo0FrHuhMQwRVrWm4F6ySaERkK1g7HNzO/cS04bG6t +5OEBZIMFY84JdZJD/Qx62mJxbRfrnhVbw68SvycVCBHo1/+6g1i +mkqmLBXEmK7vJTbIiLacCjYt9VLDEkLHZMi6jioimQmy+cFTfOa +UAY5i7UpZPFd/T2REGjORoeuUxI7MsjcT/O6qY2ug4yrJLVM0c +WiKBXYxnj2PR5wzagVE0cI1dzdiumIaEKty6jkQvCX14lrYuqf +1mt3dUqdZzHUYQTOIVz8OEK6nALDWgCBQnP8ApvSKMX9I4+Fq0F +lM8cwx+gzx+s0pA7 +cl +Figure 11: Comparisons of the continuum model predictions with corresponding DEM simulation results for the +transient evolution of the segregation dynamics for three cases of annular shear flow of disks: (a) Base case {𝑅/ ¯𝑑0 = +60, ˜𝑣w = 0.01, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}; (b) Lower inner-wall velocity case {𝑅/ ¯𝑑0 = 60, ˜𝑣w = 0.001, 𝑐l +0 = 0.5, 𝑑l/𝑑s = +1.5}; and (c) Smaller annular shear cell case {𝑅/ ¯𝑑0 = 40, ˜𝑣w = 0.01, 𝑐l +0 = 0.5, 𝑑l/𝑑s = 1.5}. Additional cases are +shown in Fig. 12. For each case, the first column shows spatiotemporal contours of the evolution of 𝑐l measured +in the DEM simulations. The second column shows comparisons of the DEM simulations (solid black lines) and +continuum model predictions (dashed gray lines) of the 𝑐l field at four time snapshots representing different stages of +the segregation process: ˜𝑡 = 5, 50, 200, and 550 in the sequence of top left, top right, bottom left, bottom right. The +third column shows comparisons of the quasi-steady, normalized velocity profiles at ˜𝑡 = 550 from DEM simulations +and continuum model predictions. +𝑔. For example, instead of (2.15), consider the following form for the segregation flux: +𝑤seg +𝑖 += 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕𝑔 +𝜕𝑥𝑖 +. +(6.1) +Then, applying the quasi-steady flux balance condition, +𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑥𝑖 +≈ 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕𝑔 +𝜕𝑥𝑖 +, +(6.2) +20 + +0 +10 +20 +30 +10-2 +100 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +10 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +0 +10 +20 +30 +10-2 +100 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +0 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +10 +50 +0 +0.5 +1 +0 +50 +0 +0.5 +1 +10 +20 +30 +40 +50 +0 +100 +200 +300 +400 +500 +0 +1 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AgfONMw=(a) +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlh3Jw +0SuW3Io7B1klX +kZKkKHeK351+z +FLI5SGCap1x3M +T40+oMpwJnBa +6qcaEshEdYMdS +SPU/mR+6pScW +6VPwljZkobM1 +d8TExpPY4C2x +lRM9TL3kz8z+u +kJrzxJ1wmqUHJ +FovCVBATk9nf +pM8VMiPGlCmu +L2VsCFVlBmbTs +G4C2/vEqa1Y +p3Vbm8r5ZqZ1k +ceTiBUyiDB9dQ +gzuoQwMYDOAZX +uHNEc6L8+58L +FpzTjZzDH/gfP +4Ag3iNA=(b) +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8H +icbVBNSwMxEJ +3Ur1q/qh69BIv +gqexKUY8FLx4r +2A9p15JNs21ok +l2SrFCW/govH +hTx6s/x5r8xbf +egrQ8GHu/NMDM +vTAQ31vO+UWF +tfWNzq7hd2tnd +2z8oHx61TJxqy +po0FrHuhMQwR +VrWm4F6ySaER +kK1g7HNzO/cS +04bG6t5OEBZIM +FY84JdZJD/Qx +62mJxbRfrnhVb +w68SvycVCBHo1 +/+6g1imkqmLBX +EmK7vJTbIiLa +cCjYt9VLDEkLH +ZMi6jioimQmy+ +cFTfOaUAY5i7 +UpZPFd/T2REGj +ORoeuUxI7Msjc +T/O6qY2ug4yr +JLVM0cWiKBXY +xnj2PR5wzagVE +0cI1dzdiumIaE +Kty6jkQvCX1 +4lrYuqf1mt3dU +qdZzHUYQTOIVz +8OEK6nALDWgCB +QnP8ApvSKMX9 +I4+Fq0FlM8cwx ++gzx+s0pA7 +cl +AB8nicbVDLSsNAFJ34rPVd +SO4GSyCq5JIfWyEghuXFewD0lAmk0k7dDITZm6EvoZblwo4tav +cefOG2z0NYDFw7n3Mu94Sp4AZc9tZWV1b39gsbZW3d3b39i +sHh2jMk1ZiyqhdDckhgkuWQs4CNZNSNJKFgnHN1N/c4T04Yr+ +QjlAUJGUgec0rASn4PuIhYDpPby36l6tbcGfAy8QpSRQWa/cpX +L1I0S5gEKogxvuemEOREA6eCTcq9zLCU0BEZMN9SRJmgnx28gS +fWSXCsdK2JOCZ+nsiJ4kx4yS0nQmBoVn0puJ/np9BfBPkXKYZME +ni+JMYFB4+j+OuGYUxNgSQjW3t2I6JpQsCmVbQje4svLpH1R8 +65q9Yd6tXFcxFCJ+gUnSMPXaMGukdN1EIUKfSMXtGbA86L8+58 +zFtXnGLmCP2B8/kDMkeREw=˜t = 5 +AB83icbVBNS8NAEJ34WetX1 +YvgJVgETyWR+nERCl48VrAf0ISy2WzapZtN2J0IJfRvePGgiFf/ +jDf/jds2B219MPB4b4aZeUEquEbH+bZWVtfWNzZLW+Xtnd29/c +rBYVsnmaKsRORqG5ANBNcshZyFKybKkbiQLBOMLqb+p0npjRP5 +COU+bHZCB5xClBI3kechGyHCe3l06/UnVqzgz2MnELUoUCzX7l +ywsTmsVMIhVE657rpOjnRCGngk3KXqZSuiIDFjPUElipv18dvP +EPjNKaEeJMiXRnqm/J3ISaz2OA9MZExzqRW8q/uf1Moxu/JzLNE +Mm6XxRlAkbE3sagB1yxSiKsSGEKm5utemQKELRxFQ2IbiLy+T9 +kXNvarVH+rVxnERwlO4BTOwYVraMA9NKEFJ4hld4szLrxXq3 +PuatK1YxcwR/YH3+AKRLkU0=˜t = 50 +AB9HicbVDLSgNBEJz1GeMr6 +kXwMhgET2E3BPUiBLx4jGAekCxhdrY3GTL7cKY3EJZ8hxcPinj1 +Y7z5N06SPWhiQUNR1U13l5dIodG2v6219Y3Nre3CTnF3b/gsH +R03NJxqjg0eSxj1fGYBikiaKJACZ1EAQs9CW1vdDfz2NQWsTRI +04ScEM2iEQgOEMjuT0U0ocMp7dV2+6XynbFnoOuEicnZKj0S9 +9fyYpyFEyCXTuvYCboZUyi4hGmxl2pIGB+xAXQNjVgI2s3mR0/ +phVF8GsTKVIR0rv6eyFio9ST0TGfIcKiXvZn4n9dNMbhxMxElKU +LEF4uCVFKM6SwB6gsFHOXEMaVMLdSPmSKcTQ5FU0IzvLq6RVr +ThXldpDrVw/zeMokDNyTi6JQ65JndyTBmkSTp7IM3klb9bYerHe +rY9F65qVz5yQP7A+fwAR/pGE˜t = 200 +AB9HicbVDLSgNBEOyNrxhfU +S+Cl8EgeAq7kqgXIeDFYwTzgGQJs7OTZMjsw5neQFjyHV48KOLV +j/Hm3zhJ9qCJBQ1FVTfdXV4shUb/rZya+sbm1v57cLO7t7+Qf +HwqKmjRDHeYJGMVNujmksR8gYKlLwdK04DT/KWN7qb+a0xV1pE4 +SNOYu4GdBCKvmAUjeR2Uifpzi9rVbtXrFkl+05yCpxMlKCDPVe +8avrRywJeIhMUq07jh2jm1KFgk+LXQTzWPKRnTAO4aGNODaTed +HT8m5UXzSj5SpEMlc/T2R0kDrSeCZzoDiUC97M/E/r5Ng/8ZNR +gnyEO2WNRPJMGIzBIgvlCcoZwYQpkS5lbChlRhiangnBWX5l +TQvy85VufJQKdVOsjycApncAEOXEMN7qEODWDwBM/wCm/W2Hqx +3q2PRWvOymaO4Q+szx8eKZGM˜t = 550 +AB9HicbVDLSgNBEOyNrxhfU +S+Cl8EgeAq7kqgXIeDFYwTzgGQJs7OTZMjsw5neQFjyHV48KOLV +j/Hm3zhJ9qCJBQ1FVTfdXV4shUb/rZya+sbm1v57cLO7t7+Qf +HwqKmjRDHeYJGMVNujmksR8gYKlLwdK04DT/KWN7qb+a0xV1pE4 +SNOYu4GdBCKvmAUjeR2Uifpzi9rVbtXrFkl+05yCpxMlKCDPVe +8avrRywJeIhMUq07jh2jm1KFgk+LXQTzWPKRnTAO4aGNODaTed +HT8m5UXzSj5SpEMlc/T2R0kDrSeCZzoDiUC97M/E/r5Ng/8ZNR +gnyEO2WNRPJMGIzBIgvlCcoZwYQpkS5lbChlRhiangnBWX5l +TQvy85VufJQKdVOsjycApncAEOXEMN7qEODWDwBM/wCm/W2Hqx +3q2PRWvOymaO4Q+szx8eKZGM˜t = 550 +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBN +SwMxEJ3Ur1q/qh69BIvgqexKU +Y8FLx4r2A9p15JNs21okl2SrFC +W/govHhTx6s/x5r8xbfegrQ8G +Hu/NMDMvTAQ31vO+UWFtfWNzq7 +hd2tnd2z8oHx61TJxqypo0FrH +uhMQwRVrWm4F6ySaERkK1g7HN +zO/cS04bG6t5OEBZIMFY84Jd +ZJD/Qx62mJxbRfrnhVbw68Svy +cVCBHo1/+6g1imkqmLBXEmK7vJ +TbIiLacCjYt9VLDEkLHZMi6ji +oimQmy+cFTfOaUAY5i7UpZPFd/ +T2REGjORoeuUxI7MsjcT/O6q +Y2ug4yrJLVM0cWiKBXYxnj2PR5 +wzagVE0cI1dzdiumIaEKty6jk +QvCX14lrYuqf1mt3dUqdZzHU +YQTOIVz8OEK6nALDWgCBQnP8Ap +vSKMX9I4+Fq0FlM8cwx+gzx+s +0pA7 +cl +AB8HicbVBNSwMxEJ3Ur1q/q +h69BIvgqexKUY8FLx4r2A9p15JNs21okl2SrFCW/govHhTx6s/x +5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx +61TJxqypo0FrHuhMQwRVrWm4F6ySaERkK1g7HNzO/cS04bG6t +5OEBZIMFY84JdZJD/Qx62mJxbRfrnhVbw68SvycVCBHo1/+6g1i +mkqmLBXEmK7vJTbIiLacCjYt9VLDEkLHZMi6jioimQmy+cFTfOa +UAY5i7UpZPFd/T2REGjORoeuUxI7MsjcT/O6qY2ug4yrJLVM0c +WiKBXYxnj2PR5wzagVE0cI1dzdiumIaEKty6jkQvCX14lrYuqf +1mt3dUqdZzHUYQTOIVz8OEK6nALDWgCBQnP8ApvSKMX9I4+Fq0F +lM8cwx+gzx+s0pA7 +cl +Figure 12: Comparisons of the continuum model predictions with corresponding DEM simulation results for the +transient evolution of the segregation dynamics for two cases of annular shear flow of disks: (a) More large grains +case {𝑅/ ¯𝑑0 = 60, ˜𝑣w = 0.01, 𝑐l +0 = 0.75, 𝑑l/𝑑s = 1.5} and (b) Larger size ratio case {𝑅/ ¯𝑑0 = 60, ˜𝑣w = 0.01, 𝑐l +0 = +0.5, 𝑑l/𝑑s = 3.0}. Additional cases are shown in Fig. 11. Results are organized as described in the caption of Fig. 11. +to the quasi-steady DEM data for vertical chute flow and annular shear flow of disks, we obtain the collapses shown +in Figs. 13(a) and (b), respectively.3 The solid lines represent the best linear fit using 𝐶S +seg = 0.08. The collapses +are reasonable but not as strong as those shown in Figs. 5(a) and 7 for a segregation flux based on gradients in the +shear-strain-rate, leading us to choose to work with the constitutive equation (2.15) on pragmatic grounds. +Additionally, we have tested a possible constitutive equation for the segregation flux driven by gradients in the +granular temperature, defined as 𝑇 = (𝛿𝑣)2, where 𝛿𝑣 is the velocity fluctuation.4 Then, consider the following form +for the segregation flux: +𝑤seg +𝑖 += 𝐶S +seg +√︁ +𝜌s/𝑃 ¯𝑑𝑐l(1 − 𝑐l) 𝜕𝑇 +𝜕𝑥𝑖 +, +(6.3) +where the inertial time +√︁ +𝜌s/𝑃 ¯𝑑 is included in the prefactor for dimensional reasons. As above, upon applying the +quasi-steady flux balance condition, +𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑥𝑖 +≈ 𝐶S +seg +√︁ +𝜌s/𝑃 ¯𝑑𝑐l(1 − 𝑐l) 𝜕𝑇 +𝜕𝑥𝑖 +, +(6.4) +to the quasi-steady DEM data, Figs. 13(c) and (d) show the collapses for granular-temperature-gradient-driven segre- +gation for vertical chute flow and annular shear flow of disks, respectively. In this case, the solid lines represent the +best linear fit using 𝐶S +seg = 0.7. The collapse is quite good; however, in order to utilize the constitutive equation (6.3) +in practice, an additional constitutive equation that gives the granular temperature field in terms of other continuum +3The coarse-grained values of 𝑔 and its gradient are obtained using the coarse-grained values of �𝛾 and its gradient, calculated as described in +Appendix A.2, along with 𝜇 and its gradient, calculated using (4.1) and not by coarse-graining. +4The definitions of the granular temperature 𝑇 and the velocity fluctuation 𝛿𝑣 as well as the coarse-graining method used to obtain these +quantities follow Zhang and Kamrin (2017). They are treated as field quantities just as �𝛾. Following the coarse-graining process described in (A.2), +the instantaneous 𝑇 field at a bin is the grain-area-weighted summation of the square of the difference of the grain instantaneous velocity vector and +the coarse-grained instantaneous velocity field at that grain center, for all grains that are intersected by the bin. +21 + +-1 +-0.5 +0 +0.5 +1 +10-3 +-5 +0 +5 +10-4 +-5 +0 +5 +10-3 +-5 +0 +5 +10-4 +-3 +-2 +-1 +0 +1 +2 +3 +10-3 +-5 +0 +5 +10-4 +-0.02 +-0.015 +-0.01 +-0.005 +0 +-2 +-1.5 +-1 +-0.5 +0 +10-3 +-2.5 +-2 +-1.5 +-1 +-0.5 +0 +10-3 +-2 +-1.5 +-1 +-0.5 +0 +10-3 +-8 +-6 +-4 +-2 +0 +10-3 +-2 +-1.5 +-1 +-0.5 +0 +10-3 +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK9 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AgfONMw=(a) +AB6nicbVDLSgNBEOyNrxhfU +Y9eBqMQL2E3iHoMePEY0TwgWcLspDcZMju7zMwKIeQTvHhQxKtf +5M2/cZLsQRMLGoqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxc +Ojpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+m +nGCfkQHkoecUWOlh3Jw0SuW3Io7B1klXkZKkKHeK351+zFLI5SG +Cap1x3MT40+oMpwJnBa6qcaEshEdYMdSPU/mR+6pScW6VPwlj +ZkobM1d8TExpPY4C2xlRM9TL3kz8z+ukJrzxJ1wmqUHJFovCVB +ATk9nfpM8VMiPGlCmuL2VsCFVlBmbTsG4C2/vEqa1Yp3Vbm8r +5ZqZ1kceTiBUyiDB9dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZz +DH/gfP4Ag3iNA=(b) +AB6n +icbVDLSgNBEO +yNrxhfUY9eBqM +QL2E3iHoMePEY +0TwgWcLspDcZM +ju7zMwKIeQTv +HhQxKtf5M2/cZ +LsQRMLGoqbrq +7gkRwbVz328m +trW9sbuW3Czu7 +e/sHxcOjpo5Tx +bDBYhGrdkA1Ci +6xYbgR2E4U0i +gQ2ApGtzO/9YR +K81g+mnGCfkQH +koecUWOlhzK7 +6BVLbsWdg6wSL +yMlyFDvFb+6/Z +ilEUrDBNW647m +J8SdUGc4ETgv +dVGNC2YgOsGOp +pBFqfzI/dUrOr +dInYaxsSUPm6 +u+JCY20HkeB7Y +yoGeplbyb+53V +SE974Ey6T1KBk +i0VhKoiJyexv +0ucKmRFjSyhT3 +N5K2JAqyoxNp2 +BD8JZfXiXNas +W7qlzeV0u1sy +OPJzAKZTBg2uo +wR3UoQEMBvAMr +/DmCOfFeXc+F +q05J5s5hj9wPn +8AhP2NQ=(c) +AB6nicbVBNSwMxEJ2tX7V+V +T16CVahXspuEfVY8OKxov2AdinZbLYNzSZLkhXK0p/gxYMiXv1F +3vw3pu0etPXBwO9GWbmBQln2rjut1NYW9/Y3Cpul3Z29/YPyo +dHbS1TRWiLSC5VN8CaciZoyzDaTdRFMcBp51gfDvzO09UaSbFo +5k1I/xULCIEWys9FANLwbliltz50CrxMtJBXI0B+WvfihJGlNh +CMda9zw3MX6GlWGE02mpn2qaYDLGQ9qzVOCYaj+bnzpF51YJUS +VLWHQXP09keFY60kc2M4Ym5Fe9mbif14vNdGNnzGRpIYKslgUpR +wZiWZ/o5ApSgyfWIKJYvZWREZYWJsOiUbgrf8ip12veVe3yv +l5pnOVxFOETqEKHlxDA+6gCS0gMIRneIU3hzsvzrvzsWgtOPnM +MfyB8/kDhoKNg=(d) +ACFHicbVBN +SwMxEM36WevXqkcvi1UQhLJbR +D0WvHisYD+gu5RsdrYNzSZLklV +K6Y/w4l/x4kERrx68+W/MtnvQ +1oGQlzdvJjMvTBlV2nW/raXld +W19dJGeXNre2fX3tvKZFJAk0 +imJCdECtglENTU82gk0rAScigH +Q6v83z7HqSigt/pUQpBgvucxp +RgbaiefeYT4Bok5X2fC8oj83J +aILVRMN93yCDTYO6YiYeXGr7 +jScReAVoIKaPTsLz8SJEtMT8 +KwUl3PTXUwxnl3BpOynylIMRni +PnQN5DgBFYynS02cE8NETiykO +WamKfu7YowTpUZJaJQJ1gM1n8v +J/3LdTMdXwZjy1GzGyeyjOGO +Fk7ukBNRCUSzkQGYSGpmNS5gi +YmxSZWNCd78yougVat6F9Xz21q +lflzYUKH6AidIg9dojq6Q3U +RAQ9omf0it6sJ+vFerc+ZtIlq6 +g5QH/C+vwBbCye8w=Vertical +chute +flow +ACE3icbVC7TsMwFHXKq5RXg +ZHFoiAhiqpEDAWsTAWiT6kpqoc56a16tiR7YCqP/Awq+wMIAQ +Kwsbf4P7GKDlSJaPz343hMknGnjut9Obml5ZXUtv17Y2Nza3i +nu7jW0TBWFOpVcqlZANHAmoG6Y4dBKFJA4NAMBtfjePMelGZS3 +JlhAp2Y9ASLGCXGSt3iqU9BGFBM9HwhmQjtC18JkXKifB/rPkzu +iMuHbrHklt0J8CLxZqSEZqh1i19+KGka25aUE63bnpuYTkaUYZT +DqOCnGhJCB6QHbUsFiUF3slOI3xslRBHUtljR5qovysyEms9jA +ObGRPT1/OxsfhfrJ2a6LKTMZGkBgSdfhSlHBuJxwbhkCmghg8tI +VQxOyumfaItS7pgjXBm195kTQqZe+8fHZbKVWPZnbk0QE6RCfI +Qxeoim5QDdURY/oGb2iN+fJeXHenY9pas6Z1eyjP3A+fwCNKp +56Annular +shear +flow +Figure 13: (a) Collapse of 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) versus ¯𝑑2𝑐l(1 − 𝑐l)(𝜕𝑔/𝜕𝑥) for several cases of vertical chute flow and +(b) collapse 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑟) versus ¯𝑑2𝑐l(1 − 𝑐l)(𝜕𝑔/𝜕𝑟) for several cases of annular shear flow of bidisperse disks. +(c) Collapse of 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) versus +√︁ +𝜌s/𝑃 ¯𝑑𝑐l(1 − 𝑐l)(𝜕𝑇/𝜕𝑥) for several cases of vertical chute flow and (d) +collapse of 𝐶diff ¯𝑑2 �𝛾(𝜕𝑐l/𝜕𝑥) versus +√︁ +𝜌s/𝑃 ¯𝑑𝑐l(1 − 𝑐l)(𝜕𝑇/𝜕𝑥) for several cases of annular shear flow of bidisperse +disks. Symbols represent coarse-grained, quasi-steady DEM field data, and the solid lines represent the best linear fit +using 𝐶S +seg = 0.08 for (a) and (b) and 𝐶S +seg = 0.7 for (c) and (d). +quantities–such as strain-rate, stress, and fluidity–is needed. Recent work by Kim and Kamrin (2020) offers a path +forward on this point. In summary, we acknowledge that the possibility for an alternative form for the segregation flux +𝑤seg +𝑖 +remains and that future work involving additional flow geometries will be required to conclusively judge which +constitutive equation is the most predictive. +Another question regarding the constitutive equation for the segregation flux is the 𝑐l dependence of the pre-factor. +We simply use a symmetric dependence 𝑐l(1 − 𝑐l), while other researchers (e.g., van der Vaart et al., 2015; Tunuguntla +et al., 2017) invoke expressions that depend on both 𝑐l and 𝑑l/𝑑s in a more complex manner. +Finally, in this paper, we have focused on two simple flow geometries that have two important features: (1) the +continuum fields are one-dimensional, only varying along one spatial direction, and (2) the pressure field is spatially +uniform. In order to apply the proposed continuum model in more complex flow geometries, such as heap flows +or split-bottom flow, two important steps are necessary. First, this paper solely considered shear-strain-rate-driven +size-segregation. Now that a predictive continuum model for this mechanism has been established, it remains to return +to pressure-gradient-driven size-segregation in order to incorporate this mechanism by introducing an additional flux +contribution to (2.12) to obtain a more general model. Second, a robust numerical implementation of the complex +system of coupled equations that is capable of addressing problems in general geometries, involving multi-dimensional +22 + +continuum fields, is needed. One possible approach is to utilize the finite-element method, as in previous work involving +the NGF model (Henann and Kamrin, 2016). These steps will be addressed in future works. +Acknowledgements +This work was supported by funds from NSF-CBET-1552556. +A +Discrete-element method simulations and coarse-graining procedures +A.1 +Simulated granular systems +We consider two types of simulated granular systems: two-dimensional systems consisting of a dense collection of +circular disks and three-dimensional systems consisting of a dense collection of spheres. We consider bidisperse +systems and denote the mean diameter of the large particles as 𝑑l and the mean diameter of the small particles as 𝑑s +for both types. For both disks and spheres, we take 𝑑l = 3 mm and 𝑑s = 2 mm for the base case, so that 𝑑l/𝑑s = 1.5. +For both large and small particles, the diameters of individual particles are chosen from a uniform distribution over the +range of ±10% of the respective mean diameters to prevent crystallization. In the two-dimensional granular system, 𝜌s +denotes the grain-material area-density, which we take to be 𝜌𝑠 = 3.26 kg/m2 for both large and small disks, and in the +three-dimensional granular system, 𝜌s denotes the grain-material volume-density, which is taken to be 𝜌s = 2450 kg/m3 +for both large and small spheres to eliminate density-based segregation. For two-dimensional systems, the mass of the +large disks is given by 𝑚l = (𝜋/4)𝜌s(𝑑l)2, and the mass of the small disks is given as 𝑚s = (𝜋/4)𝜌s(𝑑s)2, so that the +characteristic grain mass is 𝑚 = 𝑐l +0𝑚l + (1 − 𝑐l +0)𝑚s, where 𝑐l +0 the initial concentration of the large grains. Similarly, +for three-dimensional systems, the mass of the large spheres is 𝑚l = (𝜋/6)𝜌s(𝑑l)3, and the mass of the small spheres +is 𝑚s = (𝜋/6)𝜌s(𝑑s)3, so that the characteristic grain mass is 𝑚 = 𝑐l +0𝑚l + (1 − 𝑐l +0)𝑚s. +For the grain interaction model, the interaction force is given through a spring/dashpot contact law that accounts +for elasticity, damping, and sliding friction (da Cruz et al., 2005; Koval et al., 2009; Kamrin and Koval, 2014; Zhang +and Kamrin, 2017). The normal contact force 𝐹n is given linearly through the normal component of the contact +overlap, denoted by 𝛿n, with stiffness 𝑘n and the relative normal velocity, denoted by �𝛿n, with damping coefficient +𝑔n as 𝐹n = 𝑘n𝛿n + 𝑔n �𝛿n whenever 𝛿n ≥ 0 and 𝐹n = 0 whenever 𝛿n < 0. The normal damping coefficient is given +by 𝑔n = √𝑚𝑘n(−2 ln 𝑒)/ +√︃ +2(𝜋2 + ln2 𝑒) where 𝑒 is the coefficient of restitution for binary collisions and 𝑚 is the +characteristic grain mass discussed above. We denote the tangential stiffness and damping coefficient as 𝑘t and 𝑔t, +respectively, and take 𝑔t = 0, so that the tangential force is given as 𝐹t = 𝑘t𝛿t whenever 𝛿n ≥ 0, where 𝛿𝑡 is the tangential +component of the contact displacement. The magnitude of the tangential component of the contact force is limited by +Coulomb friction, which depends on the inter-particle friction coefficient 𝜇surf. Thus, the parameter set {𝑘n, 𝑘t, 𝑒, 𝜇surf} +fully describes the interaction properties. Throughout, the normal stiffness 𝑘n is taken to be sufficiently large so that +grains behave as stiff and nearly rigid. For two-dimensional systems, 𝑘n/𝑃 > 104, and for three-dimensional systems, +𝑘n/𝑃 ¯𝑑0 > 104, where 𝑃 is the characteristic confining pressure for a given configuration (force per unit length in +two dimensions and force per unit area in three dimensions) and ¯𝑑0 = 𝑐l +0𝑑l + (1 − 𝑐l +0)𝑑s is the characteristic grain +size. In the stiff grain regime, the only interaction parameter that significantly affects the rheology of dense grains is +𝜇surf, which we have kept constant as 𝜇surf = 0.4 throughout. The other parameters, namely the ratio 𝑘n/𝑘t and the +restitution coefficient 𝑒, have negligible effects on the rheology (Kamrin and Koval, 2014) and thus the segregation +dynamics in the stiff particle regime, so we maintain 𝑘t/𝑘n = 1/2 and 𝑒 = 0.1 throughout. Finally, the open-source +software LAMMPS (Plimpton, 1995) is used to numerically integrate the equations of motion for each particle, and we +restrict the time step for numerical integration to be 0.01-0.1 of the binary collision time 𝜏c = +√︂ +𝑚 +� +𝜋2 + ln2 𝑒 +� +/4𝑘n +for stability and accuracy of the simulation results. +A.2 +Averaging methods +In this appendix, we describe the spatial and temporal averaging methods utilized to extract continuum fields from +our DEM data. We begin with the spatial averaging procedure for a given snapshot of DEM data at a time 𝑡. All +flows considered in this work are one-dimensional, in which the continuum fields vary only along one direction–i.e., +23 + +along the 𝑧-direction in simple shear flow (Fig. 2), along the 𝑥-direction in vertical chute flow (Fig. 4), and along the +𝑟-direction in annular shear flow (Fig. 6)–and periodic along all other directions. Here, we describe the procedure for +vertical chute flow of disks in detail, which may be straightforwardly adapted to the other flow geometries. We utilize +a bin-based coarse-graining process, in which we construct a slender rectangle that spans the simulation domain along +the 𝑧-direction and is centered at a given 𝑥-position with a finite width along the 𝑥-direction.5 Then, we assign each +intersected grain 𝑖 a weight 𝐴𝑖, defined as the area of the grain 𝑖 inside the bin. Following Tunuguntla et al. (2017) for a +bidisperse system, we denote the sets of large and small grains intersected by the bin as F l and F s, respectively, so that +the set of all grains intersected by the bin is F = F l ∪ F s with F l ∩ F s = ∅. The instantaneous solid area fraction field +for species 𝛼 is 𝜙𝛼(𝑥, 𝑡) = (� +𝑖∈F𝛼 𝐴𝑖)/𝐴, where 𝐴 is the total area of the bin, and the corresponding concentration +field for species 𝛼 is 𝑐𝛼(𝑥, 𝑡) = 𝜙𝛼(𝑥, 𝑡)/𝜙(𝑥, 𝑡) with 𝜙(𝑥, 𝑡) = 𝜙l(𝑥, 𝑡) + 𝜙s(𝑥, 𝑡). With the instantaneous velocity of +each grain 𝑖 denoted as v𝑖(𝑡), the instantaneous velocity field is v(𝑥, 𝑡) = (� +𝑖∈F 𝐴𝑖v𝑖(𝑡))/(� +𝑖∈F 𝐴𝑖).6 Likewise, with +the instantaneous stress tensor associated with grain 𝑖 defined as 𝝈𝑖(𝑡) = (� +𝑗≠𝑖 r𝑖 𝑗 ⊗ f𝑖 𝑗)/(𝜋𝑑2 +𝑖 /4), where r𝑖 𝑗 is the +position vector from the center of grain 𝑖 to the center of grain 𝑗, f𝑖 𝑗 is the contact force applied on grain 𝑖 by grain 𝑗, +and 𝑑𝑖 is the diameter of grain 𝑖, the instantaneous stress field is 𝝈(𝑥, 𝑡) = (� +𝑖∈F 𝐴𝑖𝝈𝑖(𝑡))/𝐴. +In vertical chute flow of spheres, we utilize a similar spatial coarse-graining approach. Instead of two-dimensional, +rectangular bins, we use three-dimensional, rectangular-cuboidal bins that span the simulation domain along the 𝑦- +and 𝑧-directions and are centered at a given 𝑥-position with a finite width along the 𝑥-direction. Then, the weight +for each grain 𝑖 is the volume 𝑉𝑖 of the grain 𝑖 inside the bin. The instantaneous solid volume fraction field for +species 𝛼 is 𝜙𝛼(𝑥, 𝑡) = (� +𝑖∈F𝛼 𝑉𝑖)/𝑉, where 𝑉 is the total volume of the bin; the instantaneous concentration field +for species 𝛼 is 𝑐𝛼(𝑥, 𝑡) = 𝜙𝛼(𝑥, 𝑡)/𝜙(𝑥, 𝑡); the instantaneous velocity field is v(𝑥, 𝑡) = (� +𝑖∈F 𝑉𝑖v𝑖(𝑡))/(� +𝑖∈F 𝑉𝑖); the +instantaneous stress tensor associated with grain 𝑖 is 𝝈𝑖(𝑡) = (� +𝑗≠𝑖 r𝑖 𝑗 ⊗ f𝑖 𝑗)/(𝜋𝑑3 +𝑖 /6); and the instantaneous stress +field is 𝝈(𝑥, 𝑡) = (� +𝑖∈F 𝑉𝑖𝝈𝑖(𝑡))/𝑉. +Our analysis of the size-segregation process in this paper depends on obtaining accurate and high-resolution coarse- +grained 𝑐l fields from the DEM data. Therefore, the choices of the bin width and the spatial resolution of the bins are +crucial. Throughout, we take a bin width of 4 ¯𝑑0 and a spatial resolution of roughly 0.1 ¯𝑑0 for both disks and spheres. +Note that for these choices, adjacent bins overlap. We have ensured that a bin width of 4 ¯𝑑0 is sufficiently small so +that the coarse-grained data is not over-smoothed. Specifically, we have tested that the coarse-grained velocity and +stress fields are insensitive to the choice of bin width over the range of 0 to 9.6 ¯𝑑0. In the limit that the bin width goes +to zero, the coarse-graining procedure reduces to that utilized successfully in the literature for the velocity and stress +fields for both disks (e.g., da Cruz et al., 2005; Koval et al., 2009) and spheres (e.g., Zhang and Kamrin, 2017; Kim +and Kamrin, 2020). However, as shown by Weinhart et al. (2013), applying a coarse-graining procedure with a small +or zero bin width to the 𝜙 field–and hence the 𝜙l, 𝜙s, 𝑐l, and 𝑐s fields–will lead to spatial fluctuations due to particle +layering near the walls. We have ensured that a bin width of 4 ¯𝑑0 is sufficiently large so that these layering effects are +not observed in the 𝑐l field–i.e., we are within the “plateau range” (Weinhart et al., 2013) of bin widths that produce +bin-width-independent, coarse-grained continuum fields. We note that when plotting spatiotemporal contours of the +concentration fields, profiles of the concentration fields, and profiles of the velocity fields (e.g., Fig. 4), we truncate +the coarse-grained DEM data from bins centered within one-half of a bin-width (2 ¯𝑑0) from the walls. Furthermore, to +ensure that the DEM data collapses used to determine the dimensionless material parameter 𝐶S +seg (i.e., Figs. 5, 7, and +13) are representative of bulk behavior and not wall effects, we use a more conservative criterion and do not include +DEM data from bins within 6 ¯𝑑0 of the walls. +The collapses of Figs. 5, 7, and 13 are obtained in the long-time regime, in which the fields evolve slowly in time. +Since the flow is quasi-steady, the instantaneous concentration and velocity fields are simply arithmetically averaged +in time (using 152 instantaneous snapshots for vertical chute flow of disks, 1000 snapshots for vertical chute flow of +spheres, and 144 snapshots for annular shear flow of disks) to obtain fields that only depend on the spatial coordinate. +Then, the necessary first and second-order spatial derivatives of the field quantities (e.g., 𝜕𝑐l/𝜕𝑥, �𝛾 = 𝜕𝑣𝑧/𝜕𝑥, and +𝜕 �𝛾/𝜕𝑥 = 𝜕2𝑣𝑧/𝜕𝑥2 for vertical chute flow) are obtained from these time-averaged fields. We apply a spatial derivative +filter to the time-averaged DEM fields in order to obtain accurate estimates of the spatial derivatives. We have tested +using both cutoff Gaussian functions and Lucy functions (Weinhart et al., 2013; Tunuguntla et al., 2016) for the kernel +function of the derivative filter as well as a range of kernel function widths to ensure that the reported results in this +study are independent of these choices. +Unlike for the steady concentration and velocity fields, which allow for arithmetic time-averaging, the transient +5For annular shear flow of disks, the bins are thin, annular rings that are centered at a given 𝑟-position with a finite thickness along the 𝑟-direction. +6We note that the definition of the instantaneous velocity field is consistent with first defining species-specific velocity fields–i.e., v𝛼 (𝑥, 𝑡) = +(� +𝑖∈F𝛼 𝐴𝑖v𝑖 (𝑡))/(� +𝑖∈F𝛼 𝐴𝑖)–and then calculating the mixture-level field–i.e., v(𝑥, 𝑡) = 𝑐l(𝑥, 𝑡)vl(𝑥, 𝑡) + (1 − 𝑐l(𝑥, 𝑡))vs(𝑥, 𝑡). +24 + +concentration fields for dense flows of disks (e.g., Fig. 4(d)) are time-averaged in a slightly different manner using a +cutoff Gaussian filter. In particular, this process is performed by applying a normalized, cutoff Gaussian time filter to +the DEM data at each 𝑥-position. Denoting the standard deviation of the Gaussian kernel function as 𝜎t, so that the +cutoff time-width of the Gaussian kernel is 6𝜎t, the time-smoothed field quantity at a given 𝑥-position and time 𝑡 is +then given by the convolution of the DEM data over a time-period of 6𝜎t, centered at time 𝑡, with the cutoff Gaussian +kernel. We have tested a range of kernel widths 𝜎t to ensure that the coarse-grained concentration fields appearing in +this paper are insensitive to this choice. For the transient concentration fields for dense flows of spheres (second column +of Fig. 10), no additional time-smoothing is needed, and the concentration fields are simply instantaneous snapshots in +time. This is possible primarily because spatial smoothing is being done over a greater volume and a larger number of +grains, and therefore the instantaneous concentration fields are relatively more smooth. +B +Diffusion flux consistency test +Our process for determining the segregation flux–and hence the material parameter 𝐶S +seg–is based on the assumption +that the segregation and diffusion fluxes balance in the quasi-steady regime. +Therefore, it is essential that the +dimensionless material parameter 𝐶diff appearing in the constitutive equation for the diffusion flux (2.14) has been +accurately determined, so that the coarse-grained diffusion flux is accurate. In Section 3, we determined 𝐶diff for dense +flows of frictional disks to be 0.20 using mean square displacement data from DEM simulations of simple shear flow +of a well-mixed bidisperse granular system. In this Appendix, we perform an independent consistency check that tests +whether the constitutive equation for the diffusion flux (2.14) using this fitted value of 𝐶diff for disks is capable of +predicting the evolution of the 𝑐l field in a diffusion-dominated problem. +Consider homogeneous simple shear flow of an initially-segregated system with large grains (dark gray) on the +bottom and small grains (light gray) on the top, as shown in Fig. 14(a) for the case of 𝑑l/𝑑s = 1.5. The rectangular +domain has a length of 𝐿 = 60 ¯𝑑0 in the 𝑥-direction and a height of 𝐻 = 120 ¯𝑑0 in the 𝑧-direction. As in Sections 2.3 +and 3, shearing along the 𝑥-direction and normal stress along the 𝑧-direction are applied by the walls. We perform +DEM simulations of simple shearing for a nominal inertial number of (𝑣w/𝐻) +√︃ +¯𝑑2 +0𝜌s/𝑃w = 0.1. We run the DEM +simulation starting from the initially-segregated configuration, and the spatiotemporal evolution of the coarse-grained +𝑐l-field is shown in Fig. 14(b) for 𝑑l/𝑑s = 1.5. We observe that the interface between large and small grains, which is +initially sharp, becomes diffuse with a transition width that grows with time. We define a transition width at a given +point in time as the distance between the positions at which 𝑐l equals 0.1 and 0.9 in snapshots of the spatial 𝑐l profile. +This transition width as a function of the square-root of the dimensionless time ˜𝑡 = 𝑡/(𝐻/𝑣w) is plotted in Fig. 14(c) +as solid curves for grain-size-ratios of 𝑑l/𝑑s = 1.5 and 3.0, displaying roughly linear behavior–typical of diffusive +behavior–with a slight dependence on 𝑑l/𝑑s. +Next, we apply the continuum model for the evolution of 𝑐l (2.16) to this problem. As for planar shear flow of a +well-mixed bidisperse system (Fig. 2), no pressure gradient is present. Therefore, the evolution of 𝑐l is governed by +(2.16): +𝑑𝑐l +𝑑𝑡 + 𝜕 +𝜕𝑧 +� +−𝐶diff ¯𝑑2 �𝛾 𝜕𝑐l +𝜕𝑧 + 𝐶S +seg ¯𝑑2𝑐l(1 − 𝑐l) 𝜕 �𝛾 +𝜕𝑧 +� += 0, +(B.1) +where ¯𝑑 = 𝑐l𝑑l + (1 − 𝑐l)𝑑s. In this flow configuration, the shear-strain-rate is approximately constant. Therefore, +the shear-strain-rate-gradient is approximately zero throughout, and the diffusion flux is the dominant flux, which +acts to remix the flowing grains. This may be understood in the context of the local inertial rheology. Since the +stress ratio 𝜇 is spatially-constant in homogeneous planar shear, the resulting inertial number field 𝐼 is also spatially +constant. Since the inertial number depends on both ¯𝑑 and �𝛾, spatial variation in ¯𝑑 leads to spatial variation in �𝛾. +This spatial variation in �𝛾 is slight, and consequently, the magnitude of the diffusion flux is at each point in space is +much greater than the magnitude of the segregation flux. While the effect of segregation is small, we still include the +shear-strain-rate-gradient-driven segregation flux with 𝐶S +seg = 0.23 and solve (B.1) when analyzing the problem shown +in Fig. 14(a). We note that due to the small effect of segregation and the dependence of ¯𝑑 on 𝑐l, (B.1) is similar to but +not exactly the same as the linear diffusion equation in one dimension, so the solution is close to but not exactly an +error function. +We obtain predictions for the evolution of the 𝑐l-field by solving (B.1) using the fully-segregated initial condition +for 𝑐l(𝑧, 𝑡 = 0), no-flux boundary conditions at 𝑧 = 0 and 𝑧 = 𝐻, a spatially-constant value of inertial number 𝐼 +consistent with that prescribed in the DEM simulations, a given grain-size-ratio 𝑑l/𝑑s, and 𝐶diff = 0.20. 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+vV5yqMwdaJW5OKpCj6dtf/TAma +USFJhwr1XOdRHsZlpoRTqelfq +pogskYD2nPUIEjqrxsfvgUnRkl +RINYmhIazdXfExmOlJpEgemMs +B6pZW8m/uf1Uj248jImklRTQRa +LBilHOkazFDIJCWaTwzBRDJz +KyIjLDHRJquSCcFdfnmVtC+qb +r1au69VGqd5HEU4hM4BxcuoQG +30IQWEjhGV7hzXqyXqx362PR +WrDymSP4A+vzB9cmkdc= +L = 60 ¯d0 +Figure 14: (a) Initially-segregated configuration for two-dimensional DEM simulation of bidisperse simple shear flow +with 𝑑l/𝑑s = 1.5 and 8649 flowing grains. Upper and lower layers of black grains denote rough walls. Dark gray +grains indicate large flowing grains, and light gray grains indicate small flowing grains. A 10% polydispersity is +utilized for each species to prevent crystallization. (b) Spatiotemporal evolution of the large grain concentration field, +illustrating the transition width that grows with time. (c) Normalized transition width versus square root of normalized +time ˜𝑡 = 𝑡/(𝐻/𝑣w). +since 𝐼 is taken to be spatially-constant, �𝛾 = (𝐼/ ¯𝑑) +√︁ +𝑃w/𝜌s varies slightly in space due to the 𝑐l-dependence of ¯𝑑. We +extract the transition width as a function of time from continuum simulation results for 𝑑l/𝑑s = 1.5 and 3.0 and include +these results in Fig. 14(c) as dashed lines. The continuum model predictions agree well with the DEM data and are +even capable of capturing the small difference due to the grain-size-ratio. This result indicates that the expression for +the diffusion flux (2.14) and the fitted material parameter value 𝐶diff = 0.20 are indeed consistent with DEM data for +disks. +References +Artoni, R., Larcher, M., Jenkins, J.T., Richard, P., 2021. Self-diffusion scalings in dense granular flows. Soft Matter +17, 2596–2602. +Bancroft, R.S., Johnson, C.G., 2021. Drag, diffusion and segregation in inertial granular flows. J. Fluid Mech. 924, +A3. +Barker, T., Gray, J., 2017. Partial regularisation of the incompressible 𝜇(𝐼)-rheology for granular flow. J. Fluid Mech. +828, 5–32. +Barker, T., Rauter, M., Maguire, E., Johnson, C., Gray, J., 2021. Coupling rheology and segregation in granular flows. +J. Fluid Mech. 909, A22. +26 + +da Cruz, F., Emam, S., Prochnow, M., Roux, J., Chevoir, F., 2005. Rheophysics of dense granular materials: Discrete +simulation of plane shear flows. Phys. Rev. E. 72, 021309. +Duan, Y., Umbanhowar, P.B., Ottino, J.M., Lueptow, R.M., 2021. 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Lett. 114, 238001. +Weinhart, T., Hartkamp, R., Thornton, A.R., Luding, S., 2013. +Coarse-grained local and objective continuum +description of three-dimensional granular flows down an inclined surface. Phys. Fluids 25, 070605. +Zhang, Q., Kamrin, K., 2017. Microscopic description of the granular fluidity field in nonlocal flow modeling. Phys. +Rev. Lett. 118, 058001. +29 + diff --git a/OdFJT4oBgHgl3EQfHyxy/content/tmp_files/load_file.txt b/OdFJT4oBgHgl3EQfHyxy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3cd1eb3e43e4c8dbb5d066911231f0e800732c8 --- /dev/null +++ b/OdFJT4oBgHgl3EQfHyxy/content/tmp_files/load_file.txt @@ -0,0 +1,5026 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf,len=5025 +page_content='Coupled continuum modeling of size-segregation driven by shear-strain-rate gradients and flow in dense, bidisperse granular media Daren Liu†, Harkirat Singh†, and David L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Henann†† School of Engineering, Brown University, Providence, RI 02912, USA Abstract Dense granular systems that consist of particles of disparate sizes segregate based on size during flow, resulting in complex, coupled segregation and flow patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The ability to predict how granular mixtures segregate is important in the design of industrial processes and the understanding of geophysical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The two primary drivers of size-segregation are pressure gradients and shear-strain-rate gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In this work, we isolate size-segregation driven by shear-strain-rate gradients by studying two dense granular flow geometries with constant pressure fields: gravity-driven flow down a long vertical chute with rough parallel walls and annular shear flow with rough inner and outer walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' We perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in both flow geometries, while varying system parameters, such as the flow rate, flow configuration size, fraction of large/small grains, and grain-size ratio, and use DEM data to inform continuum constitutive equations for the relative flux of large and small particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' When the resulting continuum model for the dynamics of size-segregation is coupled with the nonlocal granular fluidity model–a nonlocal continuum model for dense granular flow rheology–we show that both flow fields and segregation dynamics may be simultaneously captured using this coupled, continuum system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 1 Introduction Dense granular systems in nature and industry are often non-monodisperse–i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', consisting of particles of disparate sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In non-monodisperse granular systems, the constituent grains segregate based on size during flow, forming complex patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', Shinbrot and Muzzio, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Thornton, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Hill and Fan, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan and Hill, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Schlick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Umbanhowar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The ability to predict the dynamics of segregation is important across applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' For example, in geophysics, granular size segregation can manifest in landslides and debris flows (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2012), in which larger grains segregate to the top of the flow, potentially causing more damage, while in industry, size-segregation can be an undesirable effect when blending granular constituents of various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The current understanding is that there are two driving forces for size-segregation in dense granular flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The first is pressure-gradients, which are typically induced by gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In pressure-gradient-driven size-segregation, small particles move more readily through the interstitial spaces that open and close during flow through a process referred to in the literature as “kinetic sieving,” leading to a system stratified along the direction of pressure gradients (Savage and Lun, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Thornton, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Chugunov, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Thornton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' While pressure-gradient-driven segregation has been the focus of significant study, Hill and coworkers (Hill and Fan, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan and Hill, 2010, 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Hill and Tan, 2014) demonstrated that grains can also segregate in inhomogeneous flows along directions orthogonal to gravitationally-induced pressure gradients, driven instead by gradients in the shear-strain-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' As an example, this mechanism has been observed in the split-bottom cell experiments of Hill and Fan (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In these experiments, not only do the larger particles segregate to the top of the cell, but they also segregate perpendicular to the direction of pressure gradients towards more rapidly shearing regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Shear-strain-rate-gradient-driven segregation has received comparatively less attention in modeling efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' ††Email address for correspondence: david_henann@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='11453v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='soft] 26 Jan 2023 Due to the complexity of flow and segregation patterns, developing a general, predictive, continuum model for coupled size-segregation and flow in dense granular materials remains an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Although much progress has been made over recent decades (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', Savage and Lun, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Thornton, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Chugunov, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan and Hill, 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Tunuguntla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Umbanhowar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2019), the development of continuum models that are capable of simultaneously predicting the evolution of both segregation and flow fields, based solely on the geometry of the flow configuration, applied loads, and boundary/initial conditions is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Instead, most continuum models for size-segregation require some flow field quantity, such as the velocity or stress fluctuation field, to be measured first from experiments or DEM simulations and then used as model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' A crucial reason for the incompleteness of current models is the lack of a dense granular flow rheology theory that may be coupled to segregation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' A widely-used class of viscoplastic models for steady, dense granular flow is based on the 𝜇(𝐼) rheology (MiDi, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Jop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' da Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' One recent work that couples rheology and segregation in dense granular flows is that of Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' (2021), which combines a regularized version of the 𝜇(𝐼) rheology (Barker and Gray, 2017) with a model for gravity-driven segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' However, it has been well-documented in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', Kamrin, 2019) that the 𝜇(𝐼) rheology, even in its regularized form, can break down in the presence of spatial flow inhomogeneity, which can be attributed to nonlocal effects not accounted for in the 𝜇(𝐼) rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' To address this point, significant effort has gone into the development of size-dependent, nonlocal continuum constitutive theories for dense granular flow rheology, and coupling a nonlocal rheological model with a segregation model provides a route to robust, simultaneous prediction of flow and segregation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In this paper, we focus on the nonlocal granular fluidity (NGF) model of Kamrin and coworkers (Kamrin and Koval, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Henann and Kamrin, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Kamrin, 2019), which has been successfully applied to predicting dense flows of monodisperse grains in a wide variety of flow geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Then, the overarching aim of this paper is to formulate a predictive continuum theory for simultaneous flow and size-segregation in dense granular systems by integrating the NGF model with a phenomenological size-segregation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' This is a broad goal, and in this paper, we narrow our focus to several simpler, quasi-one-dimensional flow configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In most real-world flows, both the pressure-gradient-driven and shear- strain-rate-gradient-driven segregation mechanisms are present, making it difficult to disentangle them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Therefore, our plan for this paper is to isolate and examine the shear-strain-rate-gradient-driven mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Specifically, we study the shear-strain-rate-gradient-driven segregation mechanism by considering flows of dense, bidisperse systems of both disks and spheres in flow geometries in which the pressure field is spatially uniform–namely, vertical chute flow and annular shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Therefore shear-strain-rate-gradients are the sole drivers of segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In order to inform continuum model development, we perform discrete element method (DEM) simulations using the open source software LAMMPS (Plimpton, 1995), which function as “numerical experiments.” The coupled continuum model that we develop is then validated by comparing its predictions of the transient evolution of segregation and flow fields against additional DEM simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In Section 2, we discuss the continuum model that we use to describe flow and size-segregation in bidisperse, dense granular materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Specifically, Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='2 introduce the mixture theory framework used to describe dense, bidisperse granular mixtures, and in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='3, we briefly revisit the 𝜇(𝐼) rheology and the NGF model for monodisperse granular systems and discuss their extension to bidisperse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Then in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='4, we propose a model for shear-strain-rate-gradient-driven size-segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In Sections 3 and 4, we consider granular diffusion and shear-strain-rate-gradient-driven segregation, respectively, and independently determine the two dimensionless material parameters that appear in the size-segregation model for both disks and spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Then, in Section 5, the proposed segregation model is coupled with the NGF model and applied to both vertical chute flow and annular shear flow to predict the transient evolution of the segregation dynamics, and the predicted continuum fields are compared against DEM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In the end, our model demonstrates a level of fidelity in simultaneously predicting flow and segregation dynamics that has not been previously achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' We close with a discussion of the segregation model and some concluding remarks in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2 Continuum framework In this section, we discuss the continuum framework used to describe dense, bidisperse granular systems and propose constitutive equations for rheology and size-segregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Throughout, we utilize a mixture-theory-based approach, which is common in continuum modeling of dense, bidisperse mixtures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', Gray and Thornton, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Chugunov, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan and Hill, 2011b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Umbanhowar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Barker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Bancroft and Johnson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='2 ' 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+page_content='Figure 1: A representative schematic of a dense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' bidisperse granular system consisting of two-dimensional disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2021), and we use standard component notation, which supposes an underlying set of Cartesian basis vectors {e𝑖|𝑖 = 1, 2, 3}, and in which the components of vectors, v, and tensors, 𝝈, are denoted by 𝑣𝑖 and 𝜎𝑖 𝑗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The Einstein summation convention is employed, and the Kronecker delta, 𝛿𝑖 𝑗, is utilized to denote the components of the identity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='1 Bidisperse systems We consider granular mixtures consisting of particles with two sizes–large grains with an average diameter of 𝑑l and small grains with an average diameter of 𝑑s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' We consider both two-dimensional systems of disks, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 1, and three-dimensional systems of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' To eliminate the effect of density-based segregation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', Tripathi and Khakhar, 2013) and isolate size-based segregation, all particles are made of the same material with density 𝜌s, which represents the area-density for disks and the volume-density for spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Throughout, we utilize the notational convention in which we denote large-grain quantities using a superscript l and small-grain quantities using a superscript s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The species-specific solid fractions–i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', the areas occupied by each species per unit total area for disks and the volumes occupied by each species per unit total volume for spheres–are 𝜙l and 𝜙s, respectively, and the total solid fraction is 𝜙 = 𝜙l + 𝜙s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The concentration of each species then follows as 𝑐l = 𝜙l/𝜙 and 𝑐s = 𝜙s/𝜙, so that 𝑐l + 𝑐s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The average mixture grain size is defined as the sizes of both species weighted by their concentrations, ¯𝑑 = 𝑐l𝑑l + 𝑐s𝑑s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' We make the common idealization that the total area for dense systems of disks or total volume for dense systems of spheres does not change (Savage, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Thornton, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Gray and Chugunov, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Fan and Hill, 2011b), and therefore 𝜙 is idealized as constant at each point in space and at each instant in time during the segregation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' We have verified in our DEM simulations that area (or volume) dilatation at flow initiation occurs over a much shorter time scale than the process of segregation, so that this idealization is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Throughout this study, we use 𝜙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='8 for disks, and 𝜙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='6 for spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Regarding the kinematics of flow, each species has an associated partial velocity, 𝑣l 𝑖 and 𝑣s 𝑖, and the mixture velocity is given by 𝑣𝑖 = 𝑐l𝑣l 𝑖 + 𝑐s𝑣s 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The mixture strain rate tensor is then defined using the mixture velocity in the standard way: 𝐷𝑖 𝑗 = (1/2)(𝜕𝑣𝑖/𝜕𝑥 𝑗 + 𝜕𝑣 𝑗/𝜕𝑥𝑖), where 𝐷𝑘𝑘 = 0 since we have assumed that the mixture area (or volume) does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The equivalent shear strain-rate is defined as �𝛾 = (2𝐷𝑖 𝑗𝐷𝑖 𝑗)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Then, the relative area (or volume) flux for each grain type 𝛼 = l, s is defined through the difference between its partial velocity and the mixture velocity as 𝑤𝛼 𝑖 = 𝑐𝛼 �𝑣𝛼 𝑖 − 𝑣𝑖 � , so that 𝑤l 𝑖 + 𝑤s 𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Conservation of mass for each species requires that 𝐷𝑐𝛼/𝐷𝑡 + 𝜕𝑤𝛼 𝑖 /𝜕𝑥𝑖 = 0, where 𝐷(•)/𝐷𝑡 is the material time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Due to the fact that 𝑐l + 𝑐s = 1, only one of 𝑐l and 𝑐s is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Therefore, we will utilize 𝑐l as the field variable that describes the dynamics of size-segregation in the following discussion, and the evolution of 𝑐l is governed by its conservation of mass equation: 𝐷𝑐l 𝐷𝑡 + 𝜕𝑤l 𝑖 𝜕𝑥𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='1) 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='2 Stress and the equations of motion We recognize that the symmetric Cauchy stress tensor 𝜎𝑖 𝑗 = 𝜎𝑗𝑖 represents the Cauchy stress of the mixture, rather than the partial stress of either species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Regarding stress-related quantities for the granular mixture, we define the pressure 𝑃 = −(1/3)𝜎𝑘𝑘, the stress deviator 𝜎′ 𝑖 𝑗 = 𝜎𝑖 𝑗 + 𝑃𝛿𝑖 𝑗, the equivalent shear stress 𝜏 = (𝜎′ 𝑖 𝑗𝜎′ 𝑖 𝑗/2)1/2, and the stress ratio 𝜇 = 𝜏/𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The Cauchy stress is then governed by the standard equations of motion 𝜙𝜌s 𝐷𝑣𝑖 𝐷𝑡 = 𝜕𝜎𝑖 𝑗 𝜕𝑥 𝑗 + 𝑏𝑖, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='2) where 𝜙 is the constant total solid fraction, and 𝑏𝑖 is the non-inertial body force per unit volume (typically gravitational).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' In order to close the system of equations, we require (1) rheological constitutive equations for the Cauchy stress 𝜎𝑖 𝑗 and (2) a constitutive equation for the flux 𝑤l 𝑖, each of which are discussed in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='3 Rheological constitutive equations for bidisperse mixtures In this section, we discuss the rheology of dense, bidisperse granular mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Our strategy for formulating rheological constitutive equations for bidisperse mixtures is to relate mixture-related quantities, such as the Cauchy stress 𝜎𝑖 𝑗 and the strain-rate tensor 𝐷𝑖 𝑗, instead of specifying constitutive equations for species-specific partial stresses and then combining them to obtain the mixture stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The starting point of this discussion is the local inertial, or 𝜇(𝐼), rheology (MiDi, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Jop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' da Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2005), which follows from dimensional arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' For a dense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' monodisperse system of dry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' stiff grains with mean grain diameter 𝑑 subjected to homogeneous shearing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' the local inertial rheology asserts that the stress ratio 𝜇 is given through the equivalent shear strain-rate �𝛾 and the pressure 𝑃 through the dimensionless relationship 𝜇 = 𝜇loc(𝐼),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' where 𝐼 = �𝛾 √︁ 𝑑2𝜌s/𝑃 is the inertial number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' representing the ratio of the microscopic time scale associated with particle motion √︁ 𝑑2𝜌s/𝑃 to the macroscopic time scale of applied deformation 1/ �𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' As shown by Rognon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' (2007) and Tripathi and Khakhar (2011), the inertial rheology function 𝜇loc(𝐼) may be straightforwardly generalized from monodisperse to bidisperse systems by defining the inertial number for a bidisperse system as 𝐼 = �𝛾 √︁ ¯𝑑2𝜌s/𝑃, where the average mixture grain size for a bidisperse system ¯𝑑 has been used in place of 𝑑 for a monodisperse system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Then, the same local rheology function 𝜇loc(𝐼) utilized for the monodisperse system may be used for bidisperse systems without any changes to the parameters appearing in the fitting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' This approach neglects potential effects of new dimensionless quantities that arise in a bidisperse granular system, such as the grain size ratio 𝑑l/𝑑s, but has been shown to capture DEM data well (Rognon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Tripathi and Khakhar, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' To demonstrate this point, consider DEM simulations of homogeneous, simple shearing of a dense, bidisperse system of disks, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2(a) for the case of 𝑑l/𝑑s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='5 and a system-wide large-grain concentration of 𝑐l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Details of the simulated granular systems, including grain interaction properties, for both two-dimensional disks and three-dimensional spheres are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The large particles are dark gray, and the small particles are light gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' With the system-wide mean grain size denoted by ¯𝑑0 = 𝑐l𝑑l + (1−𝑐l)𝑑s, the rectangular domain has a length of 𝐿 = 60 ¯𝑑0 in the 𝑥-direction and a height of 𝐻 = 60 ¯𝑑0 in the 𝑧-direction, which is filled with ∼ 5000 flowing grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Shearing is driven through the relative motion of two parallel, rough walls, which each consist of a thin layer of touching glued grains, which are denoted as black in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The bottom wall is fixed, and the top wall moves with a velocity 𝑣w along the 𝑥-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Following previous works in the literature (da Cruz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Koval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Kamrin and Koval, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Zhang and Kamrin, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Liu and Henann, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Kim and Kamrin, 2020), the 𝑧-position of the top wall is not fixed but continuously adjusted using a feedback scheme so that the normal stress applied by the top wall is maintained at a target value of 𝜎𝑧𝑧(𝑧 = 0) = −𝑃w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' Periodic boundary conditions are applied along the 𝑥-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' For homogeneous simple shearing, no segregation will occur since the flow is homogeneous and no pressure or strain-rate gradients are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' We utilize the DEM procedures described in detail in Liu and Henann (2018) in order to extract the relationship between 𝜇 and 𝐼 for bidisperse mixtures with grain size ratios of 𝑑l/𝑑s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='5, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='0 and 𝑐l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The simulated relationships are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' 2(b) using triangular symbols of different colors, along with the monodisperse data from Liu and Henann (2018) plotted as gray circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' The relationship between 𝜇 and 𝐼 for dense systems of disks is observed to be approximately independent of 𝑑l/𝑑s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' As for the monodisperse case, the DEM data for bidisperse mixtures of disks may be fit by a linear, Bingham-like functional form: 𝜇loc(𝐼) = 𝜇s + 𝑏𝐼, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='3) 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='2 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdFJT4oBgHgl3EQfHyxy/content/2301.11453v1.pdf'} +page_content=' +direction. c AFM topography of TmIG. d Reciprocal lattice mapping of TmIG along the (6 4 2) plane, +which allow in-plane lattice measurement. Film and substrate reflections are marked. e Cross- +sectional STEM imaging of TmIG/GGG along with the chemical assessment through electron dispersive +x-ray analysis (EDX). Inset represents a closer look at the atom columns in the TmIG. The dotted line +is drawn for TmIG/GGG interface. Scale bar, 10 nm. +Having confirmed that our TmIG films are of very good quality, we now discuss the SOT-induced +switching of the TmIG perpendicular magnetization in several set of samples with different thicknesses +of TmIG and Pt. As the interfacial DMI already observed in TmIG18,19,20, appears to be involved in the +switching mechanism of our samples, we have also measured the effective DMI using Brillouin Light +Scattering (BLS). Depending on the sample, we find DMI energy densities ranging between 3.3 and 7.4 +µJ/m2 (corresponding to a DMI effective field applied to the domain walls between 1.8 to 4 mT), in the +same range as the interfacial DMI found with TmIG in other publications18,19,20. All our quantitative BLS +results are presented in Supplementary Information, section VIII. +In our switching experiments, charge current pulses (maximum pulse magnitude Jc = 3 × 1011 A/m2) +are injected along the x-direction in the SOC material, here Pt. The SHE of Pt creates a spin current (Js), +which is injected into TmIG and propagates by magnons to generate DL and FT torques on its +magnetization, involving DL and FL effective fields, HDL ∝ m × σ and HFL ∝ σ (see Fig.2a). The +measurements of DL and FL fields are presented in the Supplementary Information, section VII. To +demonstrate the effect of these torques on the magnetization, we perform fully reversible +magnetization switching in a Hall bar using current pulses only. The magnetization is measured by + +1E1 +1E5 +a +b +c +4.7 nm +d +GGG +GGG(444) +TmIG +105 +TmIG(444) +0.58 +3.0 +(arb. +0.01° +0.02 +M103 +25.5225.5425.56 +2.0 +25.64 25.68 25.72 +0.57 +o (degree) +<1-1> +2μm +TmIG +101 +0.0 +48 +50 +52 +54 +20-0 +GGG +0.56 +e +Tm +Fe +Gd +Ga +0.55 +TmIG +GGG +-0.229-0.2285-0.228-0.2275 +10nm +Q.(A-) / [2, -2, 0]4 + +anomalous Hall effect (AHE), and we obtain magnetization loops as a function of the pulsed current +(Figure 2b). The initial magnetization state is first prepared by applying a magnetic field of 0.35 T in +the out-of-plane direction and then reducing it to zero to obtain the initial state (state A with negative +RAHE in Fig.2b). Then, still at zero field, we send a sequence of current pulses of magnitude |Jc| up to +3.5 × 1011 A/m2. After each electrical pulse (of 100 s width), a small reading current of (100µA = +6.7×109 A/m2) is used to detect the magnetization from the acquired values of the AHE. + +FIG 2. Field-free switching in TmIG/Pt. a Schematic of the current induced SOT fields on TmIG. b,c,d, +e The sequence of current pulses shown in top (bottom) of d induces the type of CW (CCW) switching +loops shown in b (c) : in b, starting from initial state A, a negative current pulse can switch TmIG from +negative AHE (A) to positive AHE (B) and only a positive pulse (third in the sequence) can switch it back +to A, with successive Clockwise (CW) loops generated by alternating negative and positive pulses. In +c, a succession of CCW loops is obtained from the same initial state A but with an initial positive current +pulse. The decider between CW (b) and CCW (c) is the sign of the first pulse. A mechanism consistent +with these observations is pictured in e with current flowing from right to left (cf. pulse at point 2 in +(b)), DMI at left edge of device helping SOT to start switching and DMI at right edge leading to the +remanence of a small non-reversed domain. An opposite pulse (at point 5 in (b)) can expend this non- +reversed domain to obtain the CW loop for the AHE (explained in the text). The blue (red) arrows +indicate the direction of the DMI-induced fields ‘HDMI’ (DL fields, toward right for negative pulse). A + +Pt +ImIG +Initial +state +Initiai +state [5 + +positive (negative) first pulse leads to a remanent DW at left (right) and to CCW (CW) loops. The FL +torque is not considered here. f Kerr imaging of the reversal by positive (negative) initial current pulses +in top (bottom) panel at zero field, with approximated features of reversal starting on the left (right) +edge and propagating to the right (left) with more complex behavior at the crossing of the Hall +contacts. Colors black and white represent as magnetization up and down, respectively. Images were +recorded by initializing the state with 20 mT before each higher current pulse. +In the pulse sequence starting with a negative pulse in the schematic of the top panel in Figure 2d, +when the negative pulse exceeds a critical value, RAHE switches abruptly from negative to positive +(from (2) to (3)) and goes to state B at the end of the pulse in Fig.2b, (switch of mz from positive to +negative). In the continuation of the sequence in the top panel in Figure 2d, switching back to RAHE <0 +can be obtained only by positive pulses, as in the succession of the experimental CW loops Fig.2b. We +note that the switching ratio 𝑅𝑆𝑂𝑇 +𝑃𝑢𝑙𝑠𝑒/𝑅𝐴𝐻𝐸 = 0.52/0.55 is found to be ∼95%, which means almost +complete switching (RAHE is taken from the AHE measurements, see Supplementary Information, +section VI). The results are quite different with the second type of pulse sequence (bottom panel in +Fig. 2d) starting from the same initial state (A) with first positive pulses. An abrupt switching from A +to B (from positive to negative mz) is now obtained with the first positive pulse, from (2) to (3) in Fig.2d, +and the system comes back to A with negative pulse, from (5) to (1) in the CCW loop in Fig.2c. With +the first sweep of positive current pulses, CCW loops replace the CW loops obtained with a negative +first pulse in Fig.2b. Thus, after the same magnetic initialization, the system behaves differently +depending on its first current-induced switching pulse. +We recall the usual situation in which the addition of an applied in-plane field along x is needed to +break the in-plane symmetry and switch a perpendicular magnetization by SOT3,4. In such well-known +experimental protocol, the direction of this field along x decides that the switching loop is CW or +CCW3,4. In our switching results free of any external field, the decider of the choice between CW and +CCW is the direction of the initial current pulses. Such a behavior has already been found10 and is +ascribed to the chiral remanence imprinted by the first pulse in systems with a DMI-induced field HDMI +tilting the spins at the edges25. Figure 2e describes an example of mechanism of this type, which +involves HDL and DMI and is consistent with our results. For a given direction of the current pulse and +the corresponding direction of HDL (red arrow) in Fig.2e, the switching to down starts on the left edge +where HDMI (blue arrow) helps HDL (red arrow) and propagates to right by Néel DW motion. This motion +and the corresponding switching to down stops at the approach of the opposite edge where HDMI +(blue) hinders HDL (red), what imprints a remanent non-switched up domain close to the edge (a chiral +remanence). An opposite pulse reverses HDL and enlarges the remanent up domain to the right. It gives +a CCW loop for mz (CW for RAHE) in this case. With a first pulse of opposite amplitude, the switching +starts at the left edge and it is easy to see that it leads to a CW loop (CCW for RAHE) of opposite +polarities. Adding HFL, crystal field or more complicated shape of the device leads to some variants of +this type of mechanism, as discussed later. In samples with defects, we could also consider the +possibility of chiral remanences on defects in the bulk of the layer26. +Kerr imaging (Fig.2f) shows that the experimental behavior is similar close to the scenario in Fig.2e +with, for the direction of the current pulse of the top (bottom), a reversal starting on the left (right) of +the device (Fig.2f, top panel) and propagating to the right (left) with, however, a somewhat complex +behavior when the domain wall arrives in the wider region of the Hall-cross. These images are +recorded in zero in-plane applied magnetic field (only in earth’s magnetic field). See Fig. S10 to rule +any effect of spurious magnetic fields present in the measurement setup. + +6 + + +FIG. 3. Current-field map of magnetization switching. Experimental magnetization switching map +recorded in the in-plane magnetic fields (along or opposite to the current direction) after initializing +the magnetization with +0.35 T magnetic field. Dotted line is drawn at zero-field crossover. +In order to fully characterize the magnetization reversal process, we also discuss the influence of a +magnetic field, as reported in Fig.3. In zero field, starting from an initial state with magnetization up +(RAHE < 0, point A in Fig.2b), the system switches from up to down (blue to red) at a negative current +of about 7 mA (2.5 1011 A/m2 in Fig.2b). The application of a positive field 0 Hx helps to switch and +the switching current decreases in absolute value down to about -6 mA at 10 mT. In contrast, a +negative field appears to hinder the switching by a negative current and suppresses it above about - +0.9 mT. For positive fields, the switching from up to down (blue to red) occurs in positive currents, +which correspond to a change of the polarity of the loops, from the CW type of Fig.2b to the CCW if +Fig.2c. The decrease of the switching current as the field increases expresses that negative fields help +the switching of this polarity. We thus find that, in addition to the results obtained at zero field, an +applied field can help or hinder the switching and even change the polarity of the loops. It is interesting +to note that the field changing the polarity (-0.9 mT) for negative current is close to the DMI field +derived from our BLS measurements (1.8 mT) and that the fields efficient to change the switching +currents27 are also in the same range. We also observed, depending on the Hall bar as presented in +Supplementary Information, section XI, some asymmetry in the switching current polarity we relate +to extrinsic mechanisms linked to the nucleation process (defect, shape, inhomogeneity etc..). A +perfect control of the shape and the edges of the ferromagnetic insulator may consider for further +development. + +7 + + +FIG. 4. Role of magnetocrystalline anisotropy. a Schematic of the [111] orientation of the TmIG crystal +lattice with three vectors [1 0 0] [0 1 0] [0 0 1] of the magnetocrystalline anisotropy. The [1 1 -2] and +[1 -1 0] are two in-plane direction vectors (as also depicted for substrate plane). The  is the angle of +TmIG cubic crystal under strain from substrate. b, Patterned devices with different azimuthal angles. +Critical current density plot for various devices patterned at different azimuthal angles for a positive +mz initialization. +Although the field free switching we observed with a loop polarity controlled by the sign of the first +current pulse in Fig.2 can be consistently explained by the conjunction of SOT and DMI in the +mechanism summarized in Fig.2e, additional results indicate that the cubic anisotropy (see orientation +of crystal axes at (111) surface in Fig.4) has also some influences onto the switching process. To +understand the role of the cubic anisotropy experimentally, we patterned devices with different +azimuthal angles as shown in inset of Figure 4b. The critical switching current recorded in these devices +is found to be dependent on the crystallographic orientation, while the anomalous Hall effect signals +were found to be identical with the same coercivity (not shown here), ruling out any non-uniformity +in the samples. The orientation dependence of the critical current indicates the additional influence +of the cubic anisotropy. We also measured the magnetization switching by varying the TmIG and Pt +thicknesses (see Supplementary Information, section X): some variation in the anomalous Hall signal +can be seen however field free switching is always observed. Further, micromagnetic simulations were +performed at different combination of parameters demonstrating the critical role of DMI and cubic +anisotropy in the magnetization reversal process by SOT (see Supplementary Information, section XII). +In summary, we have observed the magnetization switching of perpendicularly magnetized epitaxial +Tm3Fe5O12 (TmIG) thin films in TmIG/Pt bilayers for which we have the advantage of the deep +propagation of spin currents by magnons for efficient SOT and the absence of shunting in TmIG. +Another advantage is the existence of interfacial DMI that we determined by Brillouin Light Scattering. +We observe a reproducible and robust field-free switching at moderate current density. Starting from +a saturated magnetic state, the sign of the first switching current pulse decides if the subsequent +switching loops are CW or CCW, in the same way as, in in-plane-field-assisted switching, the “decision” +is taken by the sign of the external applied field. The main features of the field free switching results +are consistent with a mechanism in which the conjunction of efficient SOT (DL) and DMI nucleates +reversed domain on one edge of the sample and imprints a chiral remanence on the opposite edge. +Kerr imaging is also consistent with such a mechanism. In experiments in which a field is applied, we +find that a field in the range of the DMI fields helps or hinders the switching observed at zero field and + +b +5 +a +[111] +M +2.5 +Devices +[010] m, +m1 [100] +0 +[001] +[11-2 +[11-2] +m3 +-2.5 +5 +0 +60 +120 180 240 300 +360 +β (degrees)8 + +can even inverse the polarity of the loops. Additional experiments show that the cubic anisotropy +plays an intriguing role in the switching at zero-field. While there remain outstanding questions as to +the exact origin of field-free switching in TmIG, one plausible explanation we propose here is the DW +nucleation at the edges due to the DMI and cubic anisotropy, which follows the current pulse direction. +Future experimental and theoretical work should seek to further understand the role of different +parameters of TmIG, such as different crystal orientations. In conclusion, the finding of field-free +magnetization switching by SOTs in a magnetic insulator is an important milestone for future +applications in spin-orbitronics. +Acknowledgements +DARPA TEE program grant (MIPR#HR0011831554) is acknowledged for their financial support. This +work is supported by a public grant overseen by the French National Research Agency (ANR) as part +of the “Investissements d’Avenir” program (Labex NanoSaclay, reference: ANR-10-LABX-0035). ERC +AdG FRESCO (#833973) is also acknowledged. +Methods +Thin film growth: Thulium iron garnet ‘Tm3Fe5O12 (TmIG)’ ferrimagnetic insulator (FIMI) thin films +were deposited on (1 1 1) oriented Gd3Ga5O12 (GGG) substrates by off-axis sputtering. Before +deposition, substrates were treated by acetone and isopropyl alcohol in ultrasonication and +subsequently annealed at 1000°C for 5 hours in a flow of pure oxygen (O2) at atmospheric pressure. +The substrates were transferred in air into the sputtering chamber for TmIG deposition. Thin films +were deposited at room temperature in flow of Ar (40 sccm) and O2 (20 sccm) with dynamic pressure +of 4.2 mbar (base pressure is lower than ∼2×10−8 mbar). To promote the crystallinity, these films were +post-annealed (in ex-situ furnace) at 650 °C for 4 hours in a flow of pure O2 at atmospheric pressure. +Further, Pt layer of 6nm thick (unless otherwise stated) was deposited by on-axis magnetron +sputtering at room temperature. TmIG film surfaces were cleaned by O2 plasma (40 eV) before Pt +deposition. +Structural characterization: X-Ray diffraction in symmetric (2−) or asymmetric (reciprocal space +mapping, RSM) geometry were recorded by Philips X’pert-PRO Empyrean diffractometer. For XRD, +measurements were performed in Bragg-Brentano reflection mode. For RLM, the diffraction along the +(6 4 2) plane direction is used, which allows to gain the information about in-plane epitaxy relation +along [2 -2 0] direction. Topography of the substrate and film were recorded by atomic force +microscopy (AFM) using a Dimension Icon system with ScanAsyst(Bruker Dimension Icon, Billerica, +MA, USA). Images were collected in tapping mode (in air) using a tip with nominal radius <10 nm. +Atomic-scale imaging was performed by cross-sectional scanning transmission electron microscopy +(STEM). The sample investigated by STEM was prepared by a focused ion beam machine (FEI Helios +platform) using a Ga ion beam with an accelerating voltage of first 30 kV to detach the slab, and then +of 5 kV to thin it down. STEM characterization was conducted with a Hitachi HF5000 equipped with a +cold field emission gun operated at 200 kV and a probe aberration corrector. High-angle annular dark- +field images were acquired with a probe that formed an angle of 30 mrad and a collection angle of +60–300 mrad. EDX spectra were collected using two detectors from Oxford Instruments and color- +coded elemental maps were obtained using the AZtec software. Magnetization measurements were +performed by Quantum Design SQUID magnetometer. All electron transport measurements were +performed in a home-built set-up. +AHE, SMR, SOTs and magnetization switching measurements: To perform the electron transport +experiments, 5-µm wide and 50-µm long symmetric Hall-crosses were patterned using photo- + +9 + +lithography and Ar-ion milling. For Kerr imaging, the devices were patterned in 10-µm wide and 100- +µm long Hall crosses with Au contact pads. The anomalous Hall effect and spin Hall magnetoresistance +measurements were carried out using a constant dc source. For current-induced switching +measurements, current pulses with a duration of 100 s were generated by a Keithley 6221 and +injected into the Hall bar. After each pulse, a small excitation (100 A = 3109 A/m2) current was +applied to evaluate and measure the magnetization state. For the harmonic Hall measurements, an +ac current source with an amplitude from 1 to 6 mA (root mean square) was injected with a Keithley +6221 current source. The first and second harmonic signals were measured using an SR-830 lock-in +amplifier. +DMI measurements: The Dzyaloshisnkii-Moriya interaction energy was measured by Brillouin light +scattering (BLS) using a JRS TFP-2 triple-pass tandem Fabry-Perot interferometer with quarter-wave +antireflection optics and linearly-polarized (10 mW laser with 473 nm wavelength). The spectra were +recorded in the backscattering geometry at various wave vector orientations, selected by mounting +the sample on an angle-controlled sample holder providing a range of 10° to 60° incident angles +corresponding to wave vectors, qk = (4π/λ) sinθ, lying in the range 4 to 20.4 rad./µm−1. The free spectral +range was 9.4 or 18.7 GHz and spectra were recorded with 1024 points. The in-plane magnetic field +to pull the magnetization in-plane for the Damon-Eshbach geometry is provided by permanent +magnets in order to avoid thermal drift. +REFERENCES +(1) +Dieny, B.; Prejbeanu, I. 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Strain-Tunable Interfacial Dzyaloshinskii − +Moriya Interaction and Spin-Hall Topological Hall E Ff Ect in Pt/Tm 3 Fe 5 O 12 +Heterostructures. ACS Appl. Mater. Interfaces 2022, 14, 16791. +https://doi.org/10.1021/acsami.1c22942. +(21) +Vu, N. M.; Meisenheimer, P. B.; Heron, J. T. Tunable Magnetoelastic Anisotropy in Epitaxial +(111) Tm3Fe5O12 Thin Films. Journal of Applied Physics 2020, 127 (15), 153905. +https://doi.org/10.1063/1.5142856. +(22) +Gulyaev, Y. V; Zil, P. E.; Chigarev, S. G.; Epshtein, E. M. Current Induced Exchange Switching of +Magnetic Junctions with Cubic Anisotropy of a Free Layer. Physics of the Solid State 2011, 53 +(4), 723–729. https://doi.org/10.1134/S1063783411040196. +(23) +Shumate, P. W. Domain-Wall Energy in Magnetic Garnet Bubble Materials. Journal of Applied +Physics 1973, 44 (11), 5075–5077. https://doi.org/10.1063/1.1662093. +(24) +Phys, J. A.; Meisenheimer, P. B.; Heron, J. T. Tunable Magnetoelastic Anisotropy in Epitaxial ( +111 ) Tm 3 Fe 5 O 12 Thin Films Tunable Magnetoelastic Anisotropy in Epitaxial. J. Appl. Phys. +2020, 153905 (December 2019), 153905. https://doi.org/10.1063/1.5142856. +(25) +Pizzini, S.; Vogel, J.; Rohart, S.; Buda-Prejbeanu, L. D.; Jué, E.; Boulle, O.; Miron, I. M.; Safeer, +C. K.; Auffret, S.; Gaudin, G.; Thiaville, A. Chirality-Induced Asymmetric Magnetic Nucleation +in Pt/Co/AlOx Ultrathin Microstructures. Physical Review Letters 2014, 113 (4), 047203. +https://doi.org/10.1103/PhysRevLett.113.047203. +(26) +Michels, A.; Mettus, D.; Titov, I.; Malyeyev, A.; Bersweiler, M.; Bender, P.; Peral, I.; Birringer, +R.; Quan, Y.; Hautle, P.; Kohlbrecher, J.; Honecker, D.; Fernández, J. R.; Barquín, L. F.; Metlov, +K. L. Microstructural-Defect-Induced Dzyaloshinskii-Moriya Interaction. Physical Review B +2019, 99 (1), 014416. https://doi.org/10.1103/PhysRevB.99.014416. +(27) +Kim, S.; Jang, P. H.; Kim, D. H.; Ishibashi, M.; Taniguchi, T.; Moriyama, T.; Kim, K. J.; Lee, K. J.; +Ono, T. Magnetic Droplet Nucleation with a Homochiral Néel Domain Wall. Physical Review B +2017, 95 (22), 220402(R). https://doi.org/10.1103/PhysRevB.95.220402. + + + diff --git a/SdFJT4oBgHgl3EQfLiwk/content/tmp_files/load_file.txt b/SdFJT4oBgHgl3EQfLiwk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db1664b76bdab81e37f045295db2749d167aa2cb --- /dev/null +++ b/SdFJT4oBgHgl3EQfLiwk/content/tmp_files/load_file.txt @@ -0,0 +1,1061 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf,len=1060 +page_content='1 Field-free switching of perpendicular magnetization in an ultrathin epitaxial magnetic insulator Sajid Husain1*, Olivier Fayet1, Nicholas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Prestes1, Sophie Collin1, Florian Godel1, Eric Jacquet1, Thibaud Denneulin2, Rafal E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Dunin-Borkowski2, André Thiaville3, Manuel Bibes1, Nicolas Reyren1, Henri Jaffrès1, Albert Fert1, and Jean-Marie George1 1Unité Mixte de Physique, CNRS, Thales, Université Paris-Saclay, 91767 Palaiseau, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 2Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum Jülich, 52425 Jülich, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 3Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS, 91405, Orsay, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Present address: Material Physics Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA jeanmarie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='george@cnrs-thales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='fr albert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='fert@cnrs-thales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='fr shusain@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='gov For energy efficient and fast magnetic memories, switching of perpendicular magnetization by the spin-orbit torque (SOT) appears as a very promising solution, even more using magnetic insulators that suppress electrical shunting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' This SOT switching generally requires the assistance of an in-plane magnetic field to break the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Here, we present experiments demonstrating the field-free SOT switching of perpendicularly magnetized layers of the thulium iron garnet (Tm3Fe5O12) magnetic insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The polarity of the switching loops, clockwise (CW) or counter-clockwise (CCW), is determined by the direction of the initial current pulses, in contrast with field-assisted switchings in which this polarity is controlled by the direction of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' After an independent determination of Dzyaloshinskii-Moriya interaction (DMI), we relate the field free switching to the interplay of SOT and DMI and the polarity of the loops to the imprint of a Néel domain wall induced by the first pulse, in agreement with Kerr imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Our observation and interpretation of field-free electrical switching of a magnetic insulator is an important milestone for future low power spintronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The pure electrical control of the magnetization in magnetic heterostructure employing spin-orbit torques (SOTs) is proposed as the key for developing energy efficient next generation magnetic memories in spintronics1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The manipulation of the magnetization through SOTs is usually realized using the spin current provided by the large spin-orbit coupling (SOC) found in heavy metals (HM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The spin current appears due to asymmetric spin scatterings mediated by SOC within the HM via the spin Hall effect (SHE) and/or the Rashba-Edelstein effect (REE) originating from interfacial SOC or an E-field orthogonal to the film surface (z-direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The spin relaxation in the magnetic layer gives rise to a torque on the magnetization (M = Ms\uf0d7m), only the spins transverse to m relax and contribute to the torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Mainly two torques have been identified as represented by a transverse (field-like ‘FL’) torque, τFL ∝ m×σ and a longitudinal (damping-like ‘DL’) torque, τDL ∝ m×(m×σ), where σ is directed along the y-direction orthogonal to the spin current Js.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The out-of-plane torque due to structural symmetry-breaking is not considered here due to cubic symmetry of Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The driving force for the magnetization reversal in a perpendicular magnet is essentially the DL torque, but the FL can influence the nucleation4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Additionally, an in-plane magnetic field is commonly required for breaking the inversion symmetry6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' It will favor one of the two magnetic remnant states, for a uniform 2 magnetization switching (macrospin model7), and for domain nucleation and propagation mechanism as well8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' A large effort has been dedicated to achieving field-free switching, as it would be an important milestone for low-power energy application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Field-free switching has been recently studied in several systems, but this typically requires an additional process such as breaking crystal symmetry9 to induce an additional torque, develop a chiral field gradient10, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' This usually invokes the fabrication of engineered multilayer stacks exhibiting a source of symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Moreover, in order to avoid the shunting of electrical current through metallic FMs, it is of interest to replace the metallic magnetic electrodes by a low conductivity material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' On the other hand, ferrimagnetic (FIM) insulating garnets have recently entered into field of spin-orbitronics11 opening new questions and issues about the nature of interfacial electronic exchange mechanisms when using a ferromagnetic insulator with a SHE metallic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Recently, thulium iron garnet (Tm3Fe5O12, TmIG) became one of the most attractive garnets due to its compelling properties such as perpendicular magnetic anisotropy (PMA) at room temperature11, magnetization switching by SOT12,13, large domain wall velocity14, and interfacial chiral exchange, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=', Dzyaloshinskii- Moriya interaction (DMI) as claimed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Several key features such as PMA strength and domain velocity are tunable through substrate-film strain engineering as well as through thin film growth16,17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The DMI has emerged as a debatable parameter of different possible origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For example, it was reported to arise from the interface with a HM19, from the substrate-film interface as well as from the bulk of the film16,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Further, it is reported that chiral domains can be stabilized by DMI and moved by current pulses without external field18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' A detailed analysis of DMI as well as its influence on the current induced switching in TmIG insulating garnet is still pending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' TmIG grown along [1 1 1] also display cubic magnetocrystalline anisotropy21, expected to provide an additional source for magnetization switching22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In this letter, we demonstrate the fabrication of high crystalline quality TmIG with perpendicular magnetization by off-axis sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The major result of our study is the demonstration of the magnetization reversal induced by SOTs at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We then discuss the field-free magnetization switching in light of Kerr microscopy imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We demonstrate the key role of the DMI in this system favoring a deterministic domain nucleation5,10 depending on the polarity of the current and we will discuss the role of the magnetocrystalline anisotropy in the zero-field switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' To do this, we grew series of samples of TmIG on Gd3Ga5O12 (GGG) substrates and Pt is used as a spin current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' See Methods for more description of the sample fabrication and measurement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We first perform a detailed characterization of the film structural quality, as DMI and PMA might find their origin in growth-induced strain and gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The 2θ − ω diffraction pattern of TmIG (15 nm) thin film deposited on GGG is displayed in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The corresponding peaks are identified as (4 4 4) reflections from both the TmIG and GGG substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Due to the small lattice mismatch (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='49%) between the substrate and film (both are cubic crystals), the (4 4 4) diffraction peaks partly overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The Laue oscillations visible around the main peak correspond to the film thickness fringes, indicating a high-quality growth and well-defined crystallographic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The out-of-plane lattice spacing for the (4 4 4) reflection, 𝑑𝑇𝑚𝐼𝐺 444 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='1775 nm, corresponds to the out-of-plane lattice constant a⊥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='230 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' To confirm the epitaxial quality, we recorded a rocking curve along the (4 4 4) with a full width and half maximum Δω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='02° (compared to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='01° found for the GGG substrate) suggesting an epitaxial growth with a few millidegrees mosaicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The RHEED pattern of TmIG after annealing displays a single family of streaks, indicating epitaxy and in-plane crystal coherence (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We measured the topography of the film surfaces using atomic force microscopy (AFM) (Figure 1c), and found a surface roughness smaller than 3Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Furthermore, the reciprocal space mapping (RSM) was performed near the (6 4 2) asymmetric reflection, showing a pseudomorphic growth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The 3 film peak and substrate peak at the same in-plane reciprocal lattice unit 𝑑𝑇𝑚𝐼𝐺 220 = Qx-1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='4379 nm, indicating a fully strained state with an in-plane lattice constant of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='238 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Due to the elastic deformation of the TmIG under strain, the out-of-plane lattice spacing in the TmIG film corresponding to the (4 4 4) in the RSM is 𝑑𝑇𝑚𝐼𝐺 444 = Qz-1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='1770 nm, consistent with the XRD evaluated lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Using the in-and out-of-plane lattice parameters, the angle (\uf051) between for the facets of the TmIG unit cell24 is calculated to be 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='74°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' This tilt (in top edge of the cubic crystal cell) appears due to the tensile-strain from the GGG substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Since the 15nm thick film is fully strained, we assume that all the thinner films are also strained (see Supplementary Information, section I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The stoichiometry was verified using x-ray photoelectrons spectroscopy (see Supplementary Information, section II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Further, in the cross-sectional STEM image (Figure 1e), the substrate film interface is not distinguishable due to the epitaxial alignment of the lattices and two materials show a similar Z-contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' However, the interface can be resolved through chemical analysis as marked by the dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The atomically resolved STEM image shown in the inset reveals a highly ordered TmIG grown on GGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Structural characterization of TmIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' a XRD pattern of TmIG(15 nm) grown on GGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Inset on left (right) rocking curve of GGG (TmIG) of the (4 4 4) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' b RHEED pattern of TmIG along the <1 -1> direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' c AFM topography of TmIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' d Reciprocal lattice mapping of TmIG along the (6 4 2) plane, which allow in-plane lattice measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Film and substrate reflections are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' e Cross- sectional STEM imaging of TmIG/GGG along with the chemical assessment through electron dispersive x-ray analysis (EDX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Inset represents a closer look at the atom columns in the TmIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The dotted line is drawn for TmIG/GGG interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Scale bar, 10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Having confirmed that our TmIG films are of very good quality, we now discuss the SOT-induced switching of the TmIG perpendicular magnetization in several set of samples with different thicknesses of TmIG and Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' As the interfacial DMI already observed in TmIG18,19,20, appears to be involved in the switching mechanism of our samples, we have also measured the effective DMI using Brillouin Light Scattering (BLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Depending on the sample, we find DMI energy densities ranging between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='4 µJ/m2 (corresponding to a DMI effective field applied to the domain walls between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='8 to 4 mT), in the same range as the interfacial DMI found with TmIG in other publications18,19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' All our quantitative BLS results are presented in Supplementary Information, section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In our switching experiments, charge current pulses (maximum pulse magnitude Jc = 3 × 1011 A/m2) are injected along the x-direction in the SOC material, here Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The SHE of Pt creates a spin current (Js), which is injected into TmIG and propagates by magnons to generate DL and FT torques on its magnetization, involving DL and FL effective fields, HDL ∝ m × σ and HFL ∝ σ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The measurements of DL and FL fields are presented in the Supplementary Information, section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' To demonstrate the effect of these torques on the magnetization, we perform fully reversible magnetization switching in a Hall bar using current pulses only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The magnetization is measured by 1E1 1E5 a b c 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='7 nm d GGG GGG(444) TmIG 105 TmIG(444) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='0 (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='01° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='02 M103 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='5225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='5425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='64 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='68 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='57 o (degree) <1-1> 2μm TmIG 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='0 48 50 52 54 20-0 GGG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='56 e Tm Fe Gd Ga 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='55 TmIG GGG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='229-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2285-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='228-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2275 10nm Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' (A-) / [2, -2, 0]4 anomalous Hall effect (AHE), and we obtain magnetization loops as a function of the pulsed current (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The initial magnetization state is first prepared by applying a magnetic field of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='35 T in the out-of-plane direction and then reducing it to zero to obtain the initial state (state A with negative RAHE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Then, still at zero field, we send a sequence of current pulses of magnitude |Jc| up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='5 × 1011 A/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' After each electrical pulse (of 100 \uf06ds width), a small reading current of (100µA = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='7×109 A/m2) is used to detect the magnetization from the acquired values of the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' FIG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Field-free switching in TmIG/Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' a Schematic of the current induced SOT fields on TmIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' b,c,d, e The sequence of current pulses shown in top (bottom) of d induces the type of CW (CCW) switching loops shown in b (c) : in b, starting from initial state A, a negative current pulse can switch TmIG from negative AHE (A) to positive AHE (B) and only a positive pulse (third in the sequence) can switch it back to A, with successive Clockwise (CW) loops generated by alternating negative and positive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In c, a succession of CCW loops is obtained from the same initial state A but with an initial positive current pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The decider between CW (b) and CCW (c) is the sign of the first pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' A mechanism consistent with these observations is pictured in e with current flowing from right to left (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' pulse at point 2 in (b)), DMI at left edge of device helping SOT to start switching and DMI at right edge leading to the remanence of a small non-reversed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' An opposite pulse (at point 5 in (b)) can expend this non- reversed domain to obtain the CW loop for the AHE (explained in the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The blue (red) arrows indicate the direction of the DMI-induced fields ‘HDMI’ (DL fields, toward right for negative pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' A Pt ImIG Initial state Initiai state [5 positive (negative) first pulse leads to a remanent DW at left (right) and to CCW (CW) loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The FL torque is not considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' f Kerr imaging of the reversal by positive (negative) initial current pulses in top (bottom) panel at zero field, with approximated features of reversal starting on the left (right) edge and propagating to the right (left) with more complex behavior at the crossing of the Hall contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Colors black and white represent as magnetization up and down, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Images were recorded by initializing the state with 20 mT before each higher current pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In the pulse sequence starting with a negative pulse in the schematic of the top panel in Figure 2d, when the negative pulse exceeds a critical value, RAHE switches abruptly from negative to positive (from (2) to (3)) and goes to state B at the end of the pulse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b, (switch of mz from positive to negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In the continuation of the sequence in the top panel in Figure 2d, switching back to RAHE <0 can be obtained only by positive pulses, as in the succession of the experimental CW loops Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We note that the switching ratio 𝑅𝑆𝑂𝑇 𝑃𝑢𝑙𝑠𝑒/𝑅𝐴𝐻𝐸 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='52/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='55 is found to be ∼95%, which means almost complete switching (RAHE is taken from the AHE measurements, see Supplementary Information, section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The results are quite different with the second type of pulse sequence (bottom panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 2d) starting from the same initial state (A) with first positive pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' An abrupt switching from A to B (from positive to negative mz) is now obtained with the first positive pulse, from (2) to (3) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2d, and the system comes back to A with negative pulse, from (5) to (1) in the CCW loop in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' With the first sweep of positive current pulses, CCW loops replace the CW loops obtained with a negative first pulse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Thus, after the same magnetic initialization, the system behaves differently depending on its first current-induced switching pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We recall the usual situation in which the addition of an applied in-plane field along x is needed to break the in-plane symmetry and switch a perpendicular magnetization by SOT3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In such well-known experimental protocol, the direction of this field along x decides that the switching loop is CW or CCW3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In our switching results free of any external field, the decider of the choice between CW and CCW is the direction of the initial current pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Such a behavior has already been found10 and is ascribed to the chiral remanence imprinted by the first pulse in systems with a DMI-induced field HDMI tilting the spins at the edges25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Figure 2e describes an example of mechanism of this type, which involves HDL and DMI and is consistent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For a given direction of the current pulse and the corresponding direction of HDL (red arrow) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2e, the switching to down starts on the left edge where HDMI (blue arrow) helps HDL (red arrow) and propagates to right by Néel DW motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' This motion and the corresponding switching to down stops at the approach of the opposite edge where HDMI (blue) hinders HDL (red), what imprints a remanent non-switched up domain close to the edge (a chiral remanence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' An opposite pulse reverses HDL and enlarges the remanent up domain to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' It gives a CCW loop for mz (CW for RAHE) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' With a first pulse of opposite amplitude, the switching starts at the left edge and it is easy to see that it leads to a CW loop (CCW for RAHE) of opposite polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Adding HFL, crystal field or more complicated shape of the device leads to some variants of this type of mechanism, as discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In samples with defects, we could also consider the possibility of chiral remanences on defects in the bulk of the layer26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Kerr imaging (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2f) shows that the experimental behavior is similar close to the scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2e with, for the direction of the current pulse of the top (bottom), a reversal starting on the left (right) of the device (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2f, top panel) and propagating to the right (left) with, however, a somewhat complex behavior when the domain wall arrives in the wider region of the Hall-cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' These images are recorded in zero in-plane applied magnetic field (only in earth’s magnetic field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' S10 to rule any effect of spurious magnetic fields present in the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Current-field map of magnetization switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Experimental magnetization switching map recorded in the in-plane magnetic fields (along or opposite to the current direction) after initializing the magnetization with +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='35 T magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Dotted line is drawn at zero-field crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In order to fully characterize the magnetization reversal process, we also discuss the influence of a magnetic field, as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In zero field, starting from an initial state with magnetization up (RAHE < 0, point A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b), the system switches from up to down (blue to red) at a negative current of about \uf07e7 mA (\uf07e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='5 1011 A/m2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The application of a positive field \uf06d0 Hx helps to switch and the switching current decreases in absolute value down to about -6 mA at \uf07e10 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In contrast, a negative field appears to hinder the switching by a negative current and suppresses it above about - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='9 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For positive fields, the switching from up to down (blue to red) occurs in positive currents, which correspond to a change of the polarity of the loops, from the CW type of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2b to the CCW if Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The decrease of the switching current as the field increases expresses that negative fields help the switching of this polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We thus find that, in addition to the results obtained at zero field, an applied field can help or hinder the switching and even change the polarity of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' It is interesting to note that the field changing the polarity (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='9 mT) for negative current is close to the DMI field derived from our BLS measurements (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='8 mT) and that the fields efficient to change the switching currents27 are also in the same range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We also observed, depending on the Hall bar as presented in Supplementary Information, section XI, some asymmetry in the switching current polarity we relate to extrinsic mechanisms linked to the nucleation process (defect, shape, inhomogeneity etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='.).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' A perfect control of the shape and the edges of the ferromagnetic insulator may consider for further development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Role of magnetocrystalline anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' a Schematic of the [111] orientation of the TmIG crystal lattice with three vectors [1 0 0] [0 1 0] [0 0 1] of the magnetocrystalline anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The [1 1 -2] and [1 -1 0] are two in-plane direction vectors (as also depicted for substrate plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The \uf051 is the angle of TmIG cubic crystal under strain from substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' b, Patterned devices with different azimuthal angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Critical current density plot for various devices patterned at different azimuthal angles for a positive mz initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Although the field free switching we observed with a loop polarity controlled by the sign of the first current pulse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2 can be consistently explained by the conjunction of SOT and DMI in the mechanism summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2e, additional results indicate that the cubic anisotropy (see orientation of crystal axes at (111) surface in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='4) has also some influences onto the switching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' To understand the role of the cubic anisotropy experimentally, we patterned devices with different azimuthal angles as shown in inset of Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The critical switching current recorded in these devices is found to be dependent on the crystallographic orientation, while the anomalous Hall effect signals were found to be identical with the same coercivity (not shown here), ruling out any non-uniformity in the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The orientation dependence of the critical current indicates the additional influence of the cubic anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We also measured the magnetization switching by varying the TmIG and Pt thicknesses (see Supplementary Information, section X): some variation in the anomalous Hall signal can be seen however field free switching is always observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Further, micromagnetic simulations were performed at different combination of parameters demonstrating the critical role of DMI and cubic anisotropy in the magnetization reversal process by SOT (see Supplementary Information, section XII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In summary, we have observed the magnetization switching of perpendicularly magnetized epitaxial Tm3Fe5O12 (TmIG) thin films in TmIG/Pt bilayers for which we have the advantage of the deep propagation of spin currents by magnons for efficient SOT and the absence of shunting in TmIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Another advantage is the existence of interfacial DMI that we determined by Brillouin Light Scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' We observe a reproducible and robust field-free switching at moderate current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Starting from a saturated magnetic state, the sign of the first switching current pulse decides if the subsequent switching loops are CW or CCW, in the same way as, in in-plane-field-assisted switching, the “decision” is taken by the sign of the external applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The main features of the field free switching results are consistent with a mechanism in which the conjunction of efficient SOT (DL) and DMI nucleates reversed domain on one edge of the sample and imprints a chiral remanence on the opposite edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Kerr imaging is also consistent with such a mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In experiments in which a field is applied, we find that a field in the range of the DMI fields helps or hinders the switching observed at zero field and b 5 a [111] M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='5 Devices [010] m, m1 [100] 0 [001] [11-2 [11-2] m3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='5 5 0 60 120 180 240 300 360 β (degrees)8 can even inverse the polarity of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Additional experiments show that the cubic anisotropy plays an intriguing role in the switching at zero-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' While there remain outstanding questions as to the exact origin of field-free switching in TmIG, one plausible explanation we propose here is the DW nucleation at the edges due to the DMI and cubic anisotropy, which follows the current pulse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Future experimental and theoretical work should seek to further understand the role of different parameters of TmIG, such as different crystal orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' In conclusion, the finding of field-free magnetization switching by SOTs in a magnetic insulator is an important milestone for future applications in spin-orbitronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Acknowledgements DARPA TEE program grant (MIPR#HR0011831554) is acknowledged for their financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (Labex NanoSaclay, reference: ANR-10-LABX-0035).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' ERC AdG FRESCO (#833973) is also acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Methods Thin film growth: Thulium iron garnet ‘Tm3Fe5O12 (TmIG)’ ferrimagnetic insulator (FIMI) thin films were deposited on (1 1 1) oriented Gd3Ga5O12 (GGG) substrates by off-axis sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Before deposition, substrates were treated by acetone and isopropyl alcohol in ultrasonication and subsequently annealed at 1000°C for 5 hours in a flow of pure oxygen (O2) at atmospheric pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The substrates were transferred in air into the sputtering chamber for TmIG deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Thin films were deposited at room temperature in flow of Ar (40 sccm) and O2 (20 sccm) with dynamic pressure of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='2 mbar (base pressure is lower than ∼2×10−8 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' To promote the crystallinity, these films were post-annealed (in ex-situ furnace) at 650 °C for 4 hours in a flow of pure O2 at atmospheric pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Further, Pt layer of 6nm thick (unless otherwise stated) was deposited by on-axis magnetron sputtering at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' TmIG film surfaces were cleaned by O2 plasma (\uf07e40 eV) before Pt deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Structural characterization: X-Ray diffraction in symmetric (2\uf071−\uf077) or asymmetric (reciprocal space mapping, RSM) geometry were recorded by Philips X’pert-PRO Empyrean diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For XRD, measurements were performed in Bragg-Brentano reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For RLM, the diffraction along the (6 4 2) plane direction is used, which allows to gain the information about in-plane epitaxy relation along [2 -2 0] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Topography of the substrate and film were recorded by atomic force microscopy (AFM) using a Dimension Icon system with ScanAsyst(Bruker Dimension Icon, Billerica, MA, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Images were collected in tapping mode (in air) using a tip with nominal radius <10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Atomic-scale imaging was performed by cross-sectional scanning transmission electron microscopy (STEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The sample investigated by STEM was prepared by a focused ion beam machine (FEI Helios platform) using a Ga ion beam with an accelerating voltage of first 30 kV to detach the slab, and then of 5 kV to thin it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' STEM characterization was conducted with a Hitachi HF5000 equipped with a cold field emission gun operated at 200 kV and a probe aberration corrector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' High-angle annular dark- field images were acquired with a probe that formed an angle of 30 mrad and a collection angle of 60–300 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' EDX spectra were collected using two detectors from Oxford Instruments and color- coded elemental maps were obtained using the AZtec software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Magnetization measurements were performed by Quantum Design SQUID magnetometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' All electron transport measurements were performed in a home-built set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' AHE, SMR, SOTs and magnetization switching measurements: To perform the electron transport experiments, 5-µm wide and 50-µm long symmetric Hall-crosses were patterned using photo- 9 lithography and Ar-ion milling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For Kerr imaging, the devices were patterned in 10-µm wide and 100- µm long Hall crosses with Au contact pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The anomalous Hall effect and spin Hall magnetoresistance measurements were carried out using a constant dc source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For current-induced switching measurements, current pulses with a duration of 100 \uf06ds were generated by a Keithley 6221 and injected into the Hall bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' After each pulse, a small excitation (100 \uf06dA = 3\uf0b4109 A/m2) current was applied to evaluate and measure the magnetization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' For the harmonic Hall measurements, an ac current source with an amplitude from 1 to 6 mA (root mean square) was injected with a Keithley 6221 current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The first and second harmonic signals were measured using an SR-830 lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' DMI measurements: The Dzyaloshisnkii-Moriya interaction energy was measured by Brillouin light scattering (BLS) using a JRS TFP-2 triple-pass tandem Fabry-Perot interferometer with quarter-wave antireflection optics and linearly-polarized (10 mW laser with 473 nm wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The spectra were recorded in the backscattering geometry at various wave vector orientations, selected by mounting the sample on an angle-controlled sample holder providing a range of 10° to 60° incident angles corresponding to wave vectors, qk = (4π/λ) sinθ, lying in the range 4 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='4 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='/µm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The free spectral range was 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='4 or 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='7 GHz and spectra were recorded with 1024 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' The in-plane magnetic field to pull the magnetization in-plane for the Damon-Eshbach geometry is provided by permanent magnets in order to avoid thermal drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' REFERENCES (1) Dieny, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Prejbeanu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Akerman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Deac, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Pirro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Adelmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Anane, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Chumak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Hirohata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Mangin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Valenzuela, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Onbaşlı, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' d’Aquino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Prenat, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Finocchio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Lopez-Diaz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Chantrell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Chubykalo-Fesenko, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Bortolotti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Opportunities and Challenges for Spintronics in the Microelectronics Industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Nature Electronics 2020 3:8 2020, 3 (8), 446–459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='1038/S41928-020-0461-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' (2) Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Yin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Bai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Zhu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Shi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Cao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Zhao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Spintronics for Energy- Efficient Computing: An Overview and Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Proceedings of the IEEE 2021, 109 (8), 1398– 1417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='org/10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Role of Dimensional Crossover on Spin-Orbit Torque Efficiency in Magnetic Insulator Thin Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Nature communications 2018, 9 (2018), 3612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='1038/s41467-018-06059-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' (14) Vélez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Schaab, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Wörnle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Müller, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Becker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Boventer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Gradauskaite, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Welter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Gutgsell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Nistor, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Gambardella, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' High-Speed Domain Wall Racetracks in a Magnetic Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Nature Communications 2019, 10 (2019), 4750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='org/10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Beran, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Quindeau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Gambardella, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Ono, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Magnetic Droplet Nucleation with a Homochiral Néel Domain Wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' Physical Review B 2017, 95 (22), 220402(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} +page_content='220402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdFJT4oBgHgl3EQfLiwk/content/2301.11469v1.pdf'} diff --git a/TdE0T4oBgHgl3EQf2QI-/vector_store/index.faiss b/TdE0T4oBgHgl3EQf2QI-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..bd750fad3a863aca5d2b2da203be14e3d4502ec9 --- /dev/null +++ b/TdE0T4oBgHgl3EQf2QI-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bf558f84ec0c9a5d4cd2d6f9778c8599b615c4413d9501280f2c3886f41e8360 +size 3670061 diff --git 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Olde Olthof,1, 2, ∗ Stijn R. de Wit,1, ∗ Shu-Ichiro Suzuki,1, 3 +Inanc Adagideli,1, 4, 5 Jason W.A. Robinson,2 and Alexander Brinkman1, † +1MESA+ Institute for Nanotechnology, University of Twente, The Netherlands +2Department of Materials Science & Metallurgy, University of Cambridge, United Kingdom +3Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan +4Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey +5T ¨UB˙ITAK Research Institute for Fundamental Sciences, 41470 Gebze, Turkey +(Dated: January 9, 2023) +Andreev bound states (ABS) occur in Josephson junctions when the total phase of the Andreev and normal +reflections is a multiple of 2π. In ballistic junctions with an applied voltage bias, a quasi-particle undergoes +multiple Andreev reflections before entering the leads, resulting in peaks in the current-voltage I(V ) curve. Here +we present a general model for Josephson junctions with spin-active interlayers i.e., magnetic or topological +materials with broken time-reversal symmetry. We investigate how ABS change the peak positions and shape +of I(V ), which becomes asymmetric for a single incident angle. We show how the angle-resolved I(V ) curve +becomes a spectroscopic tool for the chirality and degeneracy of ABS. +Andreev reflection is the conversion of an electron into a +hole with opposite spin upon reflecting from a superconduc- +tor interface [1]. Andreev bound states (ABS) arise when a +combination of a number of Andreev and normal reflections +fulfills the Bohr-Sommerfeld quantization condition in which +the total phase adds up to multiples of 2π. Renowned exam- +ples include ABS that carry the supercurrent between two su- +perconducting leads across a normal metal [2], the Yu-Shiba- +Rusinov bound states that involve scattering from a magnetic +impurity [3–5], and the Caroli-de Gennes-Matricon bound +state in the core of an Abrikosov vortex [6]. Generally, the +required phase quantization is either fulfilled by incorporating +spin-active scattering with different phases for the reflection +of different spins [7–9], or by picking up a phase difference +due to an anisotropic order parameter in the superconductor +(an unconventional superconductor) [10–12]. +At the surface or interface of an unconventional supercon- +ductor, surface ABS at zero energy (relative to the Fermi en- +ergy) arise when the phase difference between the energy- +dependent Andreev reflection of the electron and hole is π. +This has been measured at the surface of a 45◦ grain bound- +ary junction involving a dx2−y2 cuprate superconductor [12] +and predicted for the surface of a chiral p-wave superconduc- +tor [13]. Surface ABS become (chiral) Majorana bound states +upon lifting the spin degeneracy by breaking time-reversal +symmetry in a topological superconductor [14], either by a +vortex [15] or an external magnetic field [16]. The distin- +guishing feature of a subgap ABS is the zero-bias conduc- +tance peak in the tunneling conductance [10, 17], which can +even become quantized in the Majorana case [14, 18]. +Here, we theoretically study the influence of (chiral) in- +terface ABS on the current-voltage characteristics of Joseph- +son junctions. Besides a zero-voltage supercurrent, Joseph- +son junctions are characterized by a subgap structure in the +∗ These authors contributed equally +† a.brinkman@utwente.nl +finite-bias conductance that arises from multiple Andreev re- +flections (MAR), usually providing peaks at 2∆/n, where ∆ +is the superconducting energy gap, and n is the integer num- +ber of times that the electrons or holes traverse the junction +before entering the leads [19]. If a topological insulator (TI) +interlayer featuring a magnetic (MTI) barrier is used instead +of a normal metal barrier, the subgap state opens an extra +conduction channel and the peaks in a one-dimensional (1D) +S/TI/MTI/TI/S junction are located at ∆/n [20]. +We present a generalized model for two-dimensional (2D) +Josephson junctions consisting of s-wave superconductors (S) +and spin-active layers (X) e.g., a TI, a magnetic TI (MTI) or +ferromagnetic insulator, see Fig. 1a. We structure our quan- +titative calculations around a 2D S/TI/MTI/TI/S junction of +which the S/N/S and 1D S/TI/MTI/TI/S systems are the lim- +iting cases – after which we generalize our methods to non- +topological junctions. +Our findings show a direct link be- +tween broken time-reversal symmetry, the presence of ABS +and asymmetric angle-resolved I(V ) curves. +The approach is as follows: we investigate the existence +and energy dependence of ABS in S/X/X’ and X’/X/S half- +junctions, where X’ contains a potential difference or mag- +netization. We then couple two half-junctions together into a +full S/X/S junction and calculate the I(V ) spectrum. We show +that the features in I(V ) are directly linked to the ABS, and +discuss how angle-resolved MAR can become a spectroscopic +tool for the chiral nature and degeneracy of ABS. Throughout +this paper, the interface normal is along the x-axis and we +use periodic boundary conditions along y (see Fig. 1a). We +assume the junction length to be smaller than the coherence +length and the elastic mean free path, making the transport +coherent and ballistic. +Quasi-particles undergo normal (Andreev) reflection at the +X/X’ (S/X) interface. In the X’/X/S junction in Fig. 1c, we +consider an incoming electron with angle θ consecutively un- +dergoing Andreev (reh), normal (rhh), Andreev (rhe), and +normal (ree) reflection. +Fig. 1b shows the equivalent pro- +cess in the other half-junction. To generalize the reflection +arXiv:2301.02270v1 [cond-mat.supr-con] 5 Jan 2023 + +2 +d +S +X +X’ +X +S +z +x +y +k +V +V+ +- +(a) +X’ +S +X +X’ +X +S +(c) +(b) +(e) +(d) +0 +1 +0 +-1 +0 +1 +0 +-1 +FIG. 1. (a) Schematic illustration of the S/X/S junction setup, where +S is a superconductor and X is a spin-active layer, e.g. +a ferro- +magnetic insulator or a (magnetic) topological insulator. The junc- +tion is modelled as S/X/X’/X/S where Andreev reflection occurs in +the two X regions which are coupled via a scattering region X’ of +width d. (b)-(c) The top view of the S/X/X’/X/S junction split in +two half-junctions S/X/X’ and X’/X/S, where normal (ree, rhh) and +Andreev (reh, rhe) reflection occurs. tan θ = ky/kx for kx, ky +being the components of the plane wave momenta. (d)-(e) In the +case of a topological insulator junction (X=TI) with a magnetic tun- +nel barrier (X’=MTI), the two half-junctions host chiral Majorana +modes of opposite chirality χ±. Their bound state levels E as a +function of the incident angle θ are shown for a (d) MTI/TI/S and +a (e) S/TI/MTI junction. The dashed, dotted, and solid lines corre- +spond to µTI/mz = 0.5, 1, 2, respectively. The other parameters are +mz = 300∆0, µMTI = 0, µTI = µS. +processes and incorporate phase differences due to topology +and/or magnetism, we introduce a so-called reflection asym- +metry phase eiχ := rhh/r∗ +ee as the ratio between the hole-hole +and electron-electron reflection coefficients of X’. We com- +pute rhh and ree, by imposing the continuity of the wave func- +tion across the junction. The spinor part of the wave function +is derived from the Bogoliubov-de Gennes Hamiltonian in the +basis (u↑, u↓, v↓, −v↑)T , +ˆH = +�ˆh(k) +ˆ∆ +ˆ∆∗ +−σyˆh∗(−k)σy +� +, +(1) +where ˆh(k) = f(k) + σzmz − µjσz is the single-particle +Hamiltonian. +The dispersion, f(k), can include a kinetic +term or spin-orbit physics. µj with j = S, X, X’ sets the +chemical potential in the three regions, and mz sets the +induced magnetic gap. +ˆ∆ = σ0∆0, with ∆0 ∈ R is the +s-wave superconducting gap. ∆0 and mz are only nonzero in +their respective regions. The matrices σi for i = 0, x, y, z are +the Pauli matrices. +The Hamiltonian (1) obeys particle-hole symmetry. In the +absence of a magnetic barrier (mz = 0), it is also time- +reversal symmetric. +This means that the system is placed +in symmetry class BDI when both particle-hole and time- +reversal symmetry are present, while it is in class D when +time-reversal symmetry is broken [21]. +Based on the sys- +tem’s symmetries, the topological invariant Q (the number +of symmetry-protected edge states present at the Fermi level) +per spatial dimension can be calculated through the reflec- +tion block of the scattering matrix [22] for the X/X’ inter- +face, ˆr = diag(ree, rhh) ≡ diag(ree, eiχr∗ +ee). +In 2D, the +topological invariant is a winding number, given by Q2D = +1 +2πi +� 2π +0 +dk d +dk log(det ˆr), where det ˆr += +eiχ. Details on +the symmetry classes and calculation of Q2D are provided in +Sec. S1 of the Supplemental Materials [23]. +We compute Q for the case of a topological half-junction +(X = TI) with and without time-reversal symmetry. In time- +reversal symmetric junctions (mz = 0), we find r∗ +ee = rhh, +such that eiχ = 1 and Q2D = 0, implying that the system +is topologically trivial and there are no edge modes. In the +case of broken time-reversal symmetry (mz ̸= 0), we obtain +Q2D = −1 (Q2D = +1) for the S/X/X’ (X’/X/S) half junc- +tion. A topological invariant of ±1 means that a topologically +protected chiral edge mode is present. Importantly, the sign +difference of Q between the two half-junctions indicates op- +posite chirality (winding direction), as illustrated in the inset +of Fig. 1d,e. The protected chiral edge mode in 2D is the +nonzero energy chiral Majorana mode originating from the +localized zero-energy Majorana bound state present in the 1D +channel at the symmetry point θ = 0. +Chiral Majorana modes have been predicted in MTI/S +junctions [15], and their bound state energies EABS(θ) were +previously found as poles in the conduction [24]. We compute +EABS(θ) as the energy when the Bohr-Sommerfeld quantiza- +tion condition, αree + αreh + αrhh + αrhe = 2πn, n ∈ Z, is +satisfied for the reflection coefficients depicted in Fig. 1b,c. +For subgap energies, |E| < ∆0, the quantization condition +can be written in terms of χ as −2 arccos (E/∆0)+χ = 2πn +(Sec. S2 of the Supplemental Materials [23]). +Since +2 arccos (E/∆0) is bound between 0 and 2π, the condition +is met for a nonzero χ. So, the value of χ dictates whether +ABS exist. In time-reversal symmetric systems (for instance, +when X’ is a Fermi surface mismatch barrier), χ = 0 and +no ABS forms. Whereas in time-reversal symmetry breaking +systems, χ is nonzero. The bound state energies vs incident +angle for magnetic S/X/X’ and X’/X/S junctions are shown +in Figs. 1d,e. At θ = 0 (i.e. the 1D limit), the ABS is located +at zero energy and is therefore a Majorana bound state. For +nonzero angles, the ABS moves away from zero energy and +obtains a chirality. +We recall that the two half-junctions +have opposite chirality (Q = ±1), which results in the + +3 +EABS having a different sign for a fixed nonzero value of θ. +Crucially, this means that in the coupled S/X/X’/X/S junction, +for a fixed θ, there are bound states of opposite energy on the +left and right side of the X’ barrier. +To consider the transport in the S/X/X’/X/S junction, we +construct the left and right moving wave functions in the two +X regions, as eigenfunctions of Eq. (1). We then couple the +S/X/X’ and X’/X/S junctions via a scattering region X’, which +is governed by the scattering matrices for electrons and holes +Se = +�r +t +t −r∗t +t∗ +� +, +Sh = +�eiχr∗ +t∗ +t∗ +−e−iχ rt∗ +t +� +, +(2) +where r ≡ ree and t ≡ tee are the electron-electron reflection +and transmission coefficients for the barrier X’ and we +used rhh/r∗ +ee = eiχ. +Two known limits of the scattering +matrices are eiχ = 1 for a S/N/S junction [19] and eiχ = −1 +for a 1D ferromagnetic S/TI/MTI/TI/S junction [20]. +We +consider a potential difference eV in the X’ region, such +that in the MAR picture [19, 20], every time an electron +passes from left to right, crossing X’, its energy increases +by eV , while the hole energy increases when it passes in +the opposite direction. +Consequently, the wave functions +are superpositions of states with energy E + 2neV where +E is the quasi-particle energy and n the number of Andreev +reflections. +The Andreev reflection coefficient an changes +accordingly to an ≡ reh(E + neV ). We note that the choice +of basis results in equal Andreev reflection coefficients an +at the left and right S interface [25], which is crucial for +the MAR calculations. We derive the scattering equations, +generalize the MAR recurrence relations [19] and compute +I(V ) based on the amplitudes of the superimposed wave +functions. This approach is the key technical finding in this +work; details are provided in Sec. S3 of the Supplemental +Materials [23]. +First, we investigate the angle-resolved MAR spectra for +a 2D S/TI/MTI/TI/S Josephson junction (Fig. 2). In a trivial +junction (e.g. S/N/S), there are no states inside the gap, elec- +trons (holes) undergo MAR until they have gained enough en- +ergy to leave the gap at eV = +(−)2∆0 [19]. The presence +of an ABS in the gap gives rise to extra conduction channels +[20]. When the ABS aligns with the ABS on the other side +(eV = 2EABS) [26] or the continuum (eV = ∆0 + |EABS|), +additional features appear in the I(V ) curve. Furthermore, +due to the nature of MAR, higher-order features appear for +successive Andreev reflections. +Generally, features in the +I(V ) curve are expected at eV = (∆0 + |EABS|)/n, with +n ∈ Z, stemming from alignment of the ABS with the con- +tinuum; and at eV = 2EABS/m, for odd m ∈ N, stemming +from alignment of the two ABSs. These alignments are illus- +trated in Fig. 2. We note that the modes with opposite chiral- +ity on the left and right side of the MTI barrier are retained in +the coupled Josephson junction (Sec. S1 of the Supplemental +Materials [23]). The energy asymmetry resulting from these +modes of opposite chirality dictates that m must be odd. This +can be seen by considering an electron initially incoming from +3 +2 +1 +0 +-2 +-1 +0 +1 +2 +(b) +(d) +(c) +(e) +(a) +c +e +d +FIG. 2. (a) An asymmetric I(V ) curve of a S/TI/MTI/TI/S junc- +tion for a single incident angle θ = 0.45π, with corresponding +EABS/∆0 = 0.75. IDC is normalized by I∆ = De∆0/h, for a +transparency D = 0.1. The other parameters are µTI/mz ∼ 0.7 +with mz/∆0 = 300, µS = µTI, µMTI = 0, and the MTI barrier width +is d = 1.5ℏvF /mz. (b) The density of states of the two supercon- +ductors including the subgap ABS. The filled (empty) ABS positions +correspond to a positive (negative) incident angle. The bias voltage +eV shifts the density of states of the right superconductor relatively +downwards. When considering only the filled ABS for θ > 0, trans- +port occurs with the alignment of (c) ABS-ABS, (d) ABS-continuum +and (e) continuum-continuum. +the left subgap state. For it to scatter to the empty subgap state +on the other side of the barrier it can only traverse the system +an odd number of times, gaining an odd multiple of eV in +energy. +The asymmetry of the bound state energies due to the op- +posite chirality for the left and right half-junctions in Fig. 1d,e +gives rise to the asymmetric I(V ) curve in Fig. 2. For a fixed +θ, a positive bias voltage eV aligns different levels (associated +with higher order MAR resonances) than a negative bias. In +the latter case, the density of states in Fig. 2c-e shift in the +opposite direction, the eV = 2EABS levels never align and the +associated features in I(V ) are absent for eV < 0. The I(V ) +curve for −θ is the vertical mirror image of Fig. 2. +The value eiχ in the scattering matrices (2) indicates + +4 +3 +2 +1 +0 +-1 +-2 +-2.5 +-1.5 +-0.5 +1.5 +0.5 +1 +2 +2.5 +0 +1 +0 +0 +1 +2 +6 +4 +2 +0 +1 +0.5 +0.8 +0.9 +FIG. 3. +The angle-resolved asymmetric I(V ) curves for a S/TI/MTI/TI/S junction normalized by I∆ = De∆0/h. Each curve corresponds +to a single incident angle θ ∈ (0, π/2), with corresponding bound state energy EABS(θ). The black dashed line is the angle average obtained +in the lateral 2D junction limit. The parameters are µTI/mz ∼ 0.7, with mz/∆0 = 300, µS = µTI, µMTI = 0, the transparency ranges +from D = 0.005 − 0.2, and the MTI barrier width is d = 1.5ℏvF /mz. Inset: Nanowire limit. Each I(V ) curve corresponds to a single +(normalized) quantized py = sin θ channel where the current is estimated by I(−θ) + I(θ). The graph is identical for ±eV/∆0. +whether ABS are present; eiχ = 1 means there are no ABS +and results in a trivial I(V ) curve (meaning no additional sub- +gap features), whereas eiχ = −1 indicates a non-trivial I(V ) +curve with asymmetric peaks, as shown in Fig. 2a. By chang- +ing θ, we smoothly transition from the trivial to non-trivial +regime [27]. +Fig. 3 shows the I(V ) spectra of a S/TI/MTI/TI/S junction +for positive θ ranging from 0 to π +2 , with corresponding posi- +tive EABS [26]. Two limiting cases are the red curve with the +ABS near the continuum (EABS = ∆0 and eiχ = 1) for which +we observe the strong resonance step near 2∆0 as in the triv- +ial S/N/S case [19]; and the navy curve describing ABS posi- +tions in the middle of the gap (EABS = 0 and eiχ = −1) for +which we obtain the topological 1D S/TI/S limit [20]. The +intermediate curves look strikingly different. +For negative +voltages, there is a gradual transition from one limit to the +other, whereas the positive eV -side features the distinct addi- +tional 2EABS/m peaks for odd m due to the asymmetric ABS. +Again, for negative incident angles, we obtain the vertical mir- +ror image of Fig. 3. +We now consider the consequences of the presence of ABS +and their effect on the I(V ) spectra in realistic experimen- +tal setups, e.g. +lateral junctions and nanowires. +Special- +ized setups for measuring particular Andreev-reflection an- +gles [28, 29] could reveal asymmetry in I(V ). In 2D samples +(lateral junctions on thin films), however, one generally mea- +sures the angle-averaged I(V ) and the asymmetric features +disappear (see the black dashed line in Fig. 3). Throughout +this work, we assumed an infinite junction in the y-direction, +meaning that there is a continuum of py channels and every +incident angle θ is allowed. To apply the MAR scheme to +nanowires [16, 30, 31], we instead consider a cylindrical ge- +ometry, where, due to the confinement, we obtain a set of al- +lowed quantized py values. Per confined py value, only the +corresponding θ and −θ channels are present, and we estimate +the current through the nanowire by ∼ I(θ)+I(−θ). Contrary +to the lateral junction, the subgap resonances at eV = 2EABS +are retained in the nanowire (see the inset of Fig. 3). +The proposed MAR scheme generalizes to non-topological +Josephson junctions. Subgap states are present in any s-wave +Josephson junction with broken time-reversal symmetry, but +the (a)symmetric nature of the ABS is not universal. In topo- +logically trivial systems the ABS are degenerate (on both sides +of the barrier), and thus no energy asymmetry is present in the +junction. The corresponding angle-resolved I(V ) curves are +therefore symmetric, as seen in, e.g. ferromagnetic Josephson +junctions [7–9]. In topological systems, a single mode of a +separated pair of chiral Majorana modes is confined topolog- +ically to either the top or bottom surface of the TI. Since the +considered junction is located on the top surface (see Fig. 1a), +a single subgap state is present on each side of the barrier, +giving rise to the asymmetric I(V ) curves. +The MAR scheme can also be generalized to Josephson +junctions with unconventional superconductors, which are +characterized by an anisotropic order parameter with a phase +– e.g. px-wave as in the Kitaev chain [32] and d-wave high-Tc +cuprates [11, 12]. Unconventional superconductors can have +a range of exotic properties such as intrinsic chirality – which +ensures the existence of chiral bound states – or intrinsically +broken time-reversal symmetry – which eliminates the +need for a magnetic barrier. To implement unconventional +superconductivity in our model, we recall that the choice of +basis is crucial to get equal Andreev reflection coefficients +an at the left and right S interface. +One can construct a +unitary transformation to transfer the phase from the order +parameter to eiχ such that the an remains equal at both S +interfaces and the MAR scheme is still valid, see Sec. S4 of +the Supplemental Materials [23] for details. +In conclusion, we have investigated the emergence of (chi- +ral) ABS in 2D Josephson junctions with magnetic and/or +topological interlayers and studied their effect on calculated + +5 +I(V ) spectra. +Any Josephson junction with broken time- +reversal symmetry features ABS in the density of states. +When these align with ABS on the other side or the contin- +uum, a conduction channel opens which appears as a peak in +the I(V ) curve. This directly links the I(V ) curve to the ABS +energies. +In topological systems, a single topologically protected +ABS is present, which obtains a chirality (winding number) +in 2D. The S/TI/MTI/TI/S junction features bound states of +opposite chirality on either side of the MTI barrier and the +corresponding bound state energies are inverted. This energy +asymmetry is responsible for the asymmetric I(V ) curve. We +have investigated two limits of the S/TI/MTI/TI/S I(V ) curve. +In lateral 2D junctions where one experimentally obtains an +angle-averaged I(V ) curve, the asymmetry disappears but +non-trivial steps related to the presence of subgap states re- +main. In the nanowire limit, the distinct peaks, which are an +artefact of the present asymmetric chiral Majorana modes, are +robust for quantized py channels. +The concept of non-reciprocity has regained interest in the +field of superconductivity as a potential probe for broken sym- +metries [33]. In the S/TI/MTI/TI/S case, the non-reciprocity +arises from the energy asymmetry as a function of θ between +the two emergent bound states of opposite chirality on either +end of the MTI barrier. Particle-hole symmetry is not violated +in this case since the energy of the subgap state also inverts as +θ is inverted. We propose angle-resolved ABS spectroscopy +to resolve the predicted asymmetry. +L.A.B.O.O. and J.W.A.R. were supported by the EPSRC +through the Core-to-Core International Network “Oxide Su- +perspin” (EP/P026311/1) and the “Superconducting Spintron- +ics” Programme Grant (EP/N017242/1). L.A.B.O.O. also ac- +knowledges support from the Doctoral Training Partnership +Grant (EP/N509620/1). S.-I. S. is supported by JSPS Postdoc- +toral Fellowship for Overseas Researchers and a Grant-in-Aid +for JSPS Fellows (JSPS KAKENHI Grant No. JP19J02005). +We acknowledge useful discussions with Alexander Golubov. +[1] A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964). +[2] I. O. Kulik, Sov. Phys. JETP 30, 944 (1970). +[3] L. Yu, Acta Phys. Sin. 21, 75 (1965). +[4] H. Shiba, Prog. Theor. Phys 40, 435 (1968). +[5] A. Rusinov, JETP Lett. 9, 146 (1969). +[6] C. Caroli, P. De Gennes, and J. Matricon, Phys. Lett 9, 307 +(1964). +[7] M. Eschrig, Phil. Trans. R. Soc. A 376 (2018). +[8] D. Beckmann, F. H¨ubler, M. J. Wolf, and H. v. L¨ohneysen, Phil. +Trans. R. Soc. A 376 (2018). +[9] M. Eschrig, J. Kopu, J. C. Cuevas, and G. Sch¨on, Phys. Rev. +Lett. 90, 137003 (2003). +[10] C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994). +[11] Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995). +[12] S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000). +[13] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000). +[14] C. W. J. Beenakker, Annu. Rev. Condens. Matter Phys. 4, 113 +(2013). +[15] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). +[16] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. +Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012). +[17] Y. Tanaka, S. Kashiwaya, and T. Yokoyama, Phys. Rev. B 71, +094513 (2005). +[18] K. T. Law, P. A. Lee, +and T. K. Ng, Phys. Rev. Lett. 103, +237001 (2009). +[19] D. Averin and A. Bardas, Phys. Rev. Lett. 75, 1831 (1995). +[20] D. M. Badiane, M. Houzet, and J. S. Meyer, Phys. Rev. Lett. +107, 177002 (2011). +[21] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, +New J. Phys. 12, 065010 (2010). +[22] I. C. Fulga, F. Hassler, and A. R. Akhmerov, Phys. Rev. B 85, +165409 (2012). +[23] See Supplemental Materials at [url] for derivations and details +of the model. +[24] Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev. Lett. 103, +107002 (2009). +[25] In our chosen basis, an is equal to the reflection coefficients +reh = rhe = (E − +� +E2 − ∆2 +0)/(∆0), see Sec. S2 of the +Supplemental Materials [23]. +[26] The bound state energy EABS is defined in the MTI/TI/S half- +junction. In the full S/TI/MTI/TI/S junction EABS is positive for +θ > 0. +[27] The value at which the transition happens depends on the bar- +rier strength µTI/mz. In the strong barrier limit (low µTI/mz), +the system resembles two mostly isolated half-junctions with a +large non-trivial regime, whereas for a weak barrier the non- +trivial regime is confined to θ = 0. +[28] G. Xiao and D. Xue, Appl. Phys. Lett. 60, 504 (1992). +[29] N. A. Mortensen, K. Flensberg, and A.-P. Jauho, Phys. Rev. B +59, 10176 (1999). +[30] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, +177002 (2010). +[31] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and +H. Q. Xu, Nano Letters 12, 6414 (2012). +[32] A. Y. Kitaev, Phys.-Uspekhi 44, 131 (2001). +[33] Y. Tokura and N. Nagaosa, Nature Comm. 9, 3740 (2018). + diff --git a/W9E0T4oBgHgl3EQfVwD1/content/tmp_files/load_file.txt b/W9E0T4oBgHgl3EQfVwD1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..843dc82180c060345ebf0980ed50396c7efd4fc1 --- /dev/null +++ b/W9E0T4oBgHgl3EQfVwD1/content/tmp_files/load_file.txt @@ -0,0 +1,466 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf,len=465 +page_content='Multiple Andreev reflections in two-dimensional Josephson junctions with broken time-reversal symmetry Linde A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Olde Olthof,1, 2, ∗ Stijn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' de Wit,1, ∗ Shu-Ichiro Suzuki,1, 3 Inanc Adagideli,1, 4, 5 Jason W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Robinson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='2 and Alexander Brinkman1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' † 1MESA+ Institute for Nanotechnology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' University of Twente,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The Netherlands 2Department of Materials Science & Metallurgy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' United Kingdom 3Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Nagoya University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Nagoya 464-8603,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Japan 4Faculty of Engineering and Natural Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Sabanci University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Istanbul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Turkey 5T ¨UB˙ITAK Research Institute for Fundamental Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 41470 Gebze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Turkey (Dated: January 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2023) Andreev bound states (ABS) occur in Josephson junctions when the total phase of the Andreev and normal reflections is a multiple of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In ballistic junctions with an applied voltage bias, a quasi-particle undergoes multiple Andreev reflections before entering the leads, resulting in peaks in the current-voltage I(V ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Here we present a general model for Josephson junctions with spin-active interlayers i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=', magnetic or topological materials with broken time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We investigate how ABS change the peak positions and shape of I(V ), which becomes asymmetric for a single incident angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We show how the angle-resolved I(V ) curve becomes a spectroscopic tool for the chirality and degeneracy of ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Andreev reflection is the conversion of an electron into a hole with opposite spin upon reflecting from a superconduc- tor interface [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Andreev bound states (ABS) arise when a combination of a number of Andreev and normal reflections fulfills the Bohr-Sommerfeld quantization condition in which the total phase adds up to multiples of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Renowned exam- ples include ABS that carry the supercurrent between two su- perconducting leads across a normal metal [2], the Yu-Shiba- Rusinov bound states that involve scattering from a magnetic impurity [3–5], and the Caroli-de Gennes-Matricon bound state in the core of an Abrikosov vortex [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Generally, the required phase quantization is either fulfilled by incorporating spin-active scattering with different phases for the reflection of different spins [7–9], or by picking up a phase difference due to an anisotropic order parameter in the superconductor (an unconventional superconductor) [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' At the surface or interface of an unconventional supercon- ductor, surface ABS at zero energy (relative to the Fermi en- ergy) arise when the phase difference between the energy- dependent Andreev reflection of the electron and hole is π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' This has been measured at the surface of a 45◦ grain bound- ary junction involving a dx2−y2 cuprate superconductor [12] and predicted for the surface of a chiral p-wave superconduc- tor [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Surface ABS become (chiral) Majorana bound states upon lifting the spin degeneracy by breaking time-reversal symmetry in a topological superconductor [14], either by a vortex [15] or an external magnetic field [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The distin- guishing feature of a subgap ABS is the zero-bias conduc- tance peak in the tunneling conductance [10, 17], which can even become quantized in the Majorana case [14, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Here, we theoretically study the influence of (chiral) in- terface ABS on the current-voltage characteristics of Joseph- son junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Besides a zero-voltage supercurrent, Joseph- son junctions are characterized by a subgap structure in the ∗ These authors contributed equally † a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='brinkman@utwente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='nl finite-bias conductance that arises from multiple Andreev re- flections (MAR), usually providing peaks at 2∆/n, where ∆ is the superconducting energy gap, and n is the integer num- ber of times that the electrons or holes traverse the junction before entering the leads [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' If a topological insulator (TI) interlayer featuring a magnetic (MTI) barrier is used instead of a normal metal barrier, the subgap state opens an extra conduction channel and the peaks in a one-dimensional (1D) S/TI/MTI/TI/S junction are located at ∆/n [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We present a generalized model for two-dimensional (2D) Josephson junctions consisting of s-wave superconductors (S) and spin-active layers (X) e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=', a TI, a magnetic TI (MTI) or ferromagnetic insulator, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We structure our quan- titative calculations around a 2D S/TI/MTI/TI/S junction of which the S/N/S and 1D S/TI/MTI/TI/S systems are the lim- iting cases – after which we generalize our methods to non- topological junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Our findings show a direct link be- tween broken time-reversal symmetry, the presence of ABS and asymmetric angle-resolved I(V ) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The approach is as follows: we investigate the existence and energy dependence of ABS in S/X/X’ and X’/X/S half- junctions, where X’ contains a potential difference or mag- netization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We then couple two half-junctions together into a full S/X/S junction and calculate the I(V ) spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We show that the features in I(V ) are directly linked to the ABS, and discuss how angle-resolved MAR can become a spectroscopic tool for the chiral nature and degeneracy of ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Throughout this paper, the interface normal is along the x-axis and we use periodic boundary conditions along y (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We assume the junction length to be smaller than the coherence length and the elastic mean free path, making the transport coherent and ballistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Quasi-particles undergo normal (Andreev) reflection at the X/X’ (S/X) interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In the X’/X/S junction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1c, we consider an incoming electron with angle θ consecutively un- dergoing Andreev (reh), normal (rhh), Andreev (rhe), and normal (ree) reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1b shows the equivalent pro- cess in the other half-junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' To generalize the reflection arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='02270v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='supr-con] 5 Jan 2023 2 d S X X’ X S z x y k V V+ (a) X’ S X X’ X S (c) (b) (e) (d) 0 1 0 1 0 1 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' (a) Schematic illustration of the S/X/S junction setup, where S is a superconductor and X is a spin-active layer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' a ferro- magnetic insulator or a (magnetic) topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The junc- tion is modelled as S/X/X’/X/S where Andreev reflection occurs in the two X regions which are coupled via a scattering region X’ of width d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' (b)-(c) The top view of the S/X/X’/X/S junction split in two half-junctions S/X/X’ and X’/X/S, where normal (ree, rhh) and Andreev (reh, rhe) reflection occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' tan θ = ky/kx for kx, ky being the components of the plane wave momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' (d)-(e) In the case of a topological insulator junction (X=TI) with a magnetic tun- nel barrier (X’=MTI), the two half-junctions host chiral Majorana modes of opposite chirality χ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Their bound state levels E as a function of the incident angle θ are shown for a (d) MTI/TI/S and a (e) S/TI/MTI junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The dashed, dotted, and solid lines corre- spond to µTI/mz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5, 1, 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The other parameters are mz = 300∆0, µMTI = 0, µTI = µS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' processes and incorporate phase differences due to topology and/or magnetism, we introduce a so-called reflection asym- metry phase eiχ := rhh/r∗ ee as the ratio between the hole-hole and electron-electron reflection coefficients of X’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We com- pute rhh and ree, by imposing the continuity of the wave func- tion across the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The spinor part of the wave function is derived from the Bogoliubov-de Gennes Hamiltonian in the basis (u↑, u↓, v↓, −v↑)T , ˆH = �ˆh(k) ˆ∆ ˆ∆∗ −σyˆh∗(−k)σy � , (1) where ˆh(k) = f(k) + σzmz − µjσz is the single-particle Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The dispersion, f(k), can include a kinetic term or spin-orbit physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' µj with j = S, X, X’ sets the chemical potential in the three regions, and mz sets the induced magnetic gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' ˆ∆ = σ0∆0, with ∆0 ∈ R is the s-wave superconducting gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' ∆0 and mz are only nonzero in their respective regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The matrices σi for i = 0, x, y, z are the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The Hamiltonian (1) obeys particle-hole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In the absence of a magnetic barrier (mz = 0), it is also time- reversal symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' This means that the system is placed in symmetry class BDI when both particle-hole and time- reversal symmetry are present, while it is in class D when time-reversal symmetry is broken [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Based on the sys- tem’s symmetries, the topological invariant Q (the number of symmetry-protected edge states present at the Fermi level) per spatial dimension can be calculated through the reflec- tion block of the scattering matrix [22] for the X/X’ inter- face, ˆr = diag(ree, rhh) ≡ diag(ree, eiχr∗ ee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In 2D, the topological invariant is a winding number, given by Q2D = 1 2πi � 2π 0 dk d dk log(det ˆr), where det ˆr = eiχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Details on the symmetry classes and calculation of Q2D are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S1 of the Supplemental Materials [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We compute Q for the case of a topological half-junction (X = TI) with and without time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In time- reversal symmetric junctions (mz = 0), we find r∗ ee = rhh, such that eiχ = 1 and Q2D = 0, implying that the system is topologically trivial and there are no edge modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In the case of broken time-reversal symmetry (mz ̸= 0), we obtain Q2D = −1 (Q2D = +1) for the S/X/X’ (X’/X/S) half junc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' A topological invariant of ±1 means that a topologically protected chiral edge mode is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Importantly, the sign difference of Q between the two half-junctions indicates op- posite chirality (winding direction), as illustrated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1d,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The protected chiral edge mode in 2D is the nonzero energy chiral Majorana mode originating from the localized zero-energy Majorana bound state present in the 1D channel at the symmetry point θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Chiral Majorana modes have been predicted in MTI/S junctions [15], and their bound state energies EABS(θ) were previously found as poles in the conduction [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We compute EABS(θ) as the energy when the Bohr-Sommerfeld quantiza- tion condition, αree + αreh + αrhh + αrhe = 2πn, n ∈ Z, is satisfied for the reflection coefficients depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' For subgap energies, |E| < ∆0, the quantization condition can be written in terms of χ as −2 arccos (E/∆0)+χ = 2πn (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S2 of the Supplemental Materials [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Since 2 arccos (E/∆0) is bound between 0 and 2π, the condition is met for a nonzero χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' So, the value of χ dictates whether ABS exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In time-reversal symmetric systems (for instance, when X’ is a Fermi surface mismatch barrier), χ = 0 and no ABS forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Whereas in time-reversal symmetry breaking systems, χ is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The bound state energies vs incident angle for magnetic S/X/X’ and X’/X/S junctions are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1d,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' At θ = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' the 1D limit), the ABS is located at zero energy and is therefore a Majorana bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' For nonzero angles, the ABS moves away from zero energy and obtains a chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We recall that the two half-junctions have opposite chirality (Q = ±1), which results in the 3 EABS having a different sign for a fixed nonzero value of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Crucially, this means that in the coupled S/X/X’/X/S junction, for a fixed θ, there are bound states of opposite energy on the left and right side of the X’ barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' To consider the transport in the S/X/X’/X/S junction, we construct the left and right moving wave functions in the two X regions, as eigenfunctions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We then couple the S/X/X’ and X’/X/S junctions via a scattering region X’, which is governed by the scattering matrices for electrons and holes Se = �r t t −r∗t t∗ � , Sh = �eiχr∗ t∗ t∗ −e−iχ rt∗ t � , (2) where r ≡ ree and t ≡ tee are the electron-electron reflection and transmission coefficients for the barrier X’ and we used rhh/r∗ ee = eiχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Two known limits of the scattering matrices are eiχ = 1 for a S/N/S junction [19] and eiχ = −1 for a 1D ferromagnetic S/TI/MTI/TI/S junction [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We consider a potential difference eV in the X’ region, such that in the MAR picture [19, 20], every time an electron passes from left to right, crossing X’, its energy increases by eV , while the hole energy increases when it passes in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Consequently, the wave functions are superpositions of states with energy E + 2neV where E is the quasi-particle energy and n the number of Andreev reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The Andreev reflection coefficient an changes accordingly to an ≡ reh(E + neV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We note that the choice of basis results in equal Andreev reflection coefficients an at the left and right S interface [25], which is crucial for the MAR calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We derive the scattering equations, generalize the MAR recurrence relations [19] and compute I(V ) based on the amplitudes of the superimposed wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' This approach is the key technical finding in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' details are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S3 of the Supplemental Materials [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' First, we investigate the angle-resolved MAR spectra for a 2D S/TI/MTI/TI/S Josephson junction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In a trivial junction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S/N/S), there are no states inside the gap, elec- trons (holes) undergo MAR until they have gained enough en- ergy to leave the gap at eV = +(−)2∆0 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The presence of an ABS in the gap gives rise to extra conduction channels [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' When the ABS aligns with the ABS on the other side (eV = 2EABS) [26] or the continuum (eV = ∆0 + |EABS|), additional features appear in the I(V ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Furthermore, due to the nature of MAR, higher-order features appear for successive Andreev reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Generally, features in the I(V ) curve are expected at eV = (∆0 + |EABS|)/n, with n ∈ Z, stemming from alignment of the ABS with the con- tinuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' and at eV = 2EABS/m, for odd m ∈ N, stemming from alignment of the two ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' These alignments are illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We note that the modes with opposite chiral- ity on the left and right side of the MTI barrier are retained in the coupled Josephson junction (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S1 of the Supplemental Materials [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The energy asymmetry resulting from these modes of opposite chirality dictates that m must be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' This can be seen by considering an electron initially incoming from 3 2 1 0 2 1 0 1 2 (b) (d) (c) (e) (a) c e d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' (a) An asymmetric I(V ) curve of a S/TI/MTI/TI/S junc- tion for a single incident angle θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='45π, with corresponding EABS/∆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' IDC is normalized by I∆ = De∆0/h, for a transparency D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The other parameters are µTI/mz ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='7 with mz/∆0 = 300, µS = µTI, µMTI = 0, and the MTI barrier width is d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5ℏvF /mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' (b) The density of states of the two supercon- ductors including the subgap ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The filled (empty) ABS positions correspond to a positive (negative) incident angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The bias voltage eV shifts the density of states of the right superconductor relatively downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' When considering only the filled ABS for θ > 0, trans- port occurs with the alignment of (c) ABS-ABS, (d) ABS-continuum and (e) continuum-continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' the left subgap state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' For it to scatter to the empty subgap state on the other side of the barrier it can only traverse the system an odd number of times, gaining an odd multiple of eV in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The asymmetry of the bound state energies due to the op- posite chirality for the left and right half-junctions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1d,e gives rise to the asymmetric I(V ) curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' For a fixed θ, a positive bias voltage eV aligns different levels (associated with higher order MAR resonances) than a negative bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In the latter case, the density of states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2c-e shift in the opposite direction, the eV = 2EABS levels never align and the associated features in I(V ) are absent for eV < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The I(V ) curve for −θ is the vertical mirror image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The value eiχ in the scattering matrices (2) indicates 4 3 2 1 0 1 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 1 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 0 1 0 0 1 2 6 4 2 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The angle-resolved asymmetric I(V ) curves for a S/TI/MTI/TI/S junction normalized by I∆ = De∆0/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Each curve corresponds to a single incident angle θ ∈ (0, π/2), with corresponding bound state energy EABS(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The black dashed line is the angle average obtained in the lateral 2D junction limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The parameters are µTI/mz ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='7, with mz/∆0 = 300, µS = µTI, µMTI = 0, the transparency ranges from D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='005 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='2, and the MTI barrier width is d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='5ℏvF /mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Inset: Nanowire limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Each I(V ) curve corresponds to a single (normalized) quantized py = sin θ channel where the current is estimated by I(−θ) + I(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The graph is identical for ±eV/∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' whether ABS are present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' eiχ = 1 means there are no ABS and results in a trivial I(V ) curve (meaning no additional sub- gap features), whereas eiχ = −1 indicates a non-trivial I(V ) curve with asymmetric peaks, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' By chang- ing θ, we smoothly transition from the trivial to non-trivial regime [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 3 shows the I(V ) spectra of a S/TI/MTI/TI/S junction for positive θ ranging from 0 to π 2 , with corresponding posi- tive EABS [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Two limiting cases are the red curve with the ABS near the continuum (EABS = ∆0 and eiχ = 1) for which we observe the strong resonance step near 2∆0 as in the triv- ial S/N/S case [19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' and the navy curve describing ABS posi- tions in the middle of the gap (EABS = 0 and eiχ = −1) for which we obtain the topological 1D S/TI/S limit [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The intermediate curves look strikingly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' For negative voltages, there is a gradual transition from one limit to the other, whereas the positive eV -side features the distinct addi- tional 2EABS/m peaks for odd m due to the asymmetric ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Again, for negative incident angles, we obtain the vertical mir- ror image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We now consider the consequences of the presence of ABS and their effect on the I(V ) spectra in realistic experimen- tal setups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' lateral junctions and nanowires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Special- ized setups for measuring particular Andreev-reflection an- gles [28, 29] could reveal asymmetry in I(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In 2D samples (lateral junctions on thin films), however, one generally mea- sures the angle-averaged I(V ) and the asymmetric features disappear (see the black dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Throughout this work, we assumed an infinite junction in the y-direction, meaning that there is a continuum of py channels and every incident angle θ is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' To apply the MAR scheme to nanowires [16, 30, 31], we instead consider a cylindrical ge- ometry, where, due to the confinement, we obtain a set of al- lowed quantized py values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Per confined py value, only the corresponding θ and −θ channels are present, and we estimate the current through the nanowire by ∼ I(θ)+I(−θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Contrary to the lateral junction, the subgap resonances at eV = 2EABS are retained in the nanowire (see the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The proposed MAR scheme generalizes to non-topological Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Subgap states are present in any s-wave Josephson junction with broken time-reversal symmetry, but the (a)symmetric nature of the ABS is not universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In topo- logically trivial systems the ABS are degenerate (on both sides of the barrier), and thus no energy asymmetry is present in the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The corresponding angle-resolved I(V ) curves are therefore symmetric, as seen in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' ferromagnetic Josephson junctions [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In topological systems, a single mode of a separated pair of chiral Majorana modes is confined topolog- ically to either the top or bottom surface of the TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Since the considered junction is located on the top surface (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' 1a), a single subgap state is present on each side of the barrier, giving rise to the asymmetric I(V ) curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The MAR scheme can also be generalized to Josephson junctions with unconventional superconductors, which are characterized by an anisotropic order parameter with a phase – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' px-wave as in the Kitaev chain [32] and d-wave high-Tc cuprates [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Unconventional superconductors can have a range of exotic properties such as intrinsic chirality – which ensures the existence of chiral bound states – or intrinsically broken time-reversal symmetry – which eliminates the need for a magnetic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' To implement unconventional superconductivity in our model, we recall that the choice of basis is crucial to get equal Andreev reflection coefficients an at the left and right S interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' One can construct a unitary transformation to transfer the phase from the order parameter to eiχ such that the an remains equal at both S interfaces and the MAR scheme is still valid, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S4 of the Supplemental Materials [23] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In conclusion, we have investigated the emergence of (chi- ral) ABS in 2D Josephson junctions with magnetic and/or topological interlayers and studied their effect on calculated 5 I(V ) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Any Josephson junction with broken time- reversal symmetry features ABS in the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' When these align with ABS on the other side or the contin- uum, a conduction channel opens which appears as a peak in the I(V ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' This directly links the I(V ) curve to the ABS energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In topological systems, a single topologically protected ABS is present, which obtains a chirality (winding number) in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The S/TI/MTI/TI/S junction features bound states of opposite chirality on either side of the MTI barrier and the corresponding bound state energies are inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' This energy asymmetry is responsible for the asymmetric I(V ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We have investigated two limits of the S/TI/MTI/TI/S I(V ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In lateral 2D junctions where one experimentally obtains an angle-averaged I(V ) curve, the asymmetry disappears but non-trivial steps related to the presence of subgap states re- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In the nanowire limit, the distinct peaks, which are an artefact of the present asymmetric chiral Majorana modes, are robust for quantized py channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' The concept of non-reciprocity has regained interest in the field of superconductivity as a potential probe for broken sym- metries [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' In the S/TI/MTI/TI/S case, the non-reciprocity arises from the energy asymmetry as a function of θ between the two emergent bound states of opposite chirality on either end of the MTI barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Particle-hole symmetry is not violated in this case since the energy of the subgap state also inverts as θ is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We propose angle-resolved ABS spectroscopy to resolve the predicted asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' were supported by the EPSRC through the Core-to-Core International Network “Oxide Su- perspin” (EP/P026311/1) and the “Superconducting Spintron- ics” Programme Grant (EP/N017242/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' also ac- knowledges support from the Doctoral Training Partnership Grant (EP/N509620/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' is supported by JSPS Postdoc- toral Fellowship for Overseas Researchers and a Grant-in-Aid for JSPS Fellows (JSPS KAKENHI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' JP19J02005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' We acknowledge useful discussions with Alexander Golubov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Andreev, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' JETP 19, 1228 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E0T4oBgHgl3EQfVwD1/content/2301.02270v1.pdf'} +page_content=' Kulik, 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BERG,3 ANNE JASKOT,5 DANIEL SCHAERER,4 GÁBOR WORSECK,6 RICARDO O. AMORÍN,7 HAKIM ATEK,8 +MATTHEW HAYES,9 ZHIYUAN JI,10 GÖRAN ÖSTLIN,9 ALBERTO SALDANA-LOPEZ,4 AND TRINH THUAN11 +1Center for Astrophysical Sciences, Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA +2Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +3Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA +4Department of Astronomy, University of Geneva, 51 Chemin Pegasi, 1290 Versoix, Switzerland +5Department of Astronomy, Williams College, Williamstown, MA 01267, United States +6Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany +7Instituto de Investigacion Multidisciplinar en Ciencia y Tecnologia, Universidad de La Serena, Raul Bitran 1305, La Serena, Chile +8Institut d’astrophysique de Paris, CNRS, Sorbonne Université, 98bis Boulevrad Arago, F-75014, Paris, France +9Department of Astronomy, Oskar Klein Centre; Stockholm University; SE-106 91 Stockholm, Sweden +10University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA +11Astronomy Department, University of Virginia, Charlottesville, VA 22904, USA +Submitted to AASJournal ApJ +ABSTRACT +The Mg II λλ2796, 2803 doublet has been suggested to be a useful indirect indicator for the escape of Lyα and +Lyman continuum (LyC) photons in local star-forming galaxies. However, studies to date have focused on small +samples of galaxies with strong Mg II or strong LyC emission. Here we present the first study of Mg II probing +a large dynamic range of galaxy properties, using newly obtained high signal-to-noise, moderate-resolution +spectra of Mg II for a sample of 34 galaxies selected from the Low-redshift Lyman Continuum Survey. We +show that the galaxies in our sample have Mg II profiles ranging from strong emission to P-Cygni profiles, and +to pure absorption. We find there is a significant trend (with a possibility of spurious correlations of ∼ 2%) +that galaxies detected as strong LyC Emitters (LCEs) also show larger equivalent widths of Mg II emission, +and non-LCEs tend to show evidence of more scattering and absorption features in Mg II. We then find Mg II +strongly correlates with Lyα in both equivalent width and escape fraction, regardless of whether the emission +or absorption dominates the Mg II profiles. Furthermore, we present that, for galaxies categorized as Mg II +emitters (MgE), one can adopt the information of Mg II, metallicity, and dust to estimate the escape fraction of +LyC within a factor of ∼ 3. These findings confirm that Mg II lines can be used as a tool to select galaxies as +LCEs and to serve as an indirect indicator for the escape of Lyα and LyC. +Keywords: Galaxy evolution (1052), Galaxy kinematics and dynamics(602), Ultraviolet astronomy (1736), +Galaxy spectroscopy (2171) +1. INTRODUCTION +In the last decades, considerable observational efforts have +been directed towards studying the escape of Lyman contin- +uum (LyC) photons from galaxies and attempting to explain +Corresponding author: Xinfeng Xu +xinfeng@jhu.edu +the last phase transition of the universe, i.e., Cosmic Reion- +ization. Various publications point out that star-forming (SF) +galaxies can be responsible for the epoch of reionization +(EoR). These include studies of low-redshift galaxies (z ≲ +1, e.g., Heckman et al. 2001; Bergvall et al. 2006; Leitet +et al. 2013; Borthakur et al. 2014; Leitherer et al. 2016; Izo- +tov et al. 2016a;b; Puschnig et al. 2017; Izotov et al. 2018a;b; +Wang et al. 2019; 2021; Chisholm et al. 2020; Izotov et al. +arXiv:2301.04087v1 [astro-ph.GA] 10 Jan 2023 + +2 +XU ET AL. +2021; Chisholm et al. 2022; Flury et al. 2022a;b; Izotov +et al. 2022; Marques-Chaves et al. 2022a; Saldana-Lopez +et al. 2022; Xu et al. 2022), and moderate- to high-redshifted +galaxies (z ∼ 2 – 4, e.g., Robertson et al. 2015; Vanzella et al. +2016; de Barros et al. 2016; Shapley et al. 2016; Bian et al. +2017; Marchi et al. 2017; 2018; Steidel et al. 2018; Vanzella +et al. 2018; Fletcher et al. 2019; Rivera-Thorsen et al. 2019; Ji +et al. 2020; Mestri´c et al. 2020; Vielfaure et al. 2020; Naidu +et al. 2022; Begley et al. 2022; Rivera-Thorsen et al. 2022; +Marques-Chaves et al. 2022b). These galaxies are proposed +to be analogs of high-redshift (z ∼ 6 – 8) ones at EoR (e.g., +Schaerer et al. 2016; Boyett et al. 2022). +Due to the attenuation by Lyman limit systems (LLS) +and/or neutral intergalactic medium (IGM), it is challenging +to detect LyC at z ≳ 5 (Inoue et al. 2014; Worseck et al. 2014; +Becker et al. 2021; Bosman et al. 2022). Therefore, various +indirect indicators have been developed from lower redshift +analogs (see a summary in Flury et al. 2022a;b). One of the +leading indicators is the Lyα emission (e.g., Verhamme et al. +2015; Henry et al. 2015; Dijkstra et al. 2016; Verhamme et al. +2017; Jaskot et al. 2019; Gazagnes et al. 2020; Kakiichi & +Gronke 2021; Izotov et al. 2022). Given the resonance na- +ture of Lyα, the escape of Lyα photons contains information +about the neutral hydrogen in/around the galaxy, and leaves +footprints on the Lyα emission line profiles (e.g., Izotov et al. +2018b; Gazagnes et al. 2020; Flury et al. 2022b; Le Reste +et al. 2022). Nonetheless, since Lyα photons can be also ab- +sorbed by neutral IGM, the interpretation of Lyα profiles for +high-z galaxies (z ≳ 4) is non-trivial (e.g., Stark et al. 2011; +Schenker et al. 2014; Gronke et al. 2021; Hayes et al. 2021). +The Mg II λλ2796, 2803 doublet has been commonly de- +tected as a pair of absorption lines in galaxies, which trace +the galactic outflows and their feedback effects (e.g., Weiner +et al. 2009; Erb et al. 2012; Finley et al. 2017; Wang et al. +2022). +However, recently, Mg II has been found to show +strong doublet emission lines in galaxies which are classi- +fied as Lyman continuum emitter (LCEs) candidates. Vari- +ous publications suggest that the escape of Mg II correlates +with that of Lyα and LyC in SF galaxies (Henry et al. 2018; +Chisholm et al. 2020; Naidu et al. 2022; Xu et al. 2022; Izo- +tov et al. 2022; Seive et al. 2022). +Mg II emission was studied by Henry et al. (2018) in a sam- +ple of 10 compact SF galaxies. By the first time, they showed +that the escape of Mg II correlates with the escape of Lyα. +This result is interpreted as evidence that both Mg II and Lyα +photons escape from the galaxies through a similar path of +low column density gas. Indeed, Chisholm et al. (2020) point +out that the flux ratios between the doublet lines of Mg II [i.e., +F(Mg II 2796)/F(Mg II 2803), hereafter, R] can be used to +trace the column density of neutral hydrogen. They and Xu +et al. (2022) combined the Mg II emission, metallicity, and +dust attenuation to predict f LyC +esc , and found that the predicted +f LyC +esc +correlates with the observed f LyC +esc +in samples of galax- +ies with strong Mg II emission lines. Xu et al. (2022) also +found that galaxies selected with strong Mg II emission lines +might be more likely to leak LyC than similar galaxies with +weaker Mg II. In an independent sample, Izotov et al. (2022) +confirmed that escaping LyC emission is detected predomi- +nantly in galaxies with R ≳ 1.3, which indicates that optical +depth of Mg II is low (i.e., τ2803≲ 0.5, Chisholm et al. 2020). +Therefore, a high R ratio can be used to select LCEs candi- +dates. +Mg II emission lines are also detected in higher redshift SF +galaxies. For example, in the stacks of bright Lyα emitters +(LAEs) at z ∼ 2, Naidu et al. (2022) found that LCE can- +didates tend to have R close to 2, and the Mg II emission is +closer to the systemic velocity (instead of redshifted in non- +LCEs). In addition, Witstok et al. (2021) found in a lensed +z ∼ 5 galaxy, the escape of Mg II photons is consistent with +that of Lyα. However, Katz et al. (2022) pointed out from +the hydro-cosmological simulations that Mg II is a useful di- +agnostic of f LyC +esc only in the optically thin regime. +Though all of these studies support the hypothesis that +strong Mg II emission lines and a high R ratio can serve as +a good indirect indicator for Lyα and LyC escape, there ex- +ist two caveats. (1) Existing samples are commonly small +and only focused on SF galaxies with strong Mg II emission +lines and/or galaxies as strong LyC emitters. The latter also +results in substantial Mg II emission lines. Similar SF galax- +ies with Mg II as weaker emission lines, P-Cygni profiles, or +absorption lines have not been systematically studied in ob- +servations. (2) Some of these studies only have low signal- +to-noise (S/N) Mg II spectra (e.g., Naidu et al. 2022; Xu et al. +2022; Izotov et al. 2022). Their determinations of R, and the +subsequent inference of Mg II optical depth, have more scat- +ter due to the low S/N spectra. +In this paper, we further study Mg II as an indirect indi- +cator for the escape of Lyα and LyC, while we attempt to +mitigate the two caveats noted above. We focus on galaxies +from Low-redshift Lyman Continuum Survey (LzLCS, Flury +et al. 2022a), where a large sample of 66 LCEs candidates +are selected without reference to their Mg II emission lines. +For 34 out of 66 LzLCS galaxies, we present ground-based +follow-up spectroscopy of the Mg II feature. These data pro- +vide higher spectral resolution and S/N than SDSS/BOSS, +along with coverage of Mg II in galaxies where it is blue- +ward of the SDSS bandpass. As we will show, these data +achieve more secure measurements of the Mg II properties, +ultimately validating the use of Mg II as an indirect indicator +for the escape of Lyα and LyC. +The structure of the paper is as follows. In Section 2,we +introduce the observations, data reduction, and basic mea- +surements of optical emission lines. In Section 3, we present +how we derive various important parameters from the ob- + +MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY +3 +served Mg II doublet. We show the main results in Section +4, including the comparisons of Mg II with Lyα and LyC. +We conclude the paper in Section 5. +2. OBSERVATIONS, DATA REDUCTIONS, AND BASIC +ANALYSES +2.1. Ultraviolet Spectra for LyC and Lyα Regions +In this study, we focus on LCE candidates from the Low- +redshift Lyman Continuum Survey (LzLCS, Flury et al. +2022a), which contains 66 SF galaxies at z ∼ 0.3. The survey +obtained rest-frame ultraviolet (UV) spectra for each galaxy +with the G140L grating on Hubble Space Telescope/Cosmic +Origins Spectrograph (HST/COS) under program GO 15626 +(PI: Jaskot). Both LyC and Lyα regions have been covered. +The detailed data reductions of G140L data as well as the +analysis of Lyα and LyC escape are presented in Flury et al. +(2022a). In this paper, we adopt their derived quantities for +LyC and Lyα. These mainly include their escape fractions +and the equivalent widths (EW) for Lyα. +2.2. Optical Spectra for Mg II Regions +The galaxies in the LzLCS sample already have SDSS or +BOSS spectra, where the blue wavelength coverages end at +3800 Å and 3650 Å, respectively. Thus, the Mg II features +are observed by SDSS or BOSS only for galaxies with z +≳ 0.35 or 0.30, respectively. Even for the cases with ex- +isting Mg II spectra from SDSS/BOSS, as discussed in Xu +et al. (2022), the S/N of the intrinsically weak Mg II lines +and the spectral resolution (∼ 1500) are commonly too low. +In this case, the measurements of Mg II, especially R, have +large error bars. Therefore, we have obtained higher S/N and +higher spectral resolution observations for 34 galaxies from +the LzLCS sample. These galaxies are observed by either the +Multiple Mirror Telescope (MMT) or the Very Large Tele- +scope (VLT), or the Hobby-Eberly Telescope (HET). In this +paper, we focus on studying the Mg II properties from this +subsample of 34 galaxies. The observation details and data +reductions are listed in Table 1 and discussed below. +2.2.1. MMT observations and Data Reductions +A total of 24 galaxies from the LzLCS sample have been +observed by MMT. We adopt the blue channel spectrograph +using a 1′′ slit with the 832 lines/mm grating at the second +order. This leads to a spectral resolution of ∼ 1 Å (∼ 90 km +s−1 near the Mg II region). The observations were conducted +on 6 nights in 3 different semesters (2019A, 2020A, 2021A). +The exposure time is between 30 mins to 180 mins, depend- +ing on the brightness of the target (Table 1). We stay towards +lower airmass (≲ 1.3) and, for every exposure, we reset the +slit at the parallactic angle. We reduce the data following the +methodology described in Henry et al. (2018) using IDL + +IRAF routines. The wavelength calibration is applied from +the HeArHgCd arc lamps. By matching the arc lines, we find +the root-mean-square of the residuals is < 0.1 Å (∼ 10 km −1 +around Mg II spectral regions). +Given the short wavelength coverage of the blue channel +spectrograph in MMT (∼ 3100 – 4100 Å), the only major +line covered is the Mg II doublet. Therefore, we also adopt +SDSS/BOSS spectra for measuring other optical lines (Sec- +tion 2.3). For each galaxy, we calculate its u-band magnitude +from the MMT spectra and scale it to the galaxy’s u-band +magnitude from the SDSS photometry. This accounts for any +slit losses between MMT and SDSS observations. +2.2.2. VLT observations and Data Reductions +We also include 10 LzLCS sources observed by the X- +Shooter spectrograph mounted on VLT as part of the ESO +program ID 106.215K.001 (PI: Schaerer). Observations were +carried out between Fall 2020 and Spring 2022. We use 1.0′′, +0.9′′, and 0.9′′ slits in the UVB, VIS and NIR arms provid- +ing resolution power of ∼ 5400, 8900, and 5600, respec- +tively. This yields a spectral resolution of ∼ 50 km s−1 near +the Mg II regions. Observations were performed in nodding- +on-slit mode with a standard ABBA sequence and total on- +source exposure times of 46 mins or 92 min, depending on +the brightness of each source (Table 1). We reduce X-Shooter +data following the methods in Marques-Chaves et al. (2022a) +adopting the standard ESO Reflex reduction pipeline (version +2.11.5, Freudling et al. 2013). +For each galaxy, we also calculate the u-band magnitude +from the VLT spectra and match it to the galaxy’s u-band +magnitude from the SDSS photometry. Four of our galaxies +are observed by both MMT and VLT. We have checked that +the Mg II spectral profiles from the two telescopes are similar, +and the Mg II line flux ratio (i.e., R) are consistent within er- +rors. This is expected since galaxies in our sample are rather +compact (with UV half-light-radius ≲ 0.4′′) and are smaller +than the slit sizes. We finally adopt these galaxies’ VLT ob- +servations in our analyses, given their higher S/N. +2.2.3. HET observations and Data Reductions +We include 4 additional galaxies from the LzLCS sam- +ple, which are observed by the Low-Resolution Spectro- +graph (LRS2) on the Hobby-Eberly Telescope (Ramsey et al. +1998). LRS2 is an integral field spectrograph with nearly +complete spatial sampling, and a native spatial scale of 0.25′′ +× 0.25′′ spaxels with an average of 1.25′′ seeing (Chonis +et al. 2016). LRS2 has a wavelength coverage from 3600 Å +to 10,000 Å, and its spectral resolution around Mg II region +is 1.63 Å. To match our MMT and VLT slit sizes, we extract +the Mg II spectra in the central 1.0′′ × 1.0′′ aperture. We re- +duce the LRS2 data using the same methods in Seive et al. + +4 +XU ET AL. +Table 1. Follow-up Observations and Basic Properties for Galaxies in Our Sample +ID +RA +Dec +z1 +Instrument2 +Date3 +Exp.3 +SDSS-u4 +E(B−V)5 +MW +E(B−V)6 +int. +(mm/dd/yyyy) +(s) +(mag) +J0957+2357 +09:57:00 ++23:57:09 +0.2444 +MMT/Blue +04/08/2019 +4800 +18.42 +0.0287 +0.3007 +J1314+1048 +13:14:19 ++10:47:39 +0.2960 +MMT/Blue +04/08/2019 +3600 +19.69 +0.0371 +0.1621 +J1327+4218 +13:26:33 ++42:18:24 +0.3176 +MMT/Blue +04/08/2019 +3600 +20.48 +0.0173 +0.1641 +J1346+1129 +13:45:59 ++11:28:48 +0.2371 +MMT/Blue +04/08/2019 +3600 +19.00 +0.0231 +0.2022 +J1410+4345 +14:10:13 ++43:44:35 +0.3557 +MMT/Blue +04/08/2019 +7200 +21.70 +0.0196 +0.1413 +J0926+3957 +09:25:52 ++39:57:14 +0.3141 +MMT/Blue +04/09/2019 +7200 +21.27 +0.0270 +0.1364 +J1130+4935 +11:29:33 ++49:35:25 +0.3448 +MMT/Blue +04/09/2019 +7200 +21.50 +0.0292 +0.0446 +J1133+4514 +11:33:04 ++65:13:41 +0.2414 +MMT/Blue +04/09/2019 +7200 +20.14 +0.0158 +0.0886 +J1246+4449 +12:46:19 ++44:49:02 +0.3220 +MMT/Blue +04/09/2019 +4800 +20.48 +0.0200 +0.1595 +J0723+4146 +07:23:26 ++41:46:08 +0.2966 +MMT/Blue +02/19/2020 +7200 +20.89 +0.0467 +0.0097 +J0811+4141 +08:11:12 ++41:41:46 +0.3329 +MMT/Blue +02/19/2020 +7200 +21.17 +0.0383 +0.1150 +J1235+0635 +12:35:19 ++06:35:56 +0.3326 +MMT/Blue +02/19/2020 +6000 +20.72 +0.0333 +0.0782 +J0814+2114 +08:14:09 ++21:14:59 +0.2271 +MMT/Blue +02/20/2020 +1800 +18.91 +0.0336 +0.1800 +J0912+5050 +09:12:08 ++50:50:09 +0.3275 +MMT/Blue +02/20/2020 +9600 +20.56 +0.0245 +0.1088 +J1301+5104 +13:01:28 ++51:04:51 +0.3476 +MMT/Blue +02/20/2020 +2500 +20.04 +0.0222 +0.0974 +J0047+0154 +00:47:43 ++01:54:40 +0.3535 +MMT/Blue +01/09/2021 +3600 +20.28 +0.0312 +0.1760 +J0826+1820 +08:26:52 ++18:20:52 +0.2972 +MMT/Blue +01/09/2021 +7200 +21.35 +0.0328 +0.0300 +J1158+3125 +11:58:55 ++31:25:59 +0.2430 +MMT/Blue +01/09/2021 +2700 +19.27 +0.0226 +0.1037 +J1248+1234 +12:48:35 ++12:34:03 +0.2635 +MMT/Blue +01/09/2021 +6900 +20.16 +0.0519 +0.0627 +J0113+0002 +01:13:09 ++00:02:23 +0.3060 +MMT/Blue +01/10/2021 +7200 +20.56 +0.0312 +<1E-4 +J0129+1459 +01:29:10 ++14:59:35 +0.2799 +MMT/Blue +01/10/2021 +4800 +20.11 +0.0688 +0.0729 +J0917+3152 +09:17:03 ++31:52:21 +0.3003 +MMT/Blue +01/10/2021 +3300 +19.78 +0.0204 +0.1920 +J1033+6353 +10:33:44 ++63:53:17 +0.3465 +MMT/Blue +01/10/2021 +2700 +19.84 +0.0160 +0.0819 +J1038+4527 +10:38:16 ++45:27:18 +0.3256 +MMT/Blue +01/10/2021 +2700 +19.40 +0.0241 +0.2506 +J0036+0033 +00:36:01 ++00:33:07 +0.3480 +VLT/X-Shooter +11/05/2020 +2800 +21.83 +0.0278 +0.2007 +J0047+0154 +00:47:43 ++01:54:40 +0.3537 +VLT/X-Shooter +10/23/2020 +5500 +20.28 +0.0312 +0.0699 +J0113+0002 +01:13:09 ++00:02:23 +0.3062 +VLT/X-Shooter +10/23/2020 +5500 +20.56 +0.0312 +0.1144 +J0122+0520 +01:22:17 ++05:20:44 +0.3655 +VLT/X-Shooter +10/23/2020 +5500 +21.38 +0.0559 +0.1166 +J0814+2114 +08:14:09 ++21:14:59 +0.2271 +VLT/X-Shooter +12/21/2020 +2800 +18.91 +0.0335 +0.1984 +J0911+1831 +09:11:13 ++18:31:08 +0.2622 +VLT/X-Shooter +01/16/2021 +2800 +19.79 +0.0279 +0.1106 +J0958+2025 +09:58:38 ++20:25:08 +0.3016 +VLT/X-Shooter +02/04/2021 +2800 +19.86 +0.0347 +0.0650 +J1310+2148 +13:10:37 ++21:48:17 +0.2830 +VLT/X-Shooter +04/10/2021 +5500 +20.40 +0.0204 +0.0924 +J1235+0635 +12:35:19 ++06:35:56 +0.3327 +VLT/X-Shooter +01/12/2022 +5500 +20.72 +0.0333 +0.0782 +J1244+0215 +12:44:23 ++02:15:40 +0.2394 +VLT/X-Shooter +03/08/2022 +2800 +19.56 +0.0424 +0.0673 +J0834+4805 +08:34:40 ++48:05:41 +0.3425 +HET/LRS2 +12/30/2021 +5400 +20.62 +0.0383 +0.1741 +J0940+5932 +09:40:01 ++59:32:44 +0.3716 +HET/LRS2 +01/24/2022 +5400 +20.80 +0.0380 +0.4158 +J1517+3705 +15:17:07 ++37:05:12 +0.3533 +HET/LRS2 +07/22/2022 +6300 +20.87 +0.0411 +0.0005 +J1648+4957 +16:48:49 ++49:57:51 +0.3818 +HET/LRS2 +05/27/2022 +5400 +21.93 +0.0374 +0.0070 +Note. – +(1) Redshift of the objects derived from fitting the Balmer emission lines. +(2) Instruments that are used for the follow-up observations (Section 2.2). +(3) Observation start-date and exposure time in seconds, respectively. +(4) The u-band magnitudes from SDSS photometry. +(5) +Milky Way dust extinction obtained from Galactic Dust Reddening and Extinction Map (Schlafly & Finkbeiner 2011) at +NASA/IPAC Infrared Science Archive. +(6) The internal nebular dust extinction of the galaxy derived from Balmer lines (Section 2.3). +(2022), where we adopt the HET LRS2 pipeline, Panacea1, +to perform the initial reductions, including fiber extraction, +wavelength, calibration, astrometry, and flux calibration. For +each galaxy, we also calculate the u-band magnitude from the +1 https://github.com/grzeimann/Panacea +LRS2 spectra and match it to the galaxy’s u-band magnitude +from the SDSS photometry. +2.2.4. Summary of Optical Spectra +Overall, we obtained higher-quality data for 34 out of 66 +galaxies from the LzLCS sample. +We show the final re- +duced Mg II spectra for these galaxies in Figures 1 and 2, +and have ordered them by decreasing absolute escape frac- + +MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY +5 +Figure 1. The final reduced Mg II spectra for LzLCS galaxies in velocity space, with data taken from either MMT blue channel spectrograph +or VLT/X-Shooter spectrograph. For X-Shooter and LRS2 observations, we mark them with an extra ‘X’ and ‘L’ at the end of object names, +respectively. The y-axes are in units of 10−17 ergs s−1 cm−2 Å−1. The data and corresponding errors are shown in black and gray. Objects are +ordered by measured f LyC +esc values published in Flury et al. (2022a), which are also shown in the top-left corner of each panel. The blue and red +lines represent the position of v = 0 km s−1 for Mg II λ2796 and 2803, respectively. +tion of LyC (f LyC +esc ) measured from fitting the UV continuum +(reported in Flury et al. 2022a). Based on their LyC measure- +ments, these galaxies have f LyC +esc +range between 0 and 30%. +Of the 34 galaxies, 20 are classified as Lyman continuum +emitters (LCEs, sometimes referred as LyC “leakers”), which +have LyC flux detected with 97.725% confidence (Flury et al. +2022a). The other 12 galaxies are classified as non-LCEs. +We show the derived f LyC +esc +values at the top-left corners of +each panel, while we present f LyC +esc upper limits for non-LCEs. +We mark the galaxies observed by X-Shooter or LRS2 with +an extra ‘X’ or ‘L’, respectively, at the end of their object +names in Figures 1 and 2. +2.3. Measurements of Optical Emission Lines +For galaxies that have new optical spectra as described +above, we measure several optical emission lines whenever +covered, including Mg II, [O II], [O III], and Balmer lines. +For each galaxy, we first correct the spectra for Milky Way +extinction using the Galactic Dust Reddening and Extinction +Map (Schlafly & Finkbeiner 2011) at NASA/IPAC Infrared +Science Archive, assuming the extinction law from Cardelli + +6 +XU ET AL. +Figure 2. Same as Figure 1 but for galaxies with lower f LyC +esc . Upper limits of f LyC +esc are presented for galaxies that have non-detections of LyC +flux (Flury et al. 2022a). From LCEs to non-LCEs, Mg II line profiles show a clear transition from strong emission lines to P-Cygni profiles to +strong absorption lines. See more discussion in Section 3.1. + +MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY +7 +et al. (1989). The redshift of the galaxy is matched to the +peak of Balmer emission lines. +We determine the continuum flux for the Mg II spectral re- +gion by adopting a linear fit to the spectra ∼ ± 2000 km s−1 +around the systemic velocity. Then we split the spectra at +the midpoint between the two lines, i.e., 2799.1 Å, to repre- +sent the spectral regions for 2796 and 2803, separately. For +each Mg II line, we also split it into the absorption (below the +continuum) and emission (above the continuum) parts. After +that, we integrate the separate spectral regions to get the flux +and EW. The corresponding errors on these quantities are es- +timated through a Monte Carlo (MC) simulation where we +perturb the spectrum 104 times according to the observed 1σ +uncertainties. These values are reported in Table 2. Note that +we do not correct the Mg II line fluxes by internal dust ex- +tinction of the galaxy. This is because Mg II photons are res- +onantly scattered like Lyα and robust correction is difficult +(Henry et al. 2018; Chisholm et al. 2020; Xu et al. 2022). +For other optical lines, we measure their flux and EW sim- +ilarly as Mg II. However, unlike Mg II, since they are not res- +onant lines, we also correct the spectra by the internal dust +extinction for the galaxy before the measurements. The in- +ternal dust extinction (E(B −V)int.) for each galaxy is mea- +sured from Balmer lines following the methods in Xu et al. +(2022). For galaxies that only have new MMT observations, +since the MMT blue channel does not cover the Balmer lines, +we adopt E(B−V)int. derived in Flury et al. (2022a) based on +their SDSS spectra. The final E(B−V)int. values are reported +in the last column in Table 1. +In Figure 3, we compare the sub-sample adopted in this +paper (red) to the rest of the galaxies in LzLCS (gray). We +show two general observables that can be measured at high- +redshift, including O32 = flux ratio of [O III] λ5007/[O III] +λ3727 and stellar mass derived from spectral energy distribu- +tion (SED) fitting reported in Flury et al. (2022a). Our sub- +sample of galaxies is randomly selected from the LzLCS par- +ent sample to ensure a large dynamic range in galaxy proper- +ties. +3. ANALYSES +In this section, we present the methodology to derive im- +portant properties from Mg II lines. We first discuss the sig- +nificant trends in Mg II line profiles in Section 3.1. Then we +show the plausible geometry for Mg II photon escape in Sec- +tion 3.2. We present the methods to derive the Mg II escape +fractions (f MgII +esc ) from photoionization models in Section 3.3. +Finally, we discuss how to predict the escape fraction of LyC +(f LyC +esc,pd) from Mg II, metallicity, and dust attenuation in Sec- +tion 3.4. +3.1. Significant Trends of Mg II Line Profiles +In Figures 1 and 2, we show the Mg II spectra from +our galaxies in the order of decreasing f LyC +esc +derived from +Figure 3. Comparisons of the sub-sample studied in this paper (red) +to the other galaxies in the LzLCS parent sample (gray, see Section +2). Gray-filled and open symbols stand for galaxies from LzLCS, +which are classified as LyC emitter and non-emitters, respectively +(Flury et al. 2022a). Red-filled and open symbols represent galaxies +that are Mg II emitter and non-emitters, respectively (see definitions +in Section 3.2). The mean 1σ error bars are shown at the bottom-left +of the panel. +HST/COS G140L spectra (Flury et al. 2022a). There ex- +ist a significant trend that galaxies detected as strong LCEs +also show strong Mg II emission lines, and non-LCEs present +more absorption features in Mg II. We apply the Kendall τ +test between EW(Mg II) and f LyC +esc , where we have considered +the upper limits following Akritas & Siebert (1996). This +leads to the probability of a spurious correlation, p = 0.0216, +which confirms the strong trend. The former half of this trend +is consistent with previous observations of strong LCEs (Izo- +tov et al. 2022; Xu et al. 2022). Nonetheless, our sample +is the first to show that this trend indeed extends to non- +LCEs. This can be explained as LCEs have more optically +thin clouds in/around the galaxy than non-LCEs, so both the +Mg II and LyC photons can escape with less absorption and +scattering (Chisholm et al. 2022). This is also consistent with +the expectations from simulations (Katz et al. 2022). +Notably, a high f LyC +esc +(= 16.1%) was measured for galaxy +J0917+3152, but its Mg II profiles also have clear absorption +features. This can be explained by the high metallicity of +J0917+3152, i.e., 12+log(O/H) = 8.46, which is the highest +in our sample. Thus, for this object, there exist more mag- +nesium atoms given the same amount of hydrogen atoms. In +this scenario, the clouds around J0917+3152 become opti- +cally thick to Mg II when it is still optically thin to LyC pho- +tons. Overall, the significant trend for Mg II profiles from +LCEs to non-LCEs is valid for galaxies with lower metallic- +ity (in our case, 12+log(O/H) < 8.4). In these galaxies, the +surrounding gas/clouds become optically thick to Mg II and +LyC photons at similar depths (Chisholm et al. 2020). + +8 +XU ET AL. +3.2. Possible Geometry for the Escape of Mg II photons and +Constraints on Models +The escape of Mg II photons is first discussed in details in +Chisholm et al. (2020) (hereafter, the Chisholm model, see +their Section 6.4). This model assumes that Mg II photons +escape through a partial coverage geometry or sometimes re- +ferred as the picket-fence geometry (see also Gazagnes et al. +2018; Chisholm et al. 2018; Saldana-Lopez et al. 2022; Xu +et al. 2022): +f Mg II +esc += Fobs +Fint += Cf (Mg II)e−τthick +[1−Cf (Mg II)]e−τthin +(1) +where Fobs and Fint are the observed and intrinsic flux of +Mg II, respectively; C f (Mg II) is the covering fractions for +the optically thick paths of Mg II; and τthick and τthin are the +optical depths for Mg II at optically thick and thin paths, +respectively. In the optically thick paths, it is usually as- +sumed that τthick ≫ 1 such that no Mg II photons are observed +through this path. In this model, Chisholm et al. (2020) also +found: +R = F2796,obs +F2803,obs += 2e−τ2803,thin +(2) +where R is the emission line flux ratio between the Mg II dou- +blet. This model has proven to be successful in Chisholm +et al. (2020) and Xu et al. (2022) for galaxies with strong +Mg II emissions, where Mg II photons escape from the galaxy +through mostly optically thin paths. Furthermore, Katz et al. +(2022) have tested this model in their hydro-cosmological +simulations for EoR galaxy analogs. They find the actual +line-of-sight (LOS) f MgII +esc +match well with the predicted ones +from the Chisholm model for galaxies with low metallicity +(thus less dusty) and high f LyC +esc . These galaxies have Mg II +line profiles dominated by emission. +However, as described in Section 3.1, given the large dy- +namic ranges of our sample by design, our galaxies have +Mg II profiles ranging from strong emission to P-Cygni pro- +files and pure absorption. In the latter two cases, at least two +factors complicate the applications of the Chisholm model. +(1) The measurements of R from the spectra are not well- +defined due to the absorption in Mg II profiles. Thus, the +derived τ2803,thin from R has large uncertainties. (2) Mg II +doublet are resonant lines. Thus, the effect of dust for Mg II +is more substantial, which can cause strong absorption and +scattering features in the spectra (e.g., J0957+2357). How- +ever, this effect cannot be described in simple terms, thus, is +not included in the Chisholm model. Similarly, Katz et al. +(2022) comment that the Chisholm model is likely inade- +quate to predict f MgII +esc +for metal-rich (thus more dusty) galax- +ies in their simulations. +Currently, in this paper, to highlight the limitations of the +Chisholm model, we manually split galaxies in our sample +into two categories in our analyses. +Galaxies with Mg II +as strong emission, minimal absorption, and symmetric line +profiles are categorized as MgE (acronym for Mg II emitter), +while the others belong to non–MgE. This subjective classi- +fication is similar to what was adopted in Katz et al. (2022). +The Chisholm model should apply well to the former since +Mg II photons suffer little resonant scattering effects, but not +perfectly to the latter. We show the category of each galaxy +in the second to last column in Table 2. We discuss further +how we handle these two categories in Section 4.2. +3.3. The Method to Estimate the Escape Fraction of Mg II +Henry et al. (2018) have first introduced that one can de- +rive the intrinsic flux of Mg II from a correlation between +Mg II/[O III] and [O III]/[O II] (hereafter, the Henry model): +R2796 += A2 ×O2 +32 +A1 ×O32 +A0 +(3) +R2796 += log(Fint(Mg II λ2796)/Fint([O III] λ5007)) +O32 += log(Fint([O III] λ5007)/Fint([O II] λ3727)) +(4) +where the emission line fluxes are all intrinsic, i.e., before +the attenuation by dust and absorption in the LOS. Hereafter, +we use Fint to denote the intrinsic flux. A0, A1, and A2 are +coefficients that are dependent on the gas phase metallicity of +the galaxy, but little on the ionization parameters and spectral +slopes (Henry et al. 2018). Combining Fint(Mg II) with the +measured Fobs(Mg II) from the spectra, one can derive f MgII +esc . +The photoionization models in Henry et al. (2018) con- +siders ionization bounded (IB) geometry, where most of the +cloud remains neutral and is optically thick to escaping pho- +tons. However, LCEs with strong Mg II emission lines can be +partly density bounded (DB, i.e., mostly optically thin) given +their high O32 values observed (e.g., Izotov et al. 2016a;b; +2018a;b; 2021; Flury et al. 2022a;b; Xu et al. 2022). Thus, +Xu et al. (2022) update the correlation coefficients in the +Henry model to take into account the DB scenario. At a fixed +metallicity, they also find the different models from DB and +IB only move the correlation along the line defined in Equa- +tion (3). This should explain why Katz et al. (2022) find +the Henry model is a relatively good match to galaxies in +their simulations, but with moderate scatter given the differ- +ent metallicities of their simulated galaxies. +Given the derived f MgII +esc , one can solve Cf (Mg II) and τthin +from Equations (1) and (2). Note this requires robust mea- +surements of Mg II doublet flux ratio, i.e., R in Equation (2). +As discussed in Section 3.2, this can be achieved in MgE, but +hardly in non–MgE due to the absorption features in Mg II. +Since C f (Mg II) and τthin are then adopted to predict f LyC +esc , +accurately predictions are more difficult for non–MgE (see +detailed discussion in Sections 3.4 and 4.2). + +MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY +9 +Figure 4. Correlations between Mg II and Lyα properties. Galaxies that are labelled as MgE and non-MgE from our sample are shown in filled +and open symbols, respectively (Section 3.2). Left: The net EW from Mg II λ2796 and Lyα are positively correlated. Each galaxy is shown as +a dot with the cross representing its error bars. Galaxies with strong Mg II emission lines are at the top-right of the figure. The green dashed +line represents the best linear fit. Right: The escape fraction of Mg II λ2796 and Lyα are tightly correlated. The correlation coefficients from +Kendall τ test are shown at the top-left corner in each panel. The green solid line represents the 1:1 correlation. See discussion in Section 4.1. +3.4. The Method to Predict the Escape Fraction of LyC +As discussed in numerous previous publications (e.g., Za- +ckrisson et al. 2013; Reddy et al. 2016; Chisholm et al. 2020; +Kakiichi & Gronke 2021; Saldana-Lopez et al. 2022), the es- +cape of LyC photons can be described as a partial-covering +geometry: +fesc(LyC) = Cf (H I)e−τthick ×10−0.4Athick ++[1−Cf (H I)]e−τthin ×10−0.4Athin +(5) +where Cf (H I) is the covering fractions for optically thick +paths of H I which is dominated by neutral gas, and Athick +and Athin are the attenuation parameters for LyC photons at +optically thick and optically thin paths, respectively. +For galaxies in our LzLCS sample, Saldana-Lopez et al. +(2022) have found that the covering fraction of lower ion- +ization lines (LIS, including O I, C II, Si II) trace that of H I. +Given similar ionization potentials of Mg II to these lines, we +adopt their best-fit linear correlation to estimate Cf (H I) as: +Cf (H I) = (0.63±0.19)Cf (Mg II)+(0.54±0.09) +(6) +For optically thick paths, we assume no LyC photons can +escape (i.e., τthick and/or Athick ≫ 1). Therefore, the first term +in Equation (5) is negligible. Athin is related to the dust extinc- +tion at the LyC, for which we adopt the stellar extinction de- +rived from SED fittings in Saldana-Lopez et al. (2022). They +used Starburst99 template (Leitherer et al. 1999) and have +assumed the extinction law from Reddy et al. (2015; 2016). +Therefore, we can rewrite Equation (5) as: +f LyC +esc,pd = [1−Cf (H I)]e−N(H I)σph ×10−0.4E(B−V)k(912) +(7) +where f LyC +esc,pd is the predicted absolute escape fraction of LyC, +N(H I) is the column density of neutral hydrogen, σph is the +photoionization cross section of H I at 912 Å, E(B −V) is +the stellar dust extinction from Saldana-Lopez et al. (2022), +and k(912) is the total attenuation curve at the Lyman limit. +Given the Reddy extinction law adopted in Saldana-Lopez +et al. (2022), we have k(912) = 12.87. For other extinction +laws, e.g., Cardelli et al. (1989) and Calzetti et al. (2000), +k(912) = 21.32 and 16.62, respectively. +As shown in Chisholm et al. (2020) and Xu et al. (2022), +by assuming that Mg II and LyC photons escape from sim- +ilar optically thin paths, the column density of Mg II [i.e., +N(Mg II)] can be used to trace N(H I) in a large range from +DB to nearly IB regions: +N(H I) = α×N(Mg II) +(8) +where N(Mg II) can be calculated from the optical depth +of Mg II as inferred from R in Section 3.2, and α = +N(Mg II)/N(H I) is the column density ratios predicted from +CLOUDY models (Xu et al. 2022). α is dependent on the +abundance ratio of [Mg/H] and the ionization, and has typi- +cal values ∼ 104 – 105 for galaxies in our sample. Combining +Equations (5) to (8), we can calculate f LyC +esc,pd given the infor- +mation of Mg II, metallicity, and dust. +In ∼ 20 galaxies with strong Mg II emission lines, this +model predicted f LyC +esc,pd has been found to correlate well with +the actual f LyC +esc +measured from the spectra (Chisholm et al. +2020; Xu et al. 2022). Likewise, Katz et al. (2022) also found +that f LyC +esc,pd correlates with the actual f LyC +esc for simulated high- +redshift galaxies at different LOS (top-middle panel of their +Figure 17). But their correlation contains a large scatter. We + +10 +XU ET AL. +note that they adopt α as the abundance ratio of hydrogen to +oxygen, i.e., α = 46 H +O (Chisholm et al. 2020). This assumes +the Mg II emission is found in neutral gas. This is not accu- +rate since known LCEs commonly have high O32 values and, +thus, at least a fraction of the ISM (i.e., 1 - Cf (H I)) is density +bounded. Therefore, Mg II in these galaxies should originate +in regions where N(H II) is non-negligible. Our adopted α +from CLOUDY models in Equation 8 overcomes this prob- +lem (Xu et al. 2022). In Section 4.2, we compare f LyC +esc,pd with +the measured f LyC +esc +from Flury et al. (2022a) based on the +HST COS/G140L spectra and SED fittings. +4. RESULTS +4.1. Estimates of the Escape Fraction of Mg II +From Sections 3.2 to 3.3, we show how to derive f MgII +esc . +The resulting values are listed in the last column of Table +2. Furthermore, in Figure 4, we present the correlations be- +tween Mg II and Lyα. In the left panel, we compare the net +EW (i.e., the summed EW from both emission and absorp- +tion features) between Mg II and Lyα. To be consistent with +the literature, we have multiplied the net EW by -1 to al- +low galaxies with strong emission lines to be at the top-right +corner, while galaxies with strong absorption lines to be at +the bottom-left corner. We find a strong positive correlation +between the net EW of Mg II and Lyα. This is as expected +since both are resonant lines and should follow similar ra- +diative transfer processes when travelling out of the galaxies. +The best fit linear correlation is (show as the green dashed +line): +net EW(Mg II) = a+b×net EW(Lyα) +a = −0.468+0.157 +−0.157 +b = 0.091+0.003 +−0.003 +(9) +Similar but less significant trends between EW(Mg II) and +EW(Lyα) have also been published in Henry et al. (2018) and +Xu et al. (2022), where they only focused on strong Mg II +emitters. +In the right panel of Figure 4, we compare the derived f MgII +esc +with the escape fraction of Lyα (f Lyα +esc ). The latter is derived +from each galaxy’s HST/COS spectra in Flury et al. (2022a). +The correlation is significant (p < 10−6) with scatter. We +find most of the galaxies follow the 1:1 correlation shown +as the solid green line, which suggests f Lyα +esc +≃ f MgII +esc . This is +consistent with the results in Henry et al. (2018) and Xu et al. +(2022), which also found that f MgII +esc +and f Lyα +esc values are of the +same order. This supports the scenario where Mg II and Lyα +mainly escape from optically thin (or DB) holes in ISM likely +in a single flight (e.g., Gazagnes et al. 2018; Chisholm et al. +2020; Saldana-Lopez et al. 2022). Thus, the path lengths of +Mg II and Lyα photons travelling out of the galaxy are simi- +lar, and the resulting escape fractions are close for both lines. +Figure 5. Comparisons of measured f LyC +esc +with the predicted one +from Mg II λ2796 emission lines. Galaxies that are labelled as MgE +and non-MgE from our sample are shown in filled-red and open- +gray symbols, respectively (Section 3.2). Galaxies that are deter- +mined to be non-Lyman-continuum-emitter (Flury et al. 2022a) are +shown as upper limits. The orange dotted lines are to show the fac- +tor of 3 scatter around the 1:1 relationship (orange dashed line). We +also show 5 galaxies from Guseva et al. (2020) as green colors. See +discussion in Section 4.2. +One possible scenario is that there is zero dust in the optically +thin paths. Future spatially resolved observations can solve +this puzzle. This includes our Lyman-alpha and Continuum +Origins Survey (LaCOS, HST-GO 17069, PI: Hayes), which +aims to spatially resolve the Lyα emission, dust, and stellar +population for 41 out of 66 LzLCS galaxies by HST imaging. +For all figures in this section, we show Kendall τ coeffi- +cients and the probability of a spurious correlation (p val- +ues) at the top-left corner. In the Kendall test, we have ac- +counted for the upper limits (if any) following Akritas & +Siebert (1996). We have also tested the correlations between +the scatters in each figure with other galaxy properties, in- +cluding metallicity, internal dust extinction, SFR surface den- +sity, and stellar mass. However, we do not find significant +correlations. +4.2. Estimates of the Escape Fraction of LyC from Mg II +In Figure 5, we compare f LyC +esc,pd derived in Section 3.4 with +the f LyC +esc values derived from the HST/COS spectra (based on +UV continuum fittings, Flury et al. 2022a). We draw galax- +ies classified as MgE and non–MgE as filled and open sym- +bols, respectively. We also include 5 galaxies from Guseva +et al. (2020), which have high-quality VLT/X-Shooter obser- +vations as well as direct LyC measurements. We derive f LyC +esc,pd +in the same way as in Section 3.4, and remeasure their f LyC +esc +from HST/COS spectra using the same methodology in Flury +et al. (2022a). + +MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY +11 +First of all, there is a strong correlation between the pre- +dicted f LyC +esc +from Mg II and measured ones, given the proba- +bility of a spurious correlation p < 10−7. This highlights the +power of using Mg II to trace LyC. Given the scatter around +the 1:1 relationships line (orange dashed line), our predicted +f LyC +esc +values are accurate within a factor of 3. Considering +only the non-MgEs (gray-open symbols), the correlation is +less significant. This can be because their Mg II emission +lines are affected by absorption features, and the derived +τ2803,thin and C f (Mg II) from Equation (2) is more uncertain. +For some of the non–MgE, their Mg II spectra show signif- +icant resonant scattering or absorption signatures. These in- +clude galaxies showing double peaks in each Mg II emission +lines (J1310+2148, J1244+0215, J0826+1820, and maybe +in J1235+0635), and galaxies showing strong absorption in +Mg II (J0723+4146, J0940+5932, J1346+1129, J1314+1048, +J0957+2357). These galaxies should have optically thicker +clouds in/around the galaxy, and our measurements of emis- +sion line flux of Mg II are also uncertain. Thus, no meaning- +ful predictions through Mg II emission lines can be made, and +we have excluded these galaxies from Figure 5. We note that, +among these galaxies, only one (J1310+2148, f LyC +esc +∼ 1.6%) +has small amount of LyC detected from HST/COS spectra, +and others show non-detections of LyC (see Figures 1 and +2). Thus, these non-MgEs are more similar to galaxies that +are cosmologically irrelevant to EoR (f LyC +esc +≪ 1%). There- +fore, precise estimates of their f LyC +esc +are less important for +our understanding of SF galaxies contributing to the reion- +ization. Furthermore, when selecting new LyC emitters for +future observations, one can also exclude similar non-MgEs +based on the absorption and/or scattering features in Mg II +line profiles. +In the future, we plan to perform detailed radiative transfer +models to account for the escape of Mg II out of the galaxy +and link it to the escape of Lyα and LyC (Carr et al. in prepa- +ration). We can then model and separate the emission and ab- +sorption features from the observed Mg II spectra. This will +be particularly helpful to make more realistic predictions of +f LyC +esc,pd for these galaxies labelled as non–MgE. +Overall, our derived f LyC +esc,pd from Mg II, metallicity, and dust +can correctly trace the measured f LyC +esc +within a factor of ∼ 3 +for MgE. This is consistent with previous studies in Chisholm +et al. (2020); Xu et al. (2022). We conclude that Mg II emis- +sion lines along with dust can be used to predict the escape of +LyC photons in MgEs, but we need additional information to +do so in non-MgEs (e.g., detailed radiative transfer models). +5. CONCLUSION AND FUTURE WORK +We present the analyses of Mg II spectra for 34 galaxies +chosen from the LzLCS sample. These galaxies have pub- +lished HST/COS data for their LyC and Lyα spectral regions, +and we have obtained higher S/N and resolution spectra (than +SDSS) for their Mg II regions. +While previous studies of Mg II in Lyman Continuum +Emitter (LCE) candidates have only focused on Mg II emit- +ters (MgE), galaxies in our sample have Mg II profiles rang- +ing from strong emission to P-Cygni profiles, then to pure +absorption. We find there is a significant trend (p = 0.0216) +that galaxies detected as strong LCEs show larger EW(Mg II) +in emission lines, while non-LCEs present larger EW(Mg II) +in absorption. +We discuss the picket-fence geometry for the escape of +Mg II photons from galaxies. While this geometry has been +found to apply well to galaxies categorized as MgE, it has +limitations in the case of non-MgE. We then discuss how to +use the CLOUDY photoionization models to help derive the +escape fraction of Mg II (f MgII +esc ) from the optical spectra. For +all galaxies in our sample, we find f MgII +esc +correlates with the +escape fraction of Lyα. We also show that the net equivalent +width of Mg II and Lyα are tightly correlated for both MgEs +and non-MgEs. +We also discuss the methods to predict the escape fraction +of LyC (f LyC +esc,pd) from the measurements of Mg II, metallic- +ity, and dust. We show that the predicted f LyC +esc,pd correlates +well with the actual f LyC +esc derived from the HST/COS spectra +within a factor of ∼ 3. For non-MgEs, the correlation is less +significant. This is because the absorption features in Mg II +spectra for non-MgE complicate our measurements of Mg II +emission lines. Additional information, e.g., from radiative +transfer models, may help solve this problem. +In the future, one can apply the Mg II correlations to var- +ious different studies, including: 1). We will perform de- +tailed radiative transfer models to account for the escape of +Mg II from the galaxy (Carr et al. in preparation). This will +be especially helpful for the cases of non-MgE, where the +clouds in/around the galaxy are not optically thin to Mg II. +2) For high-z galaxies, one can adopt the observed Mg II fea- +tures to estimate the intrinsic amount of Lyα, which can be +severely attenuated by the neutral IGM (e.g., Mason et al. +2018). Thus, with the aid of Mg II, one can get more ac- +curate estimates of the IGM neutral fractions from Lyα. 3). +One can conduct similar analyses of Mg II in higher-redshift +LCE candidates, whose Mg II emission lines are shifted into +the observable bands of the James Webb Space Telescope +(JWST). The Lyα–Mg II correlations can be adopted to se- +lect Lyα emitters that have detectable Mg II spectra, and the +Mg II–LyC correlation can be used to predict f LyC +esc in the case +when LyC cannot be directly detected. + +12 +XU ET AL. +Table 2. Measurements from Optical Spectra for the Comparison Sample +Object +O32 +O/H +FEmi +2796 +FEmi +2803 +|EWEmi +2796| +|EWEmi +2803| +|EWAbs +2796| +|EWAbs +2803| +Label +f MgII +esc +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +(j) +(k) +J1033+6353 +3.4 +8.2 +50.2±7.0 +34.9±7.0 +4.7±0.8 +3.6±0.8 +0.9±0.3 +0.0±0.0 +MgE +0.29±0.04 +J0917+3152 +2.0 +8.5 +16.2±7.4 +13.5±6.8 +1.1±0.4 +0.9±0.6 +1.2±0.4 +0.9±0.3 +non-MgE +0.06±0.03 +J1327+4218 +3.3 +8.2 +38.2±4.4 +22.7±4.8 +6.0±0.6 +4.1±1.2 +0.3±0.08 +0.5±0.1 +MgE +0.18±0.02 +J1410+4345 +8.3 +8.0 +10.8±2.4 +7.8±2.6 +7.3±1.6 +5.6±2.0 +0.3±0.09 +0.0±0.0 +MgE +0.13±0.03 +J1158+3125 +1.8 +8.4 +75.5±5.2 +38.6±4.8 +3.2±0.2 +1.7±0.2 +0.6±0.2 +0.3±0.09 +MgE +0.18±0.01 +J1235+0635 +3.4 +8.4 +16.1±1.6 +10.6±1.4 +2.3±0.2 +1.5±0.2 +0.5±0.1 +0.0±0.0 +MgE +0.16±0.02 +J1248+1234 +3.4 +8.2 +116.3±5.0 +62.3±4.6 +9.6±0.6 +5.0±0.4 +1.3±0.4 +0.5±0.1 +MgE +0.44±0.02 +J1517+3705 +2.5 +8.3 +41.8±1.2 +26.7±1.4 +7.4±0.4 +4.2±0.2 +1.2±0.2 +0.7±0.2 +MgE +0.12±0.003 +J0122+0520 +5.7 +7.8 +27.7±2.6 +17.5±2.4 +6.4±1.0 +4.8±1.2 +0.1±0.03 +0.3±0.09 +MgE +0.59±0.06 +J1301+5104 +3.3 +8.3 +31.4±5.6 +16.4±5.0 +5.0±0.8 +3.1±1.0 +0.7±0.2 +0.3±0.09 +non-MgE +0.18±0.03 +J1648+4957 +3.3 +8.2 +35.4±0.4 +23.9±0.4 +21.3±0.2 +16.1±0.2 +3.5±0.2 +0.0±0.0 +MgE +0.49±0.006 +J0911+1831 +1.6 +8.1 +50.4±7.4 +34.4±8.2 +2.9±0.4 +2.0±0.4 +1.0±0.3 +0.5±0.2 +MgE +0.12±0.02 +J0113+0002 +2.3 +8.3 +42.9±3.0 +23.1±2.6 +5.0±0.6 +2.9±0.4 +1.1±0.3 +0.5±0.1 +MgE +0.59±0.04 +J1133+4514 +3.6 +8.0 +83.9±6.4 +43.1±6.6 +7.3±0.6 +3.8±0.6 +0.1±0.04 +0.0±0.0 +MgE +0.65±0.05 +J0811+4141 +8.1 +7.9 +45.7±4.2 +28.0±4.0 +12.9±1.2 +7.4±1.0 +1.0±0.3 +0.0±0.0 +MgE +1.00±0.23 +J0958+2025 +5.2 +7.8 +21.2±5.0 +8.4±4.2 +8.2±2.6 +4.4±2.8 +2.2±0.7 +0.5±0.1 +non-MgE +0.11±0.03 +J1310+2148 +1.6 +8.4 +18.7±2.0 +12.4±1.6 +2.1±0.4 +1.4±0.2 +1.6±0.5 +0.3±0.09 +non-MgE +0.06±0.007 +J0047+0154 +2.9 +8.0 +41.8±2.8 +31.0±3.0 +4.0±0.2 +3.1±0.2 +1.3±0.2 +0.8±0.2 +MgE +0.23±0.02 +J1038+4527 +1.5 +8.4 +59.9±8.4 +35.4±9.0 +2.8±0.4 +1.7±0.4 +0.8±0.2 +0.6±0.2 +non-MgE +0.14±0.02 +J1246+4449 +3.4 +8.0 +72.4±4.2 +41.4±4.0 +10.8±0.6 +6.2±0.6 +0.5±0.2 +0.1±0.04 +MgE +0.29±0.02 +J0834+4805 +3.6 +8.2 +55.6±1.4 +44.7±1.4 +7.4±0.4 +6.1±0.2 +1.0±0.2 +0.0±0.0 +MgE +0.09±0.002 +J1244+0215 +3.6 +8.2 +64.3±7.6 +40.4±7.4 +2.8±0.6 +1.8±0.6 +0.7±0.2 +0.5±0.2 +non-MgE +0.08±0.009 +J1130+4935 +3.4 +8.3 +15.2±2.0 +7.4±1.8 +6.3±0.8 +3.1±0.8 +1.5±0.4 +0.9±0.3 +non-MgE +0.38±0.05 +J0129+1459 +1.6 +8.4 +33.7±7.6 +15.8±8.2 +3.3±0.8 +1.7±1.2 +1.8±0.5 +1.2±0.4 +non-MgE +0.17±0.04 +J0036+0033 +10.5 +7.8 +14.4±2.2 +7.5±2.8 +7.7±1.8 +5.3±2.4 +1.3±0.4 +1.0±0.3 +non-MgE +0.20±0.03 +J0926+3957 +2.2 +8.2 +12.3±2.2 +6.1±1.8 +4.1±0.8 +1.8±0.6 +0.8±0.3 +1.0±0.3 +non-MgE +0.17±0.03 +J0826+1820 +4.0 +8.3 +7.5±2.6 +3.4±2.2 +2.8±1.0 +1.8±1.4 +0.4±0.1 +1.2±0.4 +non-MgE +0.12±0.04 +J0912+5050 +3.0 +8.2 +25.3±3.2 +19.6±3.6 +4.4±0.6 +3.5±0.6 +0.4±0.1 +0.1±0.04 +non-MgE +0.21±0.03 +J0814+2114 +1.2 +8.1 +39.4±9.2 +28.7±10.0 +1.4±0.4 +0.9±0.4 +0.7±0.2 +0.3±0.08 +non-MgE +0.06±0.01 +J0723+4146 +3.2 +8.2 +27.6±6.4 +16.9±6.6 +4.9±1.2 +3.6±1.6 +1.6±0.5 +0.9±0.3 +non-MgE +0.33±0.08 +J0940+5932 +1.5 +8.4 +... +... +. . . +. . . +5.9±0.2 +4.2±0.4 +non-MgE +. . . +J1346+1129 +1.1 +8.3 +72.1±11.0 +68.7±12.8 +2.3±0.4 +2.2±0.6 +2.6±0.4 +2.3±0.4 +non-MgE +0.11±0.02 +J1314+1048 +1.1 +8.3 +25.0±5.6 +27.4±8.0 +1.4±0.2 +1.5±0.6 +3.0±0.4 +2.2±0.4 +non-MgE +0.05±0.01 +J0957+2357 +0.5 +8.4 +... +... +. . . +. . . +2.9±0.4 +2.6±0.4 +non-MgE +. . . +Note. –Measurements from the optical spectra for galaxies in our sample. Galaxies are ordered by decreasing f LyC +esc derived in Flury +et al. (2022a) (the same order as Figures 1 and 2). The columns are: (b) Flux ratio between [O III] λ5007 and [O II] λ3727; (c) +Gas phase metallicity in the form of 12+log(O/H); (d) and (e) Measured emission line flux of Mg II λλ2796, 2803 lines in units of +10−17 ergs s−1 cm−2, respectively; (f) and (g): Measured rest-frame EW in units of Å for the emission part from the Mg II doublet +(see Section 2.3); (h) and (i) Measured rest-frame EW in units of Å for the absorption part from the Mg II doublet; (j) Labels based +on the Mg II line profiles, i.e., MgE = Mg II emitter, non-MgE = Mg II non-emitter (see Section 3.2); and (k): the derived escape +fraction for Mg II λ2796 (see Section 3.3). + +MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY +13 +X.X. and A.H. acknowledge support from NASA STScI +grants GO 15865. Observations reported here were obtained +at the MMT Observatory, a joint facility of the University of +Arizona and the Smithsonian Institution. +1 +2 +3 +4 +Support for this work was provided by NASA through +grant number HST-GO-15626 from the Space Telescope Sci- +ence Institute. This research is based on observations made +with the NASA/ESA Hubble Space Telescope obtained from +the Space Telescope Science Institute, which is operated by +the Association of Universities for Research in Astronomy, +Inc., under NASA contract NAS 5–26555. These observa- +tions are associated with program(s) 13744, 14635, 15341, +15626, 15639, and 15941. STScI is operated by the Associ- +ation of Universities for Research in Astronomy, Inc. under +NASA contract NAS 5-26555. +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Based on observations collected at the European Organisa- +tion for Astronomical Research in the Southern Hemisphere +under ESO programme 106.215K.001. +16 +17 +18 +The Low-Resolution Spectrograph 2 (LRS2) was devel- +oped and funded by the University of Texas at Austin Mc- +Donald Observatory and the Department of Astronomy and +by Pennsylvania State University. +We thank the Leibniz- +Institut für Astrophysik Potsdam (AIP) and the Institut für +Astrophysik Göttingen (IAG) for their contributions to the +construction of the integral field units. +19 +20 +21 +22 +23 +24 +25 +ASL acknowledge support from Swiss National Science +Foundation. 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+page_content=' 1290 Versoix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Switzerland 5Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Williams College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Williamstown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' MA 01267,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' United States 6Institut für Physik und Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Universität Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 24/25, D-14476 Potsdam, Germany 7Instituto de Investigacion Multidisciplinar en Ciencia y Tecnologia, Universidad de La Serena, Raul Bitran 1305, La Serena, Chile 8Institut d’astrophysique de Paris, CNRS, Sorbonne Université, 98bis Boulevrad Arago, F-75014, Paris, France 9Department of Astronomy, Oskar Klein Centre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Stockholm University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' SE-106 91 Stockholm, Sweden 10University of Massachusetts Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA 11Astronomy Department, University of Virginia, Charlottesville, VA 22904, USA Submitted to AASJournal ApJ ABSTRACT The Mg II λλ2796, 2803 doublet has been suggested to be a useful indirect indicator for the escape of Lyα and Lyman continuum (LyC) photons in local star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, studies to date have focused on small samples of galaxies with strong Mg II or strong LyC emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Here we present the first study of Mg II probing a large dynamic range of galaxy properties, using newly obtained high signal-to-noise, moderate-resolution spectra of Mg II for a sample of 34 galaxies selected from the Low-redshift Lyman Continuum Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show that the galaxies in our sample have Mg II profiles ranging from strong emission to P-Cygni profiles, and to pure absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We find there is a significant trend (with a possibility of spurious correlations of ∼ 2%) that galaxies detected as strong LyC Emitters (LCEs) also show larger equivalent widths of Mg II emission, and non-LCEs tend to show evidence of more scattering and absorption features in Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We then find Mg II strongly correlates with Lyα in both equivalent width and escape fraction, regardless of whether the emission or absorption dominates the Mg II profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Furthermore, we present that, for galaxies categorized as Mg II emitters (MgE), one can adopt the information of Mg II, metallicity, and dust to estimate the escape fraction of LyC within a factor of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These findings confirm that Mg II lines can be used as a tool to select galaxies as LCEs and to serve as an indirect indicator for the escape of Lyα and LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Keywords: Galaxy evolution (1052), Galaxy kinematics and dynamics(602), Ultraviolet astronomy (1736), Galaxy spectroscopy (2171) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' INTRODUCTION In the last decades, considerable observational efforts have been directed towards studying the escape of Lyman contin- uum (LyC) photons from galaxies and attempting to explain Corresponding author: Xinfeng Xu xinfeng@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='edu the last phase transition of the universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Cosmic Reion- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Various publications point out that star-forming (SF) galaxies can be responsible for the epoch of reionization (EoR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These include studies of low-redshift galaxies (z ≲ 1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Bergvall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2006;' metadata={'source': 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+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Mestri´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Vielfaure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Begley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Rivera-Thorsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Marques-Chaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These galaxies are proposed to be analogs of high-redshift (z ∼ 6 – 8) ones at EoR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Schaerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Boyett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Due to the attenuation by Lyman limit systems (LLS) and/or neutral intergalactic medium (IGM), it is challenging to detect LyC at z ≳ 5 (Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Worseck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Bosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, various indirect indicators have been developed from lower redshift analogs (see a summary in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' One of the leading indicators is the Lyα emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Dijkstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Jaskot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Kakiichi & Gronke 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Given the resonance na- ture of Lyα, the escape of Lyα photons contains information about the neutral hydrogen in/around the galaxy, and leaves footprints on the Lyα emission line profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Le Reste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Nonetheless, since Lyα photons can be also ab- sorbed by neutral IGM, the interpretation of Lyα profiles for high-z galaxies (z ≳ 4) is non-trivial (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Schenker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Gronke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The Mg II λλ2796, 2803 doublet has been commonly de- tected as a pair of absorption lines in galaxies, which trace the galactic outflows and their feedback effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Weiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Erb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Finley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, recently, Mg II has been found to show strong doublet emission lines in galaxies which are classi- fied as Lyman continuum emitter (LCEs) candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Vari- ous publications suggest that the escape of Mg II correlates with that of Lyα and LyC in SF galaxies (Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Izo- tov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Seive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Mg II emission was studied by Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2018) in a sam- ple of 10 compact SF galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' By the first time, they showed that the escape of Mg II correlates with the escape of Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This result is interpreted as evidence that both Mg II and Lyα photons escape from the galaxies through a similar path of low column density gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Indeed, Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020) point out that the flux ratios between the doublet lines of Mg II [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', F(Mg II 2796)/F(Mg II 2803), hereafter, R] can be used to trace the column density of neutral hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' They and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) combined the Mg II emission, metallicity, and dust attenuation to predict f LyC esc , and found that the predicted f LyC esc correlates with the observed f LyC esc in samples of galax- ies with strong Mg II emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) also found that galaxies selected with strong Mg II emission lines might be more likely to leak LyC than similar galaxies with weaker Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In an independent sample, Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) confirmed that escaping LyC emission is detected predomi- nantly in galaxies with R ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3, which indicates that optical depth of Mg II is low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', τ2803≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='5, Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, a high R ratio can be used to select LCEs candi- dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Mg II emission lines are also detected in higher redshift SF galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For example, in the stacks of bright Lyα emitters (LAEs) at z ∼ 2, Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) found that LCE can- didates tend to have R close to 2, and the Mg II emission is closer to the systemic velocity (instead of redshifted in non- LCEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In addition, Witstok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2021) found in a lensed z ∼ 5 galaxy, the escape of Mg II photons is consistent with that of Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) pointed out from the hydro-cosmological simulations that Mg II is a useful di- agnostic of f LyC esc only in the optically thin regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Though all of these studies support the hypothesis that strong Mg II emission lines and a high R ratio can serve as a good indirect indicator for Lyα and LyC escape, there ex- ist two caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (1) Existing samples are commonly small and only focused on SF galaxies with strong Mg II emission lines and/or galaxies as strong LyC emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The latter also results in substantial Mg II emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Similar SF galax- ies with Mg II as weaker emission lines, P-Cygni profiles, or absorption lines have not been systematically studied in ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2) Some of these studies only have low signal- to-noise (S/N) Mg II spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Their determinations of R, and the subsequent inference of Mg II optical depth, have more scat- ter due to the low S/N spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In this paper, we further study Mg II as an indirect indi- cator for the escape of Lyα and LyC, while we attempt to mitigate the two caveats noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We focus on galaxies from Low-redshift Lyman Continuum Survey (LzLCS, Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a), where a large sample of 66 LCEs candidates are selected without reference to their Mg II emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For 34 out of 66 LzLCS galaxies, we present ground-based follow-up spectroscopy of the Mg II feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These data pro- vide higher spectral resolution and S/N than SDSS/BOSS, along with coverage of Mg II in galaxies where it is blue- ward of the SDSS bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' As we will show, these data achieve more secure measurements of the Mg II properties, ultimately validating the use of Mg II as an indirect indicator for the escape of Lyα and LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In Section 2,we introduce the observations, data reduction, and basic mea- surements of optical emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In Section 3, we present how we derive various important parameters from the ob- MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY 3 served Mg II doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show the main results in Section 4, including the comparisons of Mg II with Lyα and LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We conclude the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' OBSERVATIONS, DATA REDUCTIONS, AND BASIC ANALYSES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Ultraviolet Spectra for LyC and Lyα Regions In this study, we focus on LCE candidates from the Low- redshift Lyman Continuum Survey (LzLCS, Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a), which contains 66 SF galaxies at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The survey obtained rest-frame ultraviolet (UV) spectra for each galaxy with the G140L grating on Hubble Space Telescope/Cosmic Origins Spectrograph (HST/COS) under program GO 15626 (PI: Jaskot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Both LyC and Lyα regions have been covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The detailed data reductions of G140L data as well as the analysis of Lyα and LyC escape are presented in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In this paper, we adopt their derived quantities for LyC and Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These mainly include their escape fractions and the equivalent widths (EW) for Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Optical Spectra for Mg II Regions The galaxies in the LzLCS sample already have SDSS or BOSS spectra, where the blue wavelength coverages end at 3800 Å and 3650 Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, the Mg II features are observed by SDSS or BOSS only for galaxies with z ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='35 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='30, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Even for the cases with ex- isting Mg II spectra from SDSS/BOSS, as discussed in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), the S/N of the intrinsically weak Mg II lines and the spectral resolution (∼ 1500) are commonly too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In this case, the measurements of Mg II, especially R, have large error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, we have obtained higher S/N and higher spectral resolution observations for 34 galaxies from the LzLCS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These galaxies are observed by either the Multiple Mirror Telescope (MMT) or the Very Large Tele- scope (VLT), or the Hobby-Eberly Telescope (HET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In this paper, we focus on studying the Mg II properties from this subsample of 34 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The observation details and data reductions are listed in Table 1 and discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' MMT observations and Data Reductions A total of 24 galaxies from the LzLCS sample have been observed by MMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We adopt the blue channel spectrograph using a 1′′ slit with the 832 lines/mm grating at the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This leads to a spectral resolution of ∼ 1 Å (∼ 90 km s−1 near the Mg II region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The observations were conducted on 6 nights in 3 different semesters (2019A, 2020A, 2021A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The exposure time is between 30 mins to 180 mins, depend- ing on the brightness of the target (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We stay towards lower airmass (≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3) and, for every exposure, we reset the slit at the parallactic angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We reduce the data following the methodology described in Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2018) using IDL + IRAF routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The wavelength calibration is applied from the HeArHgCd arc lamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' By matching the arc lines, we find the root-mean-square of the residuals is < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1 Å (∼ 10 km −1 around Mg II spectral regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Given the short wavelength coverage of the blue channel spectrograph in MMT (∼ 3100 – 4100 Å), the only major line covered is the Mg II doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, we also adopt SDSS/BOSS spectra for measuring other optical lines (Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For each galaxy, we calculate its u-band magnitude from the MMT spectra and scale it to the galaxy’s u-band magnitude from the SDSS photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This accounts for any slit losses between MMT and SDSS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' VLT observations and Data Reductions We also include 10 LzLCS sources observed by the X- Shooter spectrograph mounted on VLT as part of the ESO program ID 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='215K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='001 (PI: Schaerer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Observations were carried out between Fall 2020 and Spring 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We use 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0′′, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='9′′, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='9′′ slits in the UVB, VIS and NIR arms provid- ing resolution power of ∼ 5400, 8900, and 5600, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This yields a spectral resolution of ∼ 50 km s−1 near the Mg II regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Observations were performed in nodding- on-slit mode with a standard ABBA sequence and total on- source exposure times of 46 mins or 92 min, depending on the brightness of each source (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We reduce X-Shooter data following the methods in Marques-Chaves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a) adopting the standard ESO Reflex reduction pipeline (version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='5, Freudling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For each galaxy, we also calculate the u-band magnitude from the VLT spectra and match it to the galaxy’s u-band magnitude from the SDSS photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Four of our galaxies are observed by both MMT and VLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We have checked that the Mg II spectral profiles from the two telescopes are similar, and the Mg II line flux ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', R) are consistent within er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is expected since galaxies in our sample are rather compact (with UV half-light-radius ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4′′) and are smaller than the slit sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We finally adopt these galaxies’ VLT ob- servations in our analyses, given their higher S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' HET observations and Data Reductions We include 4 additional galaxies from the LzLCS sam- ple, which are observed by the Low-Resolution Spectro- graph (LRS2) on the Hobby-Eberly Telescope (Ramsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' LRS2 is an integral field spectrograph with nearly complete spatial sampling, and a native spatial scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='25′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='25′′ spaxels with an average of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='25′′ seeing (Chonis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' LRS2 has a wavelength coverage from 3600 Å to 10,000 Å, and its spectral resolution around Mg II region is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='63 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' To match our MMT and VLT slit sizes, we extract the Mg II spectra in the central 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0′′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0′′ aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We re- duce the LRS2 data using the same methods in Seive et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 4 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Follow-up Observations and Basic Properties for Galaxies in Our Sample ID RA Dec z1 Instrument2 Date3 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3 SDSS-u4 E(B−V)5 MW E(B−V)6 int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (mm/dd/yyyy) (s) (mag) J0957+2357 09:57:00 +23:57:09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2444 MMT/Blue 04/08/2019 4800 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0070 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' – (1) Redshift of the objects derived from fitting the Balmer emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2) Instruments that are used for the follow-up observations (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (3) Observation start-date and exposure time in seconds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (4) The u-band magnitudes from SDSS photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (5) Milky Way dust extinction obtained from Galactic Dust Reddening and Extinction Map (Schlafly & Finkbeiner 2011) at NASA/IPAC Infrared Science Archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (6) The internal nebular dust extinction of the galaxy derived from Balmer lines (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), where we adopt the HET LRS2 pipeline, Panacea1, to perform the initial reductions, including fiber extraction, wavelength, calibration, astrometry, and flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For each galaxy, we also calculate the u-band magnitude from the 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='com/grzeimann/Panacea LRS2 spectra and match it to the galaxy’s u-band magnitude from the SDSS photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Summary of Optical Spectra Overall, we obtained higher-quality data for 34 out of 66 galaxies from the LzLCS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show the final re- duced Mg II spectra for these galaxies in Figures 1 and 2, and have ordered them by decreasing absolute escape frac- MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The final reduced Mg II spectra for LzLCS galaxies in velocity space, with data taken from either MMT blue channel spectrograph or VLT/X-Shooter spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For X-Shooter and LRS2 observations, we mark them with an extra ‘X’ and ‘L’ at the end of object names, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The y-axes are in units of 10−17 ergs s−1 cm−2 Å−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The data and corresponding errors are shown in black and gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Objects are ordered by measured f LyC esc values published in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a), which are also shown in the top-left corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The blue and red lines represent the position of v = 0 km s−1 for Mg II λ2796 and 2803, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' tion of LyC (f LyC esc ) measured from fitting the UV continuum (reported in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Based on their LyC measure- ments, these galaxies have f LyC esc range between 0 and 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Of the 34 galaxies, 20 are classified as Lyman continuum emitters (LCEs, sometimes referred as LyC “leakers”), which have LyC flux detected with 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='725% confidence (Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The other 12 galaxies are classified as non-LCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show the derived f LyC esc values at the top-left corners of each panel, while we present f LyC esc upper limits for non-LCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We mark the galaxies observed by X-Shooter or LRS2 with an extra ‘X’ or ‘L’, respectively, at the end of their object names in Figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Measurements of Optical Emission Lines For galaxies that have new optical spectra as described above, we measure several optical emission lines whenever covered, including Mg II, [O II], [O III], and Balmer lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For each galaxy, we first correct the spectra for Milky Way extinction using the Galactic Dust Reddening and Extinction Map (Schlafly & Finkbeiner 2011) at NASA/IPAC Infrared Science Archive, assuming the extinction law from Cardelli 6 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Same as Figure 1 but for galaxies with lower f LyC esc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Upper limits of f LyC esc are presented for galaxies that have non-detections of LyC flux (Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' From LCEs to non-LCEs, Mg II line profiles show a clear transition from strong emission lines to P-Cygni profiles to strong absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' See more discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY 7 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The redshift of the galaxy is matched to the peak of Balmer emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We determine the continuum flux for the Mg II spectral re- gion by adopting a linear fit to the spectra ∼ ± 2000 km s−1 around the systemic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Then we split the spectra at the midpoint between the two lines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', 2799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1 Å, to repre- sent the spectral regions for 2796 and 2803, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For each Mg II line, we also split it into the absorption (below the continuum) and emission (above the continuum) parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' After that, we integrate the separate spectral regions to get the flux and EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The corresponding errors on these quantities are es- timated through a Monte Carlo (MC) simulation where we perturb the spectrum 104 times according to the observed 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These values are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Note that we do not correct the Mg II line fluxes by internal dust ex- tinction of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is because Mg II photons are res- onantly scattered like Lyα and robust correction is difficult (Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For other optical lines, we measure their flux and EW sim- ilarly as Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, unlike Mg II, since they are not res- onant lines, we also correct the spectra by the internal dust extinction for the galaxy before the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The in- ternal dust extinction (E(B −V)int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=') for each galaxy is mea- sured from Balmer lines following the methods in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For galaxies that only have new MMT observations, since the MMT blue channel does not cover the Balmer lines, we adopt E(B−V)int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' derived in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a) based on their SDSS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The final E(B−V)int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' values are reported in the last column in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In Figure 3, we compare the sub-sample adopted in this paper (red) to the rest of the galaxies in LzLCS (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show two general observables that can be measured at high- redshift, including O32 = flux ratio of [O III] λ5007/[O III] λ3727 and stellar mass derived from spectral energy distribu- tion (SED) fitting reported in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Our sub- sample of galaxies is randomly selected from the LzLCS par- ent sample to ensure a large dynamic range in galaxy proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' ANALYSES In this section, we present the methodology to derive im- portant properties from Mg II lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We first discuss the sig- nificant trends in Mg II line profiles in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Then we show the plausible geometry for Mg II photon escape in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We present the methods to derive the Mg II escape fractions (f MgII esc ) from photoionization models in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Finally, we discuss how to predict the escape fraction of LyC (f LyC esc,pd) from Mg II, metallicity, and dust attenuation in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Significant Trends of Mg II Line Profiles In Figures 1 and 2, we show the Mg II spectra from our galaxies in the order of decreasing f LyC esc derived from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Comparisons of the sub-sample studied in this paper (red) to the other galaxies in the LzLCS parent sample (gray, see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Gray-filled and open symbols stand for galaxies from LzLCS, which are classified as LyC emitter and non-emitters, respectively (Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Red-filled and open symbols represent galaxies that are Mg II emitter and non-emitters, respectively (see definitions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The mean 1σ error bars are shown at the bottom-left of the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' HST/COS G140L spectra (Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' There ex- ist a significant trend that galaxies detected as strong LCEs also show strong Mg II emission lines, and non-LCEs present more absorption features in Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We apply the Kendall τ test between EW(Mg II) and f LyC esc , where we have considered the upper limits following Akritas & Siebert (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This leads to the probability of a spurious correlation, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0216, which confirms the strong trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The former half of this trend is consistent with previous observations of strong LCEs (Izo- tov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Nonetheless, our sample is the first to show that this trend indeed extends to non- LCEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This can be explained as LCEs have more optically thin clouds in/around the galaxy than non-LCEs, so both the Mg II and LyC photons can escape with less absorption and scattering (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is also consistent with the expectations from simulations (Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Notably, a high f LyC esc (= 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1%) was measured for galaxy J0917+3152, but its Mg II profiles also have clear absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This can be explained by the high metallicity of J0917+3152, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', 12+log(O/H) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='46, which is the highest in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, for this object, there exist more mag- nesium atoms given the same amount of hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In this scenario, the clouds around J0917+3152 become opti- cally thick to Mg II when it is still optically thin to LyC pho- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Overall, the significant trend for Mg II profiles from LCEs to non-LCEs is valid for galaxies with lower metallic- ity (in our case, 12+log(O/H) < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In these galaxies, the surrounding gas/clouds become optically thick to Mg II and LyC photons at similar depths (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 8 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Possible Geometry for the Escape of Mg II photons and Constraints on Models The escape of Mg II photons is first discussed in details in Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020) (hereafter, the Chisholm model, see their Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This model assumes that Mg II photons escape through a partial coverage geometry or sometimes re- ferred as the picket-fence geometry (see also Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022): f Mg II esc = Fobs Fint = Cf (Mg II)e−τthick +[1−Cf (Mg II)]e−τthin (1) where Fobs and Fint are the observed and intrinsic flux of Mg II, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' C f (Mg II) is the covering fractions for the optically thick paths of Mg II;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' and τthick and τthin are the optical depths for Mg II at optically thick and thin paths, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the optically thick paths, it is usually as- sumed that τthick ≫ 1 such that no Mg II photons are observed through this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In this model, Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020) also found: R = F2796,obs F2803,obs = 2e−τ2803,thin (2) where R is the emission line flux ratio between the Mg II dou- blet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This model has proven to be successful in Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020) and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) for galaxies with strong Mg II emissions, where Mg II photons escape from the galaxy through mostly optically thin paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Furthermore, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) have tested this model in their hydro-cosmological simulations for EoR galaxy analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' They find the actual line-of-sight (LOS) f MgII esc match well with the predicted ones from the Chisholm model for galaxies with low metallicity (thus less dusty) and high f LyC esc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These galaxies have Mg II line profiles dominated by emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1, given the large dy- namic ranges of our sample by design, our galaxies have Mg II profiles ranging from strong emission to P-Cygni pro- files and pure absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the latter two cases, at least two factors complicate the applications of the Chisholm model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (1) The measurements of R from the spectra are not well- defined due to the absorption in Mg II profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, the derived τ2803,thin from R has large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2) Mg II doublet are resonant lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, the effect of dust for Mg II is more substantial, which can cause strong absorption and scattering features in the spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', J0957+2357).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' How- ever, this effect cannot be described in simple terms, thus, is not included in the Chisholm model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Similarly, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) comment that the Chisholm model is likely inade- quate to predict f MgII esc for metal-rich (thus more dusty) galax- ies in their simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Currently, in this paper, to highlight the limitations of the Chisholm model, we manually split galaxies in our sample into two categories in our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Galaxies with Mg II as strong emission, minimal absorption, and symmetric line profiles are categorized as MgE (acronym for Mg II emitter), while the others belong to non–MgE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This subjective classi- fication is similar to what was adopted in Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The Chisholm model should apply well to the former since Mg II photons suffer little resonant scattering effects, but not perfectly to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show the category of each galaxy in the second to last column in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We discuss further how we handle these two categories in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The Method to Estimate the Escape Fraction of Mg II Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2018) have first introduced that one can de- rive the intrinsic flux of Mg II from a correlation between Mg II/[O III] and [O III]/[O II] (hereafter, the Henry model): R2796 = A2 ×O2 32 +A1 ×O32 +A0 (3) R2796 = log(Fint(Mg II λ2796)/Fint([O III] λ5007)) O32 = log(Fint([O III] λ5007)/Fint([O II] λ3727)) (4) where the emission line fluxes are all intrinsic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', before the attenuation by dust and absorption in the LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Hereafter, we use Fint to denote the intrinsic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' A0, A1, and A2 are coefficients that are dependent on the gas phase metallicity of the galaxy, but little on the ionization parameters and spectral slopes (Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Combining Fint(Mg II) with the measured Fobs(Mg II) from the spectra, one can derive f MgII esc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The photoionization models in Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2018) con- siders ionization bounded (IB) geometry, where most of the cloud remains neutral and is optically thick to escaping pho- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, LCEs with strong Mg II emission lines can be partly density bounded (DB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', mostly optically thin) given their high O32 values observed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) update the correlation coefficients in the Henry model to take into account the DB scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' At a fixed metallicity, they also find the different models from DB and IB only move the correlation along the line defined in Equa- tion (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This should explain why Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) find the Henry model is a relatively good match to galaxies in their simulations, but with moderate scatter given the differ- ent metallicities of their simulated galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Given the derived f MgII esc , one can solve Cf (Mg II) and τthin from Equations (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Note this requires robust mea- surements of Mg II doublet flux ratio, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', R in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2, this can be achieved in MgE, but hardly in non–MgE due to the absorption features in Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Since C f (Mg II) and τthin are then adopted to predict f LyC esc , accurately predictions are more difficult for non–MgE (see detailed discussion in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Correlations between Mg II and Lyα properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Galaxies that are labelled as MgE and non-MgE from our sample are shown in filled and open symbols, respectively (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Left: The net EW from Mg II λ2796 and Lyα are positively correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Each galaxy is shown as a dot with the cross representing its error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Galaxies with strong Mg II emission lines are at the top-right of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The green dashed line represents the best linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Right: The escape fraction of Mg II λ2796 and Lyα are tightly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The correlation coefficients from Kendall τ test are shown at the top-left corner in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The green solid line represents the 1:1 correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' See discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The Method to Predict the Escape Fraction of LyC As discussed in numerous previous publications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Za- ckrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Kakiichi & Gronke 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022), the es- cape of LyC photons can be described as a partial-covering geometry: fesc(LyC) = Cf (H I)e−τthick ×10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4Athick +[1−Cf (H I)]e−τthin ×10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4Athin (5) where Cf (H I) is the covering fractions for optically thick paths of H I which is dominated by neutral gas, and Athick and Athin are the attenuation parameters for LyC photons at optically thick and optically thin paths, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For galaxies in our LzLCS sample, Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) have found that the covering fraction of lower ion- ization lines (LIS, including O I, C II, Si II) trace that of H I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Given similar ionization potentials of Mg II to these lines, we adopt their best-fit linear correlation to estimate Cf (H I) as: Cf (H I) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='63±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='19)Cf (Mg II)+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='54±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='09) (6) For optically thick paths, we assume no LyC photons can escape (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', τthick and/or Athick ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, the first term in Equation (5) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Athin is related to the dust extinc- tion at the LyC, for which we adopt the stellar extinction de- rived from SED fittings in Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' They used Starburst99 template (Leitherer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 1999) and have assumed the extinction law from Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, we can rewrite Equation (5) as: f LyC esc,pd = [1−Cf (H I)]e−N(H I)σph ×10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4E(B−V)k(912) (7) where f LyC esc,pd is the predicted absolute escape fraction of LyC, N(H I) is the column density of neutral hydrogen, σph is the photoionization cross section of H I at 912 Å, E(B −V) is the stellar dust extinction from Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), and k(912) is the total attenuation curve at the Lyman limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Given the Reddy extinction law adopted in Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), we have k(912) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For other extinction laws, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (1989) and Calzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2000), k(912) = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='32 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='62, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' As shown in Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020) and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), by assuming that Mg II and LyC photons escape from sim- ilar optically thin paths, the column density of Mg II [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', N(Mg II)] can be used to trace N(H I) in a large range from DB to nearly IB regions: N(H I) = α×N(Mg II) (8) where N(Mg II) can be calculated from the optical depth of Mg II as inferred from R in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2, and α = N(Mg II)/N(H I) is the column density ratios predicted from CLOUDY models (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' α is dependent on the abundance ratio of [Mg/H] and the ionization, and has typi- cal values ∼ 104 – 105 for galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Combining Equations (5) to (8), we can calculate f LyC esc,pd given the infor- mation of Mg II, metallicity, and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In ∼ 20 galaxies with strong Mg II emission lines, this model predicted f LyC esc,pd has been found to correlate well with the actual f LyC esc measured from the spectra (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Likewise, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022) also found that f LyC esc,pd correlates with the actual f LyC esc for simulated high- redshift galaxies at different LOS (top-middle panel of their Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' But their correlation contains a large scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We 10 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' note that they adopt α as the abundance ratio of hydrogen to oxygen, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', α = 46 H O (Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This assumes the Mg II emission is found in neutral gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is not accu- rate since known LCEs commonly have high O32 values and, thus, at least a fraction of the ISM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', 1 - Cf (H I)) is density bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Therefore, Mg II in these galaxies should originate in regions where N(H II) is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Our adopted α from CLOUDY models in Equation 8 overcomes this prob- lem (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2, we compare f LyC esc,pd with the measured f LyC esc from Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a) based on the HST COS/G140L spectra and SED fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Estimates of the Escape Fraction of Mg II From Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3, we show how to derive f MgII esc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The resulting values are listed in the last column of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Furthermore, in Figure 4, we present the correlations be- tween Mg II and Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the left panel, we compare the net EW (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', the summed EW from both emission and absorp- tion features) between Mg II and Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' To be consistent with the literature, we have multiplied the net EW by -1 to al- low galaxies with strong emission lines to be at the top-right corner, while galaxies with strong absorption lines to be at the bottom-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We find a strong positive correlation between the net EW of Mg II and Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is as expected since both are resonant lines and should follow similar ra- diative transfer processes when travelling out of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The best fit linear correlation is (show as the green dashed line): net EW(Mg II) = a+b×net EW(Lyα) a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='468+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='157 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='157 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='091+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='003 (9) Similar but less significant trends between EW(Mg II) and EW(Lyα) have also been published in Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2018) and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), where they only focused on strong Mg II emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the right panel of Figure 4, we compare the derived f MgII esc with the escape fraction of Lyα (f Lyα esc ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The latter is derived from each galaxy’s HST/COS spectra in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The correlation is significant (p < 10−6) with scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We find most of the galaxies follow the 1:1 correlation shown as the solid green line, which suggests f Lyα esc ≃ f MgII esc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is consistent with the results in Henry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2018) and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022), which also found that f MgII esc and f Lyα esc values are of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This supports the scenario where Mg II and Lyα mainly escape from optically thin (or DB) holes in ISM likely in a single flight (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Saldana-Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, the path lengths of Mg II and Lyα photons travelling out of the galaxy are simi- lar, and the resulting escape fractions are close for both lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Comparisons of measured f LyC esc with the predicted one from Mg II λ2796 emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Galaxies that are labelled as MgE and non-MgE from our sample are shown in filled-red and open- gray symbols, respectively (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Galaxies that are deter- mined to be non-Lyman-continuum-emitter (Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a) are shown as upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The orange dotted lines are to show the fac- tor of 3 scatter around the 1:1 relationship (orange dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We also show 5 galaxies from Guseva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020) as green colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' See discussion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' One possible scenario is that there is zero dust in the optically thin paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Future spatially resolved observations can solve this puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This includes our Lyman-alpha and Continuum Origins Survey (LaCOS, HST-GO 17069, PI: Hayes), which aims to spatially resolve the Lyα emission, dust, and stellar population for 41 out of 66 LzLCS galaxies by HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For all figures in this section, we show Kendall τ coeffi- cients and the probability of a spurious correlation (p val- ues) at the top-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the Kendall test, we have ac- counted for the upper limits (if any) following Akritas & Siebert (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We have also tested the correlations between the scatters in each figure with other galaxy properties, in- cluding metallicity, internal dust extinction, SFR surface den- sity, and stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' However, we do not find significant correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Estimates of the Escape Fraction of LyC from Mg II In Figure 5, we compare f LyC esc,pd derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4 with the f LyC esc values derived from the HST/COS spectra (based on UV continuum fittings, Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We draw galax- ies classified as MgE and non–MgE as filled and open sym- bols, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We also include 5 galaxies from Guseva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020), which have high-quality VLT/X-Shooter obser- vations as well as direct LyC measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We derive f LyC esc,pd in the same way as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4, and remeasure their f LyC esc from HST/COS spectra using the same methodology in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY 11 First of all, there is a strong correlation between the pre- dicted f LyC esc from Mg II and measured ones, given the proba- bility of a spurious correlation p < 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This highlights the power of using Mg II to trace LyC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Given the scatter around the 1:1 relationships line (orange dashed line), our predicted f LyC esc values are accurate within a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Considering only the non-MgEs (gray-open symbols), the correlation is less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This can be because their Mg II emission lines are affected by absorption features, and the derived τ2803,thin and C f (Mg II) from Equation (2) is more uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For some of the non–MgE, their Mg II spectra show signif- icant resonant scattering or absorption signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These in- clude galaxies showing double peaks in each Mg II emission lines (J1310+2148, J1244+0215, J0826+1820, and maybe in J1235+0635), and galaxies showing strong absorption in Mg II (J0723+4146, J0940+5932, J1346+1129, J1314+1048, J0957+2357).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These galaxies should have optically thicker clouds in/around the galaxy, and our measurements of emis- sion line flux of Mg II are also uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, no meaning- ful predictions through Mg II emission lines can be made, and we have excluded these galaxies from Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We note that, among these galaxies, only one (J1310+2148, f LyC esc ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='6%) has small amount of LyC detected from HST/COS spectra, and others show non-detections of LyC (see Figures 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, these non-MgEs are more similar to galaxies that are cosmologically irrelevant to EoR (f LyC esc ≪ 1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' There- fore, precise estimates of their f LyC esc are less important for our understanding of SF galaxies contributing to the reion- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Furthermore, when selecting new LyC emitters for future observations, one can also exclude similar non-MgEs based on the absorption and/or scattering features in Mg II line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the future, we plan to perform detailed radiative transfer models to account for the escape of Mg II out of the galaxy and link it to the escape of Lyα and LyC (Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' in prepa- ration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We can then model and separate the emission and ab- sorption features from the observed Mg II spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This will be particularly helpful to make more realistic predictions of f LyC esc,pd for these galaxies labelled as non–MgE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Overall, our derived f LyC esc,pd from Mg II, metallicity, and dust can correctly trace the measured f LyC esc within a factor of ∼ 3 for MgE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is consistent with previous studies in Chisholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We conclude that Mg II emis- sion lines along with dust can be used to predict the escape of LyC photons in MgEs, but we need additional information to do so in non-MgEs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', detailed radiative transfer models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK We present the analyses of Mg II spectra for 34 galaxies chosen from the LzLCS sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These galaxies have pub- lished HST/COS data for their LyC and Lyα spectral regions, and we have obtained higher S/N and resolution spectra (than SDSS) for their Mg II regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' While previous studies of Mg II in Lyman Continuum Emitter (LCE) candidates have only focused on Mg II emit- ters (MgE), galaxies in our sample have Mg II profiles rang- ing from strong emission to P-Cygni profiles, then to pure absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We find there is a significant trend (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0216) that galaxies detected as strong LCEs show larger EW(Mg II) in emission lines, while non-LCEs present larger EW(Mg II) in absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We discuss the picket-fence geometry for the escape of Mg II photons from galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' While this geometry has been found to apply well to galaxies categorized as MgE, it has limitations in the case of non-MgE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We then discuss how to use the CLOUDY photoionization models to help derive the escape fraction of Mg II (f MgII esc ) from the optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For all galaxies in our sample, we find f MgII esc correlates with the escape fraction of Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We also show that the net equivalent width of Mg II and Lyα are tightly correlated for both MgEs and non-MgEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We also discuss the methods to predict the escape fraction of LyC (f LyC esc,pd) from the measurements of Mg II, metallic- ity, and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We show that the predicted f LyC esc,pd correlates well with the actual f LyC esc derived from the HST/COS spectra within a factor of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' For non-MgEs, the correlation is less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This is because the absorption features in Mg II spectra for non-MgE complicate our measurements of Mg II emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Additional information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', from radiative transfer models, may help solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' In the future, one can apply the Mg II correlations to var- ious different studies, including: 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We will perform de- tailed radiative transfer models to account for the escape of Mg II from the galaxy (Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This will be especially helpful for the cases of non-MgE, where the clouds in/around the galaxy are not optically thin to Mg II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2) For high-z galaxies, one can adopt the observed Mg II fea- tures to estimate the intrinsic amount of Lyα, which can be severely attenuated by the neutral IGM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Thus, with the aid of Mg II, one can get more ac- curate estimates of the IGM neutral fractions from Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' One can conduct similar analyses of Mg II in higher-redshift LCE candidates, whose Mg II emission lines are shifted into the observable bands of the James Webb Space Telescope (JWST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The Lyα–Mg II correlations can be adopted to se- lect Lyα emitters that have detectable Mg II spectra, and the Mg II–LyC correlation can be used to predict f LyC esc in the case when LyC cannot be directly detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 12 XU ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Measurements from Optical Spectra for the Comparison Sample Object O32 O/H FEmi 2796 FEmi 2803 |EWEmi 2796| |EWEmi 2803| |EWAbs 2796| |EWAbs 2803| Label f MgII esc (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) J1033+6353 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='9±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='8 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4 non-MgE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='4 non-MgE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' –Measurements from the optical spectra for galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Galaxies are ordered by decreasing f LyC esc derived in Flury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (2022a) (the same order as Figures 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' The columns are: (b) Flux ratio between [O III] λ5007 and [O II] λ3727;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (c) Gas phase metallicity in the form of 12+log(O/H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (d) and (e) Measured emission line flux of Mg II λλ2796, 2803 lines in units of 10−17 ergs s−1 cm−2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (f) and (g): Measured rest-frame EW in units of Å for the emission part from the Mg II doublet (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (h) and (i) Measured rest-frame EW in units of Å for the absorption part from the Mg II doublet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' (j) Labels based on the Mg II line profiles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', MgE = Mg II emitter, non-MgE = Mg II non-emitter (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' and (k): the derived escape fraction for Mg II λ2796 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' MG II PROPERTIES IN LOW-Z LYMAN CONTINUUM SURVEY 13 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' acknowledge support from NASA STScI grants GO 15865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' Observations reported here were obtained at the MMT Observatory, a joint facility of the University of Arizona and the Smithsonian Institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 1 2 3 4 Support for this work was provided by NASA through grant number HST-GO-15626 from the Space Telescope Sci- ence Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=', under NASA contract NAS 5–26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' These observa- tions are associated with program(s) 13744, 14635, 15341, 15626, 15639, and 15941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' STScI is operated by the Associ- ation of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 5 6 7 8 9 10 11 12 13 14 15 Based on observations collected at the European Organisa- tion for Astronomical Research in the Southern Hemisphere under ESO programme 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='215K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' 16 17 18 The Low-Resolution Spectrograph 2 (LRS2) was devel- oped and funded by the University of Texas at Austin Mc- Donald Observatory and the Department of Astronomy and by Pennsylvania State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtE2T4oBgHgl3EQfvAjJ/content/2301.04087v1.pdf'} +page_content=' We thank the Leibniz- Institut für Astrophysik Potsdam (AIP) and the Institut für 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b/_tFKT4oBgHgl3EQfVC14/content/tmp_files/2301.11786v1.pdf.txt @@ -0,0 +1,7110 @@ +TIF-UNIMI-2023-2 +ZU-TH 04/23 +PSI-PR-23-1 +Soft-parton contributions to heavy-quark production +at low transverse momentum +Stefano Catani(a), Simone Devoto(b), +Massimiliano Grazzini(c) and Javier Mazzitelli(d) +(a)INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia, +Universit`a di Firenze, 50019 Sesto Fiorentino, Firenze, Italy +(b)Dipartimento di Fisica “Aldo Pontremoli”, University of Milano and INFN, Sezione di +Milano, I-20133 Milano, Italy +(c)Physik Institut, Universit¨at Z¨urich, 8057 Z¨urich, Switzerland +(d)Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland +Abstract +We consider QCD radiative corrections to the production of a heavy-quark pair in +hadronic collisions. We present the computation of the soft-parton contributions +at low transverse momentum of the heavy-quark pair up to second order in the +QCD coupling αS. These results complete the evaluation at the next-to-next-to- +leading order (NNLO) of the transverse-momentum resummation formula for this +process. +Moreover, they give all the ingredients that are needed for the NNLO +implementation of the qT subtraction formalism for heavy-quark production. We +discuss the details of the computation and we provide a code that can be used to +obtain the relevant results in numerical form. +January 2023 +arXiv:2301.11786v1 [hep-ph] 27 Jan 2023 + +Contents +1 +Introduction +1 +2 +Heavy-quark production at low transverse momentum +3 +2.1 +Resummation formalism for heavy-quark production . . . . . . . . . . . . . . . . +3 +2.2 +Soft contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3 +Details of the calculation +13 +3.1 +The subtracted integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.2 +Single gluon emission at tree level . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.3 +Single gluon emission at one loop . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +3.3.1 +Massive-massless contribution: I(1) +ij +. . . . . . . . . . . . . . . . . . . . . +21 +3.3.2 +Massive-massive contribution: I(1) +34 . . . . . . . . . . . . . . . . . . . . . . +26 +3.4 +Light-quark pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +3.5 +Double gluon emission +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +3.5.1 +Massless-massless contribution: ˜S12 . . . . . . . . . . . . . . . . . . . . . +39 +3.5.2 +Massless-massive contribution: ˜Sij . . . . . . . . . . . . . . . . . . . . . . +40 +3.5.3 +Massive-massive contribution: ˜Sjj . . . . . . . . . . . . . . . . . . . . . . +44 +3.5.4 +Massive-massive contribution: ˜S34 . . . . . . . . . . . . . . . . . . . . . . +45 +4 +Numerical results +55 +5 +Summary +61 +1 +Introduction +Heavy-quark pair production is one of the classic hard-scattering processes at hadron colliders. +For a sufficiently-heavy quark, the cross section is perturbatively computable as an expansion +in the QCD coupling αS(µ2 +R) where the renormalisation scale µR is of the order of the mass m +of the heavy quark. A large variety of QCD studies of heavy-quark hadroproduction have been +carried out over the years. In this context top-quark pair production plays a special role: being +the heaviest particle in the Standard Model, the top quark couples strongly to the Higgs boson +and is therefore particularly relevant for the mechanism of electroweak symmetry breaking. +As such, top-quark pair production is especially relevant in searches for physics beyond the +Standard Model, it constitutes a possible window on new physics and, at the same time, a +crucial background in many analyses. Bottom and charm quark production have also been +extensively studied at hadron colliders, and allow us to probe QCD at smaller energy scales. +For the above reasons, the study of the hadroproduction of a heavy-quark pair has attracted +the attention and the efforts of the theoretical community for decades. Next-to-leading order +1 + +(NLO) QCD corrections to this process have been available since a long time, both for the +total cross section and for differential distributions [1–5]. Nevertheless, because of the chal- +lenging complications arising at the next order in the perturbative expansion, more than 20 +years passed before next-to-next-to-leading order (NNLO) QCD corrections for top-quark pair +production were also computed [6–13]. Further progress regards the combination of QCD and +EW corrections [14] and the inclusion of top-quark decays [15]. Results by using the MS scheme +for the renormalisation of the top-quark mass are also available [16, 17]. More recently, the +NNLO calculation of Refs. [12, 13] has been extended to bottom-quark pair production [18]. +One of the two available NNLO computations for heavy-quark production [12, 13, 17, 18] is +based on the qT subtraction formalism [19]. The qT subtraction formalism is a method to handle +and cancel the IR divergences in QCD computations at NLO, NNLO and beyond. The method +uses IR subtraction counterterms that are constructed by evaluating the qT distribution of the +produced final-state system in the limit qT → 0. If the produced final-state system is composed +of colourless particles (such as vector bosons, Higgs bosons, and so forth), the behaviour of the +qT distribution in the limit qT → 0 has a universal structure that is explicitly known to the +next-to-next-to-next-to leading order (N3LO) through the formalism of transverse-momentum +resummation [20–24]. The resummation formalism can be extended to the production of final +states containing a heavy-quark pair [25–28]. +The heavy quarks do not lead to additional +collinear singularities (which are absent because of the finite heavy-quark mass) but, being +coloured, they lead to additional soft singularities that need to be properly taken into account. +The NNLO computations of Refs. [12, 13, 17, 18] rely on the explicit evaluation of such soft- +parton contributions due to the coloured massive quarks. +The purpose of this paper is to report on the details of the computation of such soft- +parton terms. +The final numerical results can be obtained by using the program attached +to the arXiv submission of this paper.1 Our calculation is performed within the transverse- +momentum resummation formalism of Ref. [27]. A similar computation, carried out within the +framework of Soft Collinear Effective Theory (SCET) used in Refs. [25, 26], has been presented +in Ref. [32]. +We note that our formalism can be extended to the production of a heavy-quark pair +accompanied by colourless particles [33]. +Such extension has been recently applied to the +evaluation of NNLO corrections to t¯tH [34] and Wb¯b [35] production. In this paper, however, +we will limit ourselves to the case of heavy-quark production, i.e., with no additional colourless +particles. The soft-parton contributions relevant for the production of a heavy-quark pair and +a colourless system will be documented elsewhere. +The paper is organised as follows. In Sect. 2 we review the resummation formalism for heavy- +quark production and we discuss the soft-parton contributions we want to compute. In Sect. 3 +we illustrate our calculation, starting from single-gluon emission at tree level in Sect. 3.2, then +going to single-gluon emission at one loop in Sect. 3.3, soft q¯q emission in Sect. 3.4 and double- +gluon emission in Sect. 3.5. Our numerical implementation and final results are presented in +Sect. 4. In Sect. 5 we summarise our findings. +1 The soft-parton contributions evaluated in this work also enter the MiNNLOPS formalism for the matching +of NNLO calculations to parton showers for heavy-quark production [29–31]. +2 + +2 +Heavy-quark production at low transverse momentum +2.1 +Resummation formalism for heavy-quark production +We consider the inclusive hard-scattering process +h1(P1) + h2(P2) → Q(p3) + ¯Q(p4) + X +(1) +where the collision of the two hadrons h1 and h2 with momenta P1 and P2 produces the heavy- +quark pair Q ¯Q, and X denotes the accompanying final-state radiation. The heavy quarks have +four-momenta p3 and p4, total momentum q = p3 + p4, invariant mass M 2 = q2 and total +transverse momentum ⃗qT = ⃗p3,T + ⃗p4,T. The rapidity of the Q ¯Q pair is y = 1/2 ln(q · P2/q · P1). +The knowledge of M, y and ⃗qT completely specifies the total momentum q of the heavy- +quark pair. The kinematics of the observed heavy quarks is fully determined by q and two +additional independent kinematical variables, that we denote by ⃗Ω. +For example, we can +choose ⃗Ω = {y3, φ3}, where y3 and φ3 are the rapidity and the azimuthal angle of the heavy +quark Q. +The hadronic cross section corresponding to Eq. (1) can be computed by convoluting par- +tonic cross sections with parton distribution functions fa/h(x, µ2 +F) (a = q, ¯q, g denotes the +massless partons) of the colliding hadrons. The partonic cross sections can be computed in +QCD perturbation theory. At the leading order (LO) only two partonic processes contribute: +quark-antiquark annihilation q¯q → Q ¯Q and gluon fusion gg → Q ¯Q. For both processes the +⃗qT dependence of the cross section at LO is simply proportional to δ(2)(⃗qT), since no radiation +is emitted at this perturbative order. At higher perturbative orders the partonic cross section +in the limit qT → 0 receives large logarithmic contributions of the form αn+2 +S +1 +q2 +T lnk(M 2/q2 +T) +(k ≤ 2n−1) that need be resummed to all orders. The resummation is customarily carried out +in impact parameter (⃗b) space, to factorise the kinematics of multiple parton emission. +The all-order structure of the logarithmically enhanced contributions can be written as [27] +dσ(P1, P2; ⃗qT, M, y, ⃗Ω) +d2⃗qT dM 2 dy d⃗Ω += +M 2 +2P1 · P2 +� +c=q,¯q,g +� +dσ(0) +c¯c +� � +d2⃗b +(2π)2ei⃗b·⃗qT Sc(M, b) +× +� +a1,a2 +� 1 +x1 +dz1 +z1 +� 1 +x2 +dz2 +z2 +[(H∆)C1C2]c¯c;a1a2fa1/h1(x1/z1, b2 +0/b2)fa2/h2(x2/z2, b2 +0/b2) , +(2) +where b0 = 2e−γE (γE = 0.5772.... is the Euler number) and the kinematic variables x1 and x2 +are defined as +x1,2 = +M +√2P1 · P2 +e±y . +(3) +The symbol +� +dσ(0) +c¯c +� +is related to the LO cross section dˆσ(0) +c¯c→Q ¯Q for the partonic process +c(p1) + ¯c(p2) → Q(p3) + ¯Q(p4), +c = q, ¯q, g +(4) +3 + +with pi = xiPi (i = 1, 2), and we have +� +dσ(0) +c¯c +� += α2 +S(M 2) +dˆσ(0) +c¯c→Q ¯Q +M 2d⃗Ω +. +(5) +We briefly recall the perturbative ingredients entering the resummation formula in Eq. (2) +(more details can be found in Ref. [27]). The formula contains process-dependent and process- +independent contributions. The functions Ci include the contribution of radiation collinear to +the initial-state partons at small momentum scales q ≲ 1/b, while the Sudakov form factor +Sc accounts for soft and flavour-conserving collinear emissions at scales 1/b ≲ q ≲ M. Since +they are originated by the spin- and qT-dependent collinear splitting kernels, the functions Ci +feature also a dependence on the azimuthal degree of freedom of ⃗b. All the information on the +process-dependent corrections is embodied in the term H∆, while the collinear functions Ci +and the Sudakov form factor Sc are universal. The radiative factor ∆ is specific of heavy-quark +pair production and is due to soft radiation from the Q ¯Q final state and from the initial-state +and final-state interference. It depends on the invariant mass M 2, on the kinematics of the +partonic process in Eq. (4) and on the impact parameter ⃗b. The azimuthal dependence can be +specified through the angle φ = φ3 −φb, where φ3 and φb are the azimuthal angles of ⃗p3,T and ⃗b, +respectively. The hard-virtual term H, which embodies virtual contributions at scale q ∼ M, +depends on the all-loop scattering amplitude Mc¯c→Q ¯Q for the partonic process c¯c → Q ¯Q. +The explicit form of the term H∆ is +(H∆)c¯c = ⟨ � +Mc¯c→Q ¯Q|∆| � +Mc¯c→Q ¯Q⟩ +α2 +S(M 2) +���M(0) +c¯c→Q ¯Q +��� +2 +. +(6) +The symbol M(0) +c¯c→Q ¯Q denotes the Born-level amplitude, while � +Mc¯c→Q ¯Q represents the all-loop +renormalised amplitude after subtraction of the IR singularities (see Eq. (16)). The amplitude +| � +Mc¯c→Q ¯Q⟩ is a vector in the colour space of {c, ¯c, Q, ¯Q} and ∆ is a colour-space operator. In the +gluon fusion channel (c = g), the Lorentz (spin) indeces of � +M in Eq. (6) are properly summed +with the corresponding indeces of the gluon collinear functions Ci (see Eqs. (11) and (13) in +Ref. [27]). +The action of the colour factor ∆ is expressed in terms of the operators2 D and V [27] +∆(⃗b, M) = V†(b, M)D(φ, αS(b2 +0/b2))V(b, M) . +(7) +The evolution factor V resums logarithmic terms αn +S(M 2) lnk(M 2b2) (with k ≤ n). It is obtained +by the exponentiation of the integral of the anomalous dimension matrix Γt, which is specific +2 Here and in the following, the additional dependence on the rapidity difference y34 = y3 − y4 is left under- +stood. +4 + +of transverse-momentum resummation for QQ production +V(b, M) = P q exp +� +− +� M2 +b2 +0/b2 +dq2 +q2 Γt(αS(q2)) +� +. +(8) +The symbol P q in Eq. (8) denotes the anti path-ordering of the exponential matrix with respect +to the integration variable q2. +The soft-parton factor D in Eq. (7) embodies the azimuthal correlations produced by the +soft radiation and it is defined [27] in such a way that it gives a trivial contribution after +integration over the azimuthal angle. We have +⟨D(φ, αS)⟩av. = 1 , +(9) +where the symbol ⟨...⟩av. denotes the average with respect to the azimuthal angle φ. +The explicit expressions of the factor H∆ up to O(αS) and of the anomalous dimension +Γt up to O(α2 +S) are given in Ref. [27].3 In this paper we present a general discussion of the +resummation factor H∆ and of its detailed origin and dependence on soft-parton contributions. +Moreover we explicitly compute H∆ up to O(α2 +S). This O(α2 +S) result is also relevant in the +context of the QCD computation of heavy-quark production at NNLO. Indeed, as recalled +below, it permits the NNLO implementation of the qT subtraction formalism for this production +process. +Within the qT-subtraction formalism, the NNLO differential cross section dσQQ +NNLO of the +process in Eq. (1) is split into a part with qT = 0 and one with qT ̸= 0 +dσQQ +NNLO = dσQQ +NNLO +�� +qT =0 + dσQQ +NNLO +�� +qT ̸=0 . +(10) +Since at the Born level the final state QQ has qT = 0, the NNLO contributions at qT ̸= 0 are +actually given by NLO contributions for the final state QQ+jets +dσQQ +NNLO +�� +qT ̸=0 = dσQQ+jets +NLO +. +(11) +At NNLO, we can hence handle the IR divergences of the qT ̸= 0 part with the available NLO +techniques. By doing so, we are nevertheless left with additional singularities of purely NNLO +origin connected to the limit qT → 0, for which we need an additional subtraction. Following +this strategy, we write the cross section as [19] +dσQQ +NNLO = HQQ +NNLO ⊗ dσQQ +LO + +� +dσQQ+jets +NLO +− dσCT +NNLO +� +. +(12) +The cancellation of the extra singularities of NNLO type is performed by introducing the +counterterm dσCT +NNLO, while the coefficient HQQ +NNLO embodies the information on the virtual +3 The explicit expressions of the corresponding resummation coefficients for the production of an arbitrary +number of heavy quarks accompanied by a colourless system is reported in Ref. [33]. Note that the expression +of the first-order contribution D(1) to D therein is mistyped. The correct expression is obtained by replacing +�b → −�b in Eqs. (25) and (26). +5 + +corrections to the process and contains the qT = 0 contribution. +The counterterm dσCT +NNLO needs to capture the singular behaviour of the amplitude in the +limit qT → 0 and it can been derived by using the knowledge on the low transverse-momentum +spectrum. In particular, it can be obtained from the NNLO perturbative expansion of the +logarithmically-enhanced contributions of the resummation formula in Eq. (2). It depends [36] +on the resummation coefficients that already appear in the case of a colourless final state, on +the additional Q ¯Q resummation coefficients at O(αS) and on the anomalous dimension Γt at +O(α2 +S). +The coefficient HQQ +NNLO contains the virtual corrections to the process in Eq. (4) and contri- +butions that compensate for the subtraction of the counterterm dσCT +NNLO. It is defined as the +NNLO truncation of the following perturbative series +HQQ = 1 + αS +π HQQ(1) + +�αS +π +�2 +HQQ(2) + . . . +(13) +where HQQ can be expressed [12, 33, 36] in terms of the functions that we just introduced in +the context of qT resummation. We have +HQQ = ⟨(HD)C1C2⟩av. , +(14) +where the average is over the azimuthal angle φ, which appears [27] both in the factor D and +through the functions Ci in the gluon channel. Analogously to Eq. (6), the explicit form of the +term HD reads +(HD)c¯c = ⟨ � +Mc¯c→Q ¯Q|D| � +Mc¯c→Q ¯Q⟩ +α2 +S(M 2) +���M(0) +c¯c→Q ¯Q +��� +2 +. +(15) +The second-order coefficient HQQ(2) can be computed with the results presented in this paper. +2.2 +Soft contributions +In our computation we regularise both ultraviolet and IR divergences by using conventional +dimensional regularisation in D = 4 − 2ϵ space-time dimensions (see, e.g., Ref. [37]). The +SU(Nc) QCD colour factors are CF = (N 2 +c −1)/(2Nc), CA = Nc, TR = 1/2 and we use Cc = CF +if c = q and Cc = CA if c = g. We consider nf flavours of massless quarks in addition to the +heavy quark Q. The QCD running coupling αS(µ2 +R) = α +(nf) +S +(µ2 +R) is introduced through MS +renormalisation at the scale µR and decoupling of the heavy quark [37]. +We start our discussion by considering the finite part � +Mc¯c→Q ¯Q of the all-order virtual +amplitude Mc¯c→Q ¯Q, which is defined through the relation [27] +| � +Mc¯c→Q ¯Q⟩ = +� +1 − �Ic¯c→Q ¯Q +� +|Mc¯c→Q ¯Q⟩ , +(16) +6 + +where the subtraction operator �Ic¯c→Q ¯Q in colour space has the following expansion +�Ic¯c→Q ¯Q(αS(M 2), ϵ; {pi}) = +∞ +� +n=1 +�αS(µ2 +R) +2π +�n +�I(n) +c¯c→Q ¯Q(ϵ, M 2/µ2 +R; {pi}) . +(17) +It is useful to introduce the subtraction operator in the simpler case in which a colourless system +F with invariant mass M is produced. In this case we can write [38] +| � +Mc¯c→F⟩ = +� +1 − �Ic +� +|Mc¯c→F⟩ , +(18) +where the subtraction operator �Ic(αS(M 2), ϵ) is now a c-number, and it can be perturbatively +expanded as in Eq. (17). +The explicit expression of the first two perturbative coefficients +�I(1) +c (ϵ, M 2/µ2 +R) and �I(2) +c (ϵ, M 2/µ2 +R) can be found in Ref. [38]. We note that �Ic depends on the +initial-state parton c, but it is completely independent of the produced colourless system F. +For later convenience we also define Vc as follows +Vc = ln(1 − �Ic) , +(19) +and we write its decomposition in IR divergent and IR finite components +Vc = V sing +c ++ V fin +c +. +(20) +The term V sing +c +, which includes the complete IR divergent contributions to Vc, is a perturbative +series in powers of αS(M 2) and the corresponding perturbative coefficients are proportional to ϵ +poles, with no additional ϵ dependence. The remaining ϵ dependence of Vc is entirely embodied +in V fin +c , which is finite in the limit ϵ → 0. The all-order virtual amplitude Mc¯c→F has IR +divergent contributions that are cancelled by �Ic, and � +Mc¯c→F in Eq. (18) is IR finite in the limit +ϵ → 0. +Comparing the transverse-momentum resummation formula in Eq. (2) with the correspond- +ing formula for the production of a colourless system F [38], we recall [27] that the factor +⟨ � +Mc¯c→Q ¯Q|∆| � +Mc¯c→Q ¯Q⟩ in Eq. (6) is analogous to the factor | � +Mc¯c→F|2 for F production and, +therefore, we can introduce the following master formula4 +⟨ � +M|∆| � +M⟩ = +� +⟨M| eV ∗ +c e2Fex(⃗b)eVc |M⟩ +� +ϵ=0 , +(21) +where we have written 1 − �Ic = eVc, according to Eq. (19). The term �Ic in Eq. (18) is due +to real emission contributions to the underlying partonic process c¯c → F. More precisely, �Ic +is produced by radiation of final-state partons that are either soft or collinear to the colliding +partons c and ¯c [38]. In the case of Q ¯Q production, the underlying partonic process is c¯c → Q ¯Q, +and the produced Q and ¯Q act as extra source of soft-parton radiation, while the accompanying +initial-state collinear radiation is the same as for F production. The amount of extra soft +4 For convenience, here and in the following the amplitudes Mc¯c→Q ¯ +Q and � +Mc¯c→Q ¯ +Q are denoted as M and +� +M, by removing the subscript c¯c → Q ¯Q. +7 + +radiation due to Q and ¯Q is embodied by the factor e2Fex in the right-hand side of Eq. (21). +This factor is the result of the integration of the soft-emission contributions after factorisation +of the initial-state emission, which is taken into account by the factor eVc. We note that Fex is +a colour space operator, which depends on the colour charges of the partons c, ¯c, Q, ¯Q. +The real-emission factor eV ∗ +c e2Fex(⃗b)eVc in Eq. (21) is IR divergent, and it cancels the IR +divergences of the virtual amplitude M. This cancellation mechanism and the ensuing structure +of the IR-finite terms ∆ and � +M are discussed in the remaining part of this Section. +The structure of the IR singular contributions in QCD amplitudes with massive partons is +discussed in Refs. [39–43]. The IR singularities of the amplitude M in Eq. (21) are factorised +in the IR divergent operator Z [43] that permits to write the IR-finite remainder Mfin of the +amplitude as follows +|Mfin(µIR)⟩ = Z−1(µIR) |M⟩ . +(22) +Both Z(µIR) and Mfin(µIR) depend on the arbitrary subtraction scale µIR. The operator Z(µIR) +is a perturbative series in powers of αS(µ2 +IR) and the corresponding perturbative coefficients are +proportional to ϵ poles, with no additional ϵ dependence. Unless otherwise stated we will use +µIR = M. +We write the operator Z as follows +Z(M) = ZexZc(M) , +(23) +where the factor Zc(M) embodies the IR divergences due to the initial-state partons c and +¯c, while Zex includes the additional IR divergences due to soft wide-angle radiation from the +colour-charged heavy quarks. Therefore Zc(M) is the IR divergent operator of the amplitude +Mc¯c→F in Eq. (18), and we also have +Zc(M) = e−V sing +c +, +(24) +since the real-emission factor eVc in Eq. (18) cancels the virtual IR divergences of Mc¯c→F. The +operator Zex can be obtained by exponentiation of the integral of the subtracted soft anomalous +dimension Γsub introduced in Ref. [27]. We have +Zex(M) = P q exp +� +−1 +2 +� M2 +0 +dq2 +q2 Γsub(αS(q2)) +� +, +(25) +where αS(q2) is the renormalised QCD coupling in D = 4 − 2ϵ dimensions and the perturbative +expansion of Γsub is +Γsub = αS +2πΓ(1) +sub + +�αS +2π +�2 +Γ(2) +sub + O(α3 +S) . +(26) +8 + +The explicit scale dependence of αS is +αS(q2) = αS(µ2) +�µ2 +q2 +�ϵ � +1 − β0 +ϵ αS(µ2) +� +1 − +�µ2 +q2 +�ϵ� ++ O(α2 +S) +� +(27) +where β0 is the first coefficient of the QCD beta function +12πβ0 = 11CA − 2nf. +(28) +Using Eq. (27), the operator Zex in Eq. (25) can be written as +Zex(M) = e−Vex(M2) , +(29) +where the explicit expression of Vex up to O(α2 +S) reads +Vex(M 2) = αS(M 2) +2π +� +− 1 +2ϵΓ(1) +sub +� ++ +�αS(M 2) +2π +�2 � 1 +ϵ2 +πβ0 +2 Γ(1) +sub − 1 +ϵ +1 +4Γ(2) +sub +� ++ O(α3 +S) . +(30) +We note that the anti-path ordered operator ¯Pq in Eq. (25) is irrelevant to evaluate Zex up to +O(α2 +S). +Using Eqs. (22)–(24) in the right-hand side of Eq. (21) we see that the colourless subtraction +operator Vc only cancels the IR singularities of M that originate from the initial-state emission +factor Zc(M). The virtual IR divergences in Zex are removed by the IR divergences in Fex, as +discussed in the following. The perturbative expansion of Fex(⃗b) can be written as +Fex(⃗b) = α0 +2πSϵ +�b2µ2 +0 +b2 +0 +�ϵ +Fex,1 (φ) + +�α0 +2πSϵ +�2 �b2µ2 +0 +b2 +0 +�2ϵ +Fex,2 (φ) + O(α3 +0) , +(31) +where α0 denotes the unrenormalised QCD coupling. In our calculation of Fex(⃗b), the renor- +malisation of the coupling constant is taken into account by using the MS scheme: the running +coupling αS is related to the bare coupling α0 via the relation +α0µ2ϵ +0 Sϵ = αS(µ2 +R)µ2ϵ +R +� +1 − αS(µ2 +R)β0 +ϵ + O(α2 +S) +� +, +(32) +where5 β0 is given in Eq. (28) and +Sϵ = (4π)ϵe−ϵγE +(33) +is the customary D-dimensional spherical factor. We note that the operator Fex(⃗b) fulfils the +relation F† +ex(⃗b) = Fex(−⃗b). We also point out that, while the function Fex(⃗b) depends on the +vector ⃗b, the dependence on b in Eq. (31) is fully embodied in the prefactors b2nϵ and, therefore, +the perturbative coefficients Fex,n(φ) only depend on the azimuthal degree of freedom φ of ⃗b. +In the following, this dependence is left understood. Each perturbative coefficient can also be +5 At the end of Sect. 3.3 we comment on the contribution to Fex(⃗b) of heavy-quark loops. +9 + +expanded in ϵ as follows +Fex,1 = 1 +ϵ F(−1) +ex,1 + F(0) +ex,1 + ϵ F(1) +ex,1 + . . . , +(34) +Fex,2 = 1 +ϵ2 F(−2) +ex,2 + 1 +ϵ F(−1) +ex,2 + F(0) +ex,2 + . . . . +(35) +The poles in Eqs. (34) and (35) are due to the soft singularities of the real-emission contributions +and, as previously mentioned, they have to cancel the virtual IR divergences due to the factor +Zex in Eq. (21). The cancellation of IR divergences leads to relations between the coefficients +F(k) +ex,n in Eqs. (34), (35) and the coefficients Γ(n) +sub of the ϵ pole contributions in Eqs. (29),(30). +We find the following relations +F(−1) +ex,1 = −1 +4 +� +Γ(1) +sub + h.c. +� +, +(36) +F(−2) +ex,2 = πβ0F(−1) +ex,1 + 1 +8 +�� +Γ(1) +sub − h.c. +� +, F(−1) +ex,1 +� +, +(37) +F(−1) +ex,2 = −1 +8 +� +Γ(2) +sub + h.c. +� ++ 2πβ0F(0) +ex,1 + 1 +4 +�� +Γ(1) +sub − h.c. +� +, F(0) +ex,1 +� +. +(38) +The explicit calculation of Fex is presented in Sect. 3. We have verified that Eqs. (36)–(38) are +fulfilled by our final result for Fex, which is an important cross-check of our computation. +We can now consider the master formula in Eq. (21), implement the cancellation of the real +and virtual IR divergences and derive the expressions of ∆ and � +M. We write +⟨ � +M|∆| � +M⟩ = +� +⟨Mfin| e−V† +ex(M2)e−V sing∗ +c +eV ∗ +c e2Fex(⃗b)eVce−V sing +c +e−Vex(M2) |Mfin⟩ +� +ϵ=0 += +� +⟨Mfin| eV fin∗ +c +e−V† +ex(M2)e2Fex(⃗b)e−Vex(M2)eV fin +c +|Mfin⟩ +� +ϵ=0 += +� +⟨Mfin| eV fin∗ +c +V† +sub(b, M)e−V† +ex(b2 +0/b2)e2Fex(⃗b)e−Vex(b2 +0/b2)Vsub(b, M)eV fin +c +|Mfin⟩ +� +ϵ=0 . +(39) +In the first line of Eq. (39) we have used Eqs. (22), (23), (24) and (29). In the second line we +have used Eq. (20), and the fact that Vc is a c-number that commutes with the other operators +in colour space. In the third line we have introduced the evolution operator Vsub, defined by +the following relation: +e−Vex(M2) = e−Vex(b2 +0/b2) ¯Pq exp +� +−1 +2 +� M2 +b2 +0/b2 +dq2 +q2 Γsub(αS(q2)) +� +≡ e−Vex(b2 +0/b2)Vsub(b, M) . +(40) +The IR poles in the third line of Eq. (39) are fully contained in the individual factors of the +operator e−V† +ex(b2 +0/b2)e2Fex(⃗b)e−Vex(b2 +0/b2). Their cancellation takes place at the operator level after +combining the exponential functions together, and it is guaranteed by the relations between +Fex and Γsub that are reported in Eqs. (36) and (38). Therefore we can safely perform the limit +10 + +ϵ → 0 and we obtain a finite reminder that, for later convenience, we define as follows +lim +ϵ→0 +� +e−V† +ex(b2 +0/b2)e2Fex(⃗b)e−Vex(b2 +0/b2)� += K†(−⃗b)K(⃗b) , +(41) +where we also used the relation F† +ex(⃗b) = Fex(−⃗b). +To recast Eqs. (39) and (41) in the form of Eqs. (6) and (7) we isolate the azimuthal +dependence of K†(−⃗b)K(⃗b) in a factor with azimuthal average equal to unity, thus identifying +the operator D(φ, αS). We write +K†(−⃗b)K(⃗b) = h(αS(b2 +0/b2))D(φ, αS)h(αS(b2 +0/b2)) , +(42) +with +h†(αS(b2 +0/b2)) = h(αS(b2 +0/b2)) , +(43) +⟨D(φ, αS)⟩av. = 1 . +(44) +The expressions for the colour operators h and D can be trivially obtained from K as follows +(h(αS(b2 +0/b2))2 = ⟨K†(−⃗b)K(⃗b)⟩av. , +(45) +D(φ, αS(b2 +0/b2)) = h−1(αS(b2 +0/b2)) +� +K†(−⃗b)K(⃗b) +� +h−1(αS(b2 +0/b2)) . +(46) +In terms of Fex and Γsub they read +h(αS) = 1 + αS +2π ⟨F(0) +ex,1⟩av. + +�αS +2π +�2 � +⟨(F(0) +ex,1)2⟩av. − 1 +2 +� +⟨F(0) +ex,1⟩av. +�2 ++ ⟨F(0) +ex,2⟩av. − 2πβ0 ⟨F(1) +ex,1⟩av. − 1 +4 +�� +Γ(1) +sub − h.c. +� +, ⟨F(1) +ex,1⟩av. +� � ++ O(α3 +S) , +(47) +D(φ, αS) = 1 + 2 αS +2π +� +F(0) +ex,1 +� +cor ++ 2 +�αS +2π +�2 �� +F(0) +ex,2 − 2πβ0F(1) +ex,1 − 1 +4 +�� +Γ(1) +sub − h.c. +� +, F(1) +ex,1 +�� +cor ++ +� +(F(0) +ex,1)2� +cor − ⟨F(0) +ex,1⟩av. +� +F(0) +ex,1 +� +cor − +� +F(0) +ex,1 +� +cor ⟨F(0) +ex,1⟩av. +� ++ O(α3 +S) , +(48) +where, to keep the notation compact, we have defined the azimuthal correlation (f)cor of an +operator f as +(f)cor = f − ⟨f⟩av. . +(49) +In the operator h of Eq. (42) the scale of αS is b2 +0/b2. The scale in h can be evolved up to the +hard scale M 2 by using the operator Vsub of Eq. (40) and by introducing the operator V of +Eq. (8) through the following relation +V(b, M) = h(αS(b2 +0/b2))Vsub(b, M)h−1(αS(M 2)) . +(50) +11 + +From here we also obtain the relation between the anomalous dimensions Γt and Γsub in Eqs. (8) +and (40). Computing the logarithmic derivative of Eq. (50) with respect to M 2 we find +Γt(αS) = 1 +2h(αS)Γsub(αS)h−1(αS) + β(αS)dh(αS) +d ln αS +h−1(αS) , +(51) +where we have introduced the QCD β function +β(αS(q2)) = d ln αS(q2) +d ln q2 += − +∞ +� +k=1 +βk−1αk +S(q2) , +(52) +with β0 given in Eq. (28). At O(α2 +S) Eq. (51) is Eq. (40) of Ref. [27] with the identification +F(1) +t += 2 ⟨F(0) +ex,1⟩av.. +We can collect all the results of our discussion by inserting Eqs. (41), (42) and (50) in the +third line of Eq. (39), and we obtain +⟨ � +M| ∆ | � +M⟩ = ⟨ � +M| V†(b, M)D(φ, αS(b2 +0/b2))V(b, M) | � +M⟩ , +(53) +where the IR finite matrix element | � +M⟩ can be expressed as +| � +M⟩ = lim +ϵ→0 +� +h(αS(M 2))eV fin +c Z−1 |M⟩ +� +, +(54) +and Z−1 |M⟩ = |Mfin⟩ can be obtained from Ref. [37].6 For convenience, we report the explicit +expression of V fin +c +in the limit ϵ → 0 +V fin +c +=Cc +� +− π2 +12 +�αS(M 2) +2π +� ++ +�αS(M 2) +2π +�2 � �607 +162 − 67 +144π2 + π4 +72 − 77 +36 +� +CA ++ +� +−41 +81 + 5 +72π2 + 7 +18ζ3 +� +nf − iπ4 +6 β0 +� ++ O(α3 +S) +� +. +(55) +In this Section we have discussed how the factors ∆ and � +M that appear in the transverse- +momentum resummation formalism of Sect. 2.1 are related to the soft-radiation contribution +Fex(⃗b). The first order resummation coefficients that were presented in Ref. [27] depend on the +first-order term Fex,1 in Eq. (31). In the following sections we illustrate the explicit computation +of the first- and second-order terms Fex,1 and Fex,2. In particular, Fex,2, controls the NNLO +contribution to the operator h (see Eq. (47)) and, through Eq. (54), it allows us to evaluate +the NNLO subtracted amplitude � +M. +6 To be precise the numerical expression of the two-loop amplitude in Ref. [37] is presented by using µIR = m +as IR subtraction scale, while in Eq. (54) the operator Z is defined at the IR subtraction scale µIR = M. +Therefore the implementation of the results of Ref. [37] in Eq. (54) requires the evolution of the numerical +result presented in Ref. [37] from the scale m to the scale M. We also note that a fully analytic result for the +two-loop amplitude in the q¯q → Q ¯Q channel became available recently [44]. +12 + +3 +Details of the calculation +3.1 +The subtracted integrals +The evaluation of the operator Fex(⃗b) introduced in the previous section requires the integration +of the soft-parton contributions after subtraction of the corresponding contribution of initial- +state emission. We can write this symbolically as +Fex(⃗b) = 1 +2 +� +FQ ¯Q − Fcolourless +� +≡ 1 +2Fsub . +(56) +At NLO we just have to consider the emission of one soft gluon, which can be described by the +customary tree-level eikonal factorisation formula, after subtraction of initial-state emission. +The relevant contribution is +I(0) +g (⃗b) = − +� +dDk +(2π)D−1δ+(k2) +��J(0) +g (k) +��2 +sub ei⃗b·⃗kT , +(57) +where the subtracted squared current +���J(0) +g (k) +��� +2 +sub is defined in Eq. (83). At NNLO we need to +consider contributions from: +• single-gluon emission at one-loop order (see Sect. 3.3); +• emission of a soft quark-antiquark pair (see Sect. 3.4); +• emission of two soft gluons (see Sect. 3.5). +The corresponding integrals read +I(1) +g (⃗b) = − +� +dDk +(2π)D−1δ+(k2) +� +J(0)† +g +(k)J(1) +g (k) + c.c. +� +sub ei⃗b·⃗kT , +(58) +I(0) +q¯q (⃗b) = +� +dDk1 +(2π)D−1 +dDk2 +(2π)D−1δ+(k2 +1)δ+(k2 +2)I(0) +q¯q (k1, k2) +�� +subei⃗b·(⃗kT 1+kT 2) , +(59) +I(0) +gg (⃗b) = 1 +2 +� +dDk1 +(2π)D−1 +dDk2 +(2π)D−1δ+(k2 +1)δ+(k2 +2)W(0) +gg (k1, k2) +�� +subei⃗b·(⃗kT 1+kT 2) , +(60) +where the soft factors +� +J(0)† +g +(k)J(1) +g (k) + c.c. +� +, I(0) +q¯q (k1, k2) and W(0) +gg (k1, k2) are explicitly given +in Eq. (104), (173) and (196), respectively. As in Eq. (57), the label “sub” in Eqs. (58)–(60) +denotes the subtraction procedure that removes the initial-state emission contributions. Details +of this procedure are given in Sects. 3.3, 3.4 and 3.5. All the integrals are computed by using +dimensional regularisation with D = 4 − 2ϵ dimensions. The relations with the perturbative +13 + +coefficients Fex,1 and Fex,2 in Eq. (31) are +2 × Sϵ +8π2 +�b2 +b2 +0 +�ϵ +Fex,1(φ) = I(0) +g (⃗b) , +(61) +2 × +S2 +ϵ +(8π2)2 +�b2 +b2 +0 +�2ϵ +Fex,2(φ) = I(1) +g (⃗b) + I(0) +q¯q (⃗b) + I(0) +gg (⃗b) . +(62) +We observe that the expression for Fex,2 (see Eq. (35)) has up to double poles in ϵ. This is +however not the case for the different contributions defined here: in particular terms 1/ϵ3 are +separately present in I(1) +g +and I(0) +gg , but they cancel in Eq. (62). +All the integrals presented so far have been written in b-space, that is, in the space of the +impact parameter b, connected to the ordinary space (qT-space) by a Fourier transform. The +transformation from a b-space integral in a qT-space one is hence obtained with the formal +substitution +δ(D−2) � +⃗qT + ⃗kT1 + ⃗kT2 +� +−→ +ei⃗b·(⃗kT 1+⃗kT 2) . +(63) +For the computation of the function h in Eq. (47), azimuthal averages are required, which +are denoted as ⟨...⟩av.. We compute the D-dimensional azimuthal average of a function F(φ) as +⟨F(φ)⟩av. = +1 +B +� 1 +2, 1 +2 − ϵ +� +� 1 +−1 +d cos φ (1 − cos2 φ)− 1 +2 −ϵF(φ) , +(64) +where B(x, y) is the Euler beta function. +We note that, when considering the azimuthally averaged result, the step from b-space to +qT-space is straightforward, and is determined by the overall dependence on b of the integral +under consideration. Given a b-space function I(⃗b) we introduce the corresponding qT-space +transform as +˜I(⃗qT) = +1 +(2π)D−2 +� +dD−2⃗b I(⃗b) e−i⃗b·⃗qT . +(65) +Performing the azimuthal average in qT space of Eq. (65) we obtain +⟨˜I(⃗qT)⟩av. = +1 +(2π)D−2 +� +dD−2⃗b ⟨I(⃗b)⟩av. e−i⃗b·⃗qT . +(66) +If the b-space function has the factorised form +I(⃗b) = f(b2)¯I(ˆb) , +(67) +where the function ¯I(ˆb) depends only on the azimuthal angle of ⃗b, Eq. (66) gives +⟨˜I(⃗qT)⟩av. = ⟨¯I(ˆb)⟩av. +1 +(2π)D−2 +� +dD−2⃗b f(b2) e−i⃗b·⃗qT . +(68) +By inspection of the structure of Eq. (31), we see that the soft integrals to be evaluated at +14 + +NnLO are of the form +I(⃗b) = b2nϵ ¯I(ˆb) . +(69) +We can then use Eq. (68) with f(b2) = b2nϵ to obtain +⟨˜I(⃗qT)⟩av. = 4nϵπ−1+ϵ +� 1 +q2 +T +�1+(n−1)ϵ Γ(1 + (n − 1)ϵ) +Γ(−nϵ) +⟨¯I(ˆb)⟩av. +(70) +We conclude this section by specifying the kinematical variables for the Born level process +in Eq. (4). The polar angle θ is defined as the angle between the beam axis and the momentum +of the final-state heavy quark in the centre-of-mass frame of the colliding partons. The variable +β is defined as +β = +√ +1 − τ , +(71) +with 0 < τ < 1 +τ = 4m2 +s +, +(72) +where s = (p1 + p2)2 = (p3 + p4)2. We also introduce the following auxiliary variables, that will +be useful in order to write our partial results in a more compact form +B = p2 +T,3 +m2 = p2 +T,4 +m2 = +β2 +1 − β2 sin2 θ , +(73) +r = +√ +1 + B , +(74) +v = +� +1 − +� +2m2 +s − 2m2 +�2 += +2β +1 + β2 , +(75) +c = 1 − β +1 + β , +(76) +cT = 1 − +√ +1 − r2 τ +1 + +√ +1 − r2 τ . +(77) +3.2 +Single gluon emission at tree level +The evaluation of the soft-gluon contributions at NLO has already been performed in Ref. [27]. +In the following, we describe the strategy adopted to carry out the calculation. We note that, +for the extension to NNLO, we need to obtain the NLO result up to O(ϵ) (see Eq. (47)). +The integral I(0) +g (⃗b) in Eq. (57) is obtained from the subtracted current +���J(0) +g (k) +��� +2 +sub, which +is constructed as follows. We start from the tree-level eikonal current J(0) +g (k) describing the +emission of a soft gluon with momentum k from the c(p1)¯c(p2) → Q(p3) ¯Q(p4) Born level +amplitude +J(0) +g,µ(k) = +4 +� +i=1 +Ti +piµ +(pi · k) . +(78) +15 + +The corresponding factorisation formula reads7 +|M(0) +c¯c→Q ¯Qg|2 ∼ (g0µϵ +0)2⟨M(0) +c¯c→Q ¯Q|J(0)† +g,µ (k) dµν(k) J(0) +g,ν(k)|M(0) +c¯c→Q ¯Q⟩ += −(g0µϵ +0)2⟨M(0) +c¯c→Q ¯Q| +��J(0) +g (k) +��2 |M(0) +c¯c→Q ¯Q⟩ , +(79) +where g0 is the bare coupling (g2 +0 = 4πα0), +dµν(k) = −gµν + gauge terms +(80) +is the spin-polarisation tensor of the soft gluon and the gauge terms give vanishing contribution +due to current conservation. The square of the current can be written in the form +��J(0) +g (k) +��2 = +� +j=3,4 +� +p2 +j +(pj · k)2T2 +j + +� +i=1,2 +2pi · pj +(pi · k)(pj · k)Ti · Tj +� ++ +2p3 · p4 +(p3 · k)(p4 · k)T3 · T4 + +2p1 · p2 +(p1 · k)(p2 · k)T1 · T2 . +(81) +From this expression we need to subtract the initial-state contribution, which is relevant for +the production of a colourless system. It reads +��J(0) +g (k) +��2 +colourless = +2p1 · p2 +(p1 · k)(p2 · k)T1 · T2 = − +(p1 · p2) +(p1 · k)(p2 · k) +� +T2 +1 + T2 +2 +� += − +� +(p1 · p2) +(p1 · k)(p1 + p2) · k + +(p1 · p2) +(p2 · k)(p1 + p2) · k +� � +T2 +1 + T2 +2 +� +, +(82) +where we used the colour conservation relation T1 + T2 = 0 for the corresponding production +process. The subtracted squared current that appears in Eq. (57) is defined as +��J(0) +g (k) +��2 +sub = +��J(0) +g (k) +��2 − +��J(0) +g (k) +��2 +colourless += +� +j=3,4 +� +m2 +(pj · k)2T2 +j + 2 +� +i=1,2 +�pi · pj +pj · k − +p1 · p2 +(p1 + p2)k +� Ti · Tj +pi · k +� ++ +2p3 · p4 +(p3 · k)(p4 · k)T3 · T4 , +(83) +We emphasise that each of the three colour contributions in Eq. (83) is separately collinear +safe. +Using Eq. (83) the evaluation of Eq. (57) is reduced to the computation of the following +7 Here +and +in +the +following +the +unrenormalised +scattering +amplitudes +are +denoted +as +Mu += +α0µ2ϵ +0 +� +M(0) + M(1) + .... +� +where M(0) is the tree-level contribution, M(1) is the one-loop virtual correction +and so forth. +16 + +integrals +Ijj(⃗b) = +� +dDk δ+(k2) +m2 +(pj · k)2 ei⃗b·⃗kT , +(84) +Iij(⃗b) = +� +dDk δ+(k2) +1 +pi · k +�pi · pj +pj · k − +p1 · p2 +(p1 + p2) · k +� +ei⃗b·⃗kT , +(85) +I34(⃗b) = +� +dDk δ+(k2) +p3 · p4 +(p3 · k)(p4 · k) ei⃗b·⃗kT , +(86) +where i = 1, 2 labels an initial-state parton, while j = 3, 4 labels one of the final-state massive +particles. In terms of these definitions, Eq. (57) reads +I(0) +g (⃗b) = − +1 +(2π)D−1 +� � +j=3,4 +� +Ijj(⃗b) T2 +j + 2 +� +i=1,2 +Iij(⃗b) Ti · Tj +� ++ 2 I34(⃗b) T3 · T4 +� +. +(87) +The simplest contribution is the one where only a massive final-state particle is involved, Ijj +defined in Eq. (84). To perform its computation, we introduce light-cone coordinates +p± = p0 ± pz +√ +2 +, +pµkµ = p+k− + p−k+ − ⃗pT · ⃗kT , +(88) +and Eq. (84) becomes +Ijj(⃗b) = +� +dk+ dk− dD−2⃗kT δ(2k+k− − ⃗k2 +T) +m2 ei⃗b·⃗kT +� +pj,+k− + pj,−k+ − ⃗pj,T · ⃗kT +�2 . +(89) +The delta function can now be used to perform the integral over k+ +Ijj(⃗b) = +� +dk− dD−2⃗kT +2 m2 k− ei⃗b·⃗kT +� +pj,−k2 +T + 2pj,+k2 +− − 2k−⃗pj,T · ⃗kT +�2 . +(90) +The leftover angular integral can be simplified by removing the angular dependence from the +denominator with an appropriate shift of the ⃗kT variable. We obtain +Ijj(⃗b) = (2π)1−ϵbϵm2 +� +dk− +2k− +p2 +j,− +e +i +k− +pj,− +⃗b·⃗pj,T +� +dkT +k1−ϵ +T +J−ϵ(bkT) +� +k2 +T + m2 k2 +− +p2 +j,− +�2 , +(91) +where Jn(x) is the Bessel function of the first kind. Including the azimuthal average, we are +now left with a three-fold integral that can be performed via standard techniques to all orders +in ϵ. +A similar strategy can be followed to compute Iij(⃗b) in Eq. (85), but this requires some +additional care. The term Iij(⃗b) involves the contribution of the initial-state emitter that is +massless, and this may lead to a collinear singularity in the region pi · k → 0. The collinear +singularity is absent in the complete integrand of Iij(⃗b), but it is present in the two separate +17 + +contributions that correspond to the two terms in the square bracket of Eq. (85). To apply the +same integration procedure used for Ijj(⃗b), the two contributions must be computed separately +and, therefore, a regulator for the collinear singularity needs be introduced. We thus multiply +the integrand by the factor [45, 46] +�pi · k +m2 +�2λ +, +(92) +where λ is a small, positive coefficient and the mass scale m has been chosen equal to the +heavy-quark mass, but it is in principle arbitrary. With the inclusion of this additional factor, +the collinear singularity is regularised, and after integration it leads to poles in λ, which cancel +with each other once the results from the two contributions are combined. +The divergence that appears in the intermediate steps of the evaluation of Iij(⃗b) is just +an artifact of the approximation used to compute the small-qT behavior. Similar divergences +arise in SCET computations and are usually called rapidity divergences [47–51]. +However, +we point out that the term Iij(⃗b) and our entire soft contributions in Eqs. (57)–(62) have no +collinear or rapidity divergences. In our computation the collinear singularities from initial- +state emission can only appear due to technical reasons, since for practical purposes we split +integrable integrands in several non-integrable terms that are evaluated separately. +We now focus on the final integral, I34(⃗b) in Eq. (86). It can be computed from Ijj(⃗b) by +using Feynman parametrisation +I34(⃗b) = +� 1 +0 +dx +� +dDk δ+(k2) +p3 · p4 +(p(x) · k)2 ei⃗b·⃗kT = p3 · p4 +m2 +� 1 +0 +dx Ijj(⃗b) +�� +pj=p(x) , +(93) +where we introduced the auxiliary momentum +pµ(x) = xpµ +3 + (1 − x)pµ +4 . +(94) +The integration over the Feynman parameter can be easily performed in terms of multiple +polylogarithms after the azimuthal average and an expansion to O(ϵ). +We now present our results for the azimuthally averaged integrals ⟨Ijj(⃗b)⟩av., ⟨Iij(⃗b)⟩av. and +⟨I34(⃗b)⟩av.. When the all order result is available, we show both the expression before and after +the ϵ expansion. We find +⟨Ijj(⃗b)⟩av. =π1−ϵ Γ(1 − ϵ) +�b2 +4 +�ϵ � +−1 +ϵ +2F1 (1, −ϵ; 1 − ϵ; −B) +� +=π1−ϵ Γ(1 − ϵ) +�b2 +4 +�ϵ � +− 1 +ϵ − ln (1 + B) + ϵ Li2 (−B) + O(ϵ2) +� +, +(95) +⟨Iij(⃗b)⟩av. = lim +λ→0 +1 +2π1−ϵ +�b2 +4 +�ϵ +Γ( λ +2 − ϵ)Γ( λ +2) +� �pi · pj +m +�λ +2F1 +�λ +2, λ +2 − ϵ; 1 − ϵ; −B +� +− +�p1 · p2 +√s +�λ +2F1 +�λ +2, λ +2 − ϵ; 1 − ϵ; −1 +s +� � +18 + +=π1−ϵΓ(1 − ϵ) +2 +�b2 +4 +�ϵ � +− 2 +ϵ ln +�2 pi · pj +√s m +� ++ Li2 (−B) + ϵLi3 (−B) + O(ϵ2) +� +, (96) +⟨I34(⃗b)⟩av. =π1−ϵ Γ(1 − ϵ) +�b2 +4 +�ϵ 1 + β2 +2β +� +−1 +ϵL0(β) − L1(β, θ) + ϵ P2(β, θ) + O(ϵ2) +� +, +(97) +where the coefficient λ is the one introduced with the collinear regulator in Eq. (92) and the +functions Ln(β, θ), Pn(β, θ) are defined as +Ln(β, θ) = (p3 · p4) +2β +1 + β2 +� 1 +0 +dx +p(x)2 lnn +� +1 + ⃗pT(x)2 +p(x)2 +� +→ +� β +−β +dz +1 − z2 lnn +�1 − z2 cos θ +1 − z2 +� +, +(98) +Pn(β) = (p3 · p4) +2β +1 + β2 +� 1 +0 +dx +p(x)2Lin +� +−⃗pT(x)2 +p(x)2 +� +→ +� β +−β +dz +1 − z2Lin +�z2 sin2 θ +z2 − 1 +� +. +(99) +The momentum pµ(x) is defined in Eq. (94), and in the last step in Eqs. (98), (99) we have +used ⃗p3 = −⃗p4. The explicit expressions of the functions L0(β), L1(β, θ) and P2(β, θ) read +L0(β) = ln +�1 + β +1 − β +� +, +(100) +L1(β, θ) = ln +�1 + β +1 − β +� +ln (1 + B) − Li2 +� +4β +(1 + β)2 +� +− 1 +2 ln2 +�1 + β +1 − β +� ++ Li2(1 − c cT) ++ Li2 +� +1 − c +cT +� ++ 1 +2 ln2 cT , +(101) +P2(β, θ) = G +� +0, 0, β − 1 +2β , sin2 +�θ +2 +�� ++ G +� +0, 1, β − 1 +2β , sin2 +�θ +2 +�� ++ G +� +0, β − 1 +2β , 0, sin2 +�θ +2 +�� ++ G +� +0, β − 1 +2β , 1, sin2 +�θ +2 +�� +− G +� +1, 0, β − 1 +2β , sin2 +�θ +2 +�� +− G +� +1, 1, β − 1 +2β , sin2 +�θ +2 +�� +− G +� +1, β − 1 +2β , 0, sin2 +�θ +2 +�� +− 2 ln(1 − β)G +� +1, 0, sin2 +�θ +2 +�� +− 2 ln(1 − β)G +� +1, 1, sin2 +�θ +2 +�� +− G +� +1, β − 1 +2β , 1, sin2 +�θ +2 +�� +− ln +�sin2(θ) +4 +� +G +� +0, β − 1 +2β , sin2 +�θ +2 +�� ++ ln +�sin2(θ) +4 +� +G +� +1, β − 1 +2β , sin2 +�θ +2 +�� ++ 2 ln(1 − β) ln +�sin2(θ) +4 +� +ln +� +cos2 +�θ +2 +�� +. +(102) +with c and cT defined in Eq. (76) and Eq. (77) respectively. +The function P2(β, θ) is expressed in terms of multiple polylogarithmic functions G. We +note that the same kind of integrals in Eqs. (98), (99) will also appear at a later stage in the +computation of the double gluon emission contribution (see Sect. 3.5): in this case though we +will need Ln and Pn up to n = 3. +19 + +3.3 +Single gluon emission at one loop +We now focus on the emission of a soft gluon at one loop order. The corresponding factorisation +formula reads [52, 53] +⟨M(0) +c¯c→Q ¯Qg|M(1) +c¯c→Q ¯Qg⟩ + c.c. ≃ − (g0µϵ +0)2 � +⟨M(0) +c¯c→Q ¯Q|J(0) +g (k) · J(0) +g (k)|M(1) +c¯c→Q ¯Q⟩ + c.c. +� ++ (g0µϵ +0)4 � +⟨M(0) +c¯c→Q ¯Q|J(0)† +g +(k) · J(1) +g (k)|M(0) +c¯c→Q ¯Q⟩ + c.c. +� +, (103) +where J(1) +g (k) is the one-loop correction to the soft-gluon current. The first contribution in +Eq. (103) factorises the tree-level squared current from the interference between the c¯c → Q ¯Q +Born and one-loop amplitudes. Such term does not lead to new soft contributions to Fex. The +product of the tree and loop soft currents can be written as +J(0)† +g +(k) · J(1) +g (k) + c.c. =2CA +� +i̸=j +� +(pi · pj) +(pi · k)(pj · k) − +m2 +(pj · k)2 +� +Rij Ti · Tj +− 4π +� +i,j,k +′ +pi · pj +(pi · k)(pj · k)Iikf abcT a +i T b +kT c +j , +(104) +where �′ +i,j,k denotes the sum over distinct indices (i ̸= j, j ̸= k, k ̸= i). The expansion in ϵ +of the Rij, Iij functions can be found in Ref. [53] and, in the case of two massive emitters, a +simplified expression has been presented in Ref. [54]. +Since we limit ourselves to considering heavy-quark production, we only need to evaluate +the contribution proportional to Rij. In fact Iij is proportional to the three-partons correlator +f abcT a +i T b +kT c +j that vanishes when acting on the tree-level amplitudes of the process8 c¯c → Q ¯Q [56– +58]. +By using colour conservation, we can apply the same procedure employed in Sect. 3.2 to +isolate the initial-state radiation and thus replace the first contribution on the right hand side +of Eq. (104) with +� +J(0) +g (k)·J(1) +g (k) + c.c. +� +sub ≡ +=2CA +� +i=1,2 +j=3,4 +�� +2(pi · pj) +(pi · k)(pj · k) − +m2 +(pj · k)2 +� +Rij − +2(pi · pj) +(pi · k)(p1 + p2) · kR12 +� +Ti · Tj ++ 2CA +� +2(p3 · p4) +(p3 · k)(p4 · k) − +m2 +(p3 · k)2 − +m2 +(p4 · k)2 +� +R34 T3 · T4 + .... +(105) +where the dots stand for the contributions proportional to Iij that will eventually vanish when +evaluated onto tree-level amplitudes. We now need to expand Rij in powers of ϵ. In order to +8 Note that this is not generally the case for processes in which the heavy-quark pair is accompanied by +particles with complex couplings (see the Note Added in Ref. [55]). +20 + +match the normalisation used in Ref. [53] we write +Rij = +� +(pi · pj) +2(pi · k)(pj · k) +�ϵ +Rij , +(106) +where +Rij = +∞ +� +n=−2 +R(n) +ij ϵn , +(107) +and R(n) +ij +with n ≤ 2 are given in Sect. 2 of Ref. [53]. The integral of the one-loop squared +current in Eq. (58) can be organised into a massless-massive and a massive-massive contribution +based on their colour factor +I(1) +g (⃗b) = − +2CA +(2π)D−1 +� +� +� +� +� +� +j=1,2 +i=3,4 +I(1) +ij (⃗b) Ti · Tj + I(1) +34 (⃗b) T3 · T4 +� +� +� +� +� +, +(108) +where the massless-massive contribution reads +I(1) +ij (⃗b) = +� +dDk δ+(k2) +�� +2(pi · pj) +(pi · k)(pj · k) − +m2 +(pj · k)2 +� +Rij − +2(pi · pj) +(pi · k)(p1 + p2) · kR12 +� +ei⃗b·⃗kT , +(109) +while the massive-massive contribution is +I(1) +34 (⃗b) = +� +dDk δ+(k2) +� +2(p3 · p4) +(p3 · k)(p4 · k) − +m2 +(p3 · k)2 − +m2 +(p4 · k)2 +� +R34 ei⃗b·⃗kT . +(110) +3.3.1 +Massive-massless contribution: I(1) +ij +By inspecting Eq. (109) we can identify three different contributions proportional to +(pi·pj) +(pi·k)(pj·k), +m2 +(pj·k)2 and +(pi·pj) +(pi·k)(p1+p2)·k, respectively. We therefore define the three auxiliary integrals +I(1) +ij,ij(⃗b) = +� +dDk δ+(k2) +(pi · pj) +(pi · k)(pj · k) Rij +� +(pi · pj) +2(pi · k)(pj · k) +�ϵ +ei⃗b·⃗kT , +(111) +I(1) +ij,jj(⃗b) = +� +dDk δ+(k2) +m2 +(pj · k)2 Rij +� +(pi · pj) +2(pi · k)(pj · k) +�ϵ +ei⃗b·⃗kT , +(112) +I(1) +ij,i(12)(⃗b) = +� +dDk δ+(k2) +(pi · pj) +(pi · k)(p1 + p2) · k R12 +� +(p1 · p2) +2(p1 · k)(p2 · k) +�ϵ +ei⃗b·⃗kT . +(113) +In terms of these auxiliary integrals, I(1) +ij +reads +I(1) +ij (⃗b) = 2 I(1) +ij,ij(⃗b) − I(1) +ij,jj(⃗b) − 2I(1) +ij,i(12)(⃗b) . +(114) +We start from I(1) +ij,ij. In this case we have a collinear singularity associated with the radiation +from the initial-state massless particle which is due to the factor (pi · k) in the denominator. +21 + +To take care of it, we can introduce a λ regulator similarly to what was done in the case of the +NLO contribution in Eq. (92) +�pi · k +m2 +�2λ +, +(115) +with λ being positive. The collinear singularity will then be translated into poles in λ, which +will cancel with analogous poles in the massless-massless contribution I(1) +ij,i(12). +From this stage, we can closely follow the procedure used in Sect. 3.2 to perform the integral +over the phase space of the emitted gluon. We obtain +I(1) +ij,ij(⃗b) =(m2)−3λπ1−ϵ(pi · pj)2λ +�b2 +4 +�2ϵ−λ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +� ∞ +0 +dw +Rij(w) +w(1 + w)1+ϵ +× +� +� +�wλ + +� +�2F1 +� +�−2ϵ, −ϵ; 1 +2; +�⃗b · ⃗pT,j +b m +�2 +w +� +� − 1 − 4ϵi⃗b · ⃗pT,j +b m +×√w Γ( 1 +2 − 2ϵ)Γ(1 + ϵ) +Γ(1 − 2ϵ)Γ( 1 +2 + ϵ) +2F1 +� +�1 +2 − 2ϵ, 1 +2 − ϵ; 3 +2; +�⃗b · ⃗pT,j +b m +�2 +w +� +� +� +� +� +� +� , +(116) +which after azimuthal average becomes +⟨I(1) +ij,ij(⃗b)⟩av. = (m2)−3λπ1−ϵ(pi · pj)2λ +�b2 +4 +�2ϵ−λ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +� ∞ +0 +dw +Rij(w) +w(1 + w)1+ϵ +× +� +wλ + Re {[2F1 (−2ϵ, −ϵ; 1 − ϵ; wB) − 1] (1 + i cot(πϵ))} +� +. +(117) +Expanding in ϵ the expression in the curly bracket of Eq. (117) we obtain +Re {[2F1 (−2ϵ, −ϵ; 1 − ϵ; wB) − 1] (1 + i cot(πϵ))} = − 2ϵ +π Im [Li2 (wB)] + O(ϵ2) +=2ϵ ln (wB) θ (wB − 1) + O(ϵ2) . +(118) +In Eqs. (116) and (117) we have defined the adimensional variable w as +w = m2 +p2 +j,− +k2 +− +k2 +T +. +(119) +The kinematical invariants can be written in terms of w as +(pi · k) = (pi · pj) +m +kT +√w , +(120) +(pj · k) = m +2 kT +� 1 +√w + √w +� +, +(121) +while the explicit expression of the coefficients R(n) +ij +presented in Ref. [53] in terms of w reads +R(−2) +ij += −1 +2 , +(122) +22 + +R(−1) +ij += 0 , +(123) +R(0) +ij += 1 +24 +� +5π2 − 6w ln2 +� +w +w + 1 +�� +, +(124) +R(1) +ij += 1 +12 +� +6(w − 1)Li3 +� +w +w + 1 +� ++ 6(w − 1)Li2 +� +1 +w + 1 +� +ln +� +w +w + 1 +� ++ 2(7 − 3w)ζ3 ++ ln +� +w +w + 1 +� � +π2(6w + 1) − 3(w − 1) ln +� +w +w + 1 +� +ln(w + 1) +� � +. +(125) +Our task is now to integrate Eq. (117) with the expansion of R defined in Eqs. (122)–(125). +The final result reads +⟨I(1) +ij,ij(⃗b)⟩av. =(m2)−3λπ1−ϵ(pi · pj)2λ +�b2 +4 +�2ϵ−λ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +� +1 +λ +� +− 1 +2ϵ2 + 5π2 +24 + 7ζ3ϵ +6 ++ O +� +ϵ2�� ++ +� +1 +ϵ +� +−Li2 +� +− 1 +B +� +− 1 +2 ln2(B) − π2 +12 +� ++ 1 +6 +� +− 6Li3 +� +B +B + 1 +� +− 6Li2 +� +− 1 +B +� +ln(B + 1) + ln3 +� 1 +B + 1 +� ++ ln3(B) + 3 ln(B) ln2(B + 1) +− 6 ln2(B) ln(B + 1) − 2π2 ln +� 1 +B + 1 +� +− 2π2 ln(B) + π2 ln(B + 1) − 6ζ3 +� ++ ϵ +� +ζ3 ln(B) + ζ3 ln(B + 1) − 1 +2Li2 +� +− 1 +B +� +2 + 1 +4π2Li2 +� +− 1 +B +� +− 2Li4 +� +− 1 +B +� +− Li4 +� +1 +B + 1 +� +− Li4 +� +B +B + 1 +� +− 1 +2Li2 +� +− 1 +B +� +ln2(B) +− Li3 +� +− 1 +B +� +ln(B) − Li3 +� +− 1 +B +� +ln(B + 1) − 2Li3 +� +1 +B + 1 +� +ln(B + 1) +− 2Li3 +� +B +B + 1 +� +ln(B + 1) + S2,2 +� +− 1 +B +� +− 1 +24 ln4(B) + 7 +24 ln4(B + 1) ++ 1 +3 ln(B + 1) ln3(B) + 1 +3 ln3 +� 1 +B + 1 +� +ln(B + 1) − 3 +4 ln2(B + 1) ln2(B) +− 73 +24π2 ln2(B) + 19 +6 π2 ln2 +� 1 +B + 1 +� +− 11 +4 π2 ln2(B + 1) +− 1 +24 ln2 +� 1 +B + 1 +� +ln2 +� +2B +2B + 1 +� +− 1 +24 ln2 +� 1 +B + 1 +� +ln2 +� +1 +2B + 1 + 1 +� ++ 1 +12 ln2 +� 1 +B + 1 +� +ln +� +2B +2B + 1 +� +ln +� +1 +2B + 1 + 1 +� ++ 35 +6 π2 ln(B + 1) ln(B) +− 2 +3π2 ln +� 1 +B + 1 +� +ln(B + 1) − 23 +240π4 +� ++ O +� +ϵ2� +� ++ O (λ) +� +. +(126) +with B defined as in Eq. (73) and S2,2 being the Nielsen generalised polylogarithm function. +23 + +We now consider the integral I(1) +ij,jj(⃗b). +The only difference with I(1) +ij,ij(⃗b) consists in the +replacement +(pi·pj) +(pi·k)(pj·k) → +m2 +(pj·k)2, which in terms of our integration variable w implies +2 +k2 +T(1 + w) −→ +2 +k2 +T(1 + w) +2w +(1 + w) , +(127) +that is, we simply need to multiply the integrand by a factor 2w/(1 + w). In addition, the +presence of only final-state emitters in the integrand implies that we can set λ = 0 throughout. +We can use this method to obtain from Eq. (116) an expression for I(1) +ij,jj as an integral over w +I(1) +ij,jj(⃗b) =π1−ϵ +�b2 +4 +�2ϵ Γ(−2ϵ) +Γ(1 + ϵ) +� ∞ +0 +Rij(w) +(1 + w)2+ϵ +�� +2F1 +� +−2ϵ, −ϵ; 1 +2; c2 +jbw +� +− 4ϵicjb +√w Γ( 1 +2 − 2ϵ)Γ(1 + ϵ) +Γ(1 − 2ϵ)Γ( 1 +2 + ϵ) +2F1 +�1 +2 − 2ϵ, 1 +2 − ϵ; 3 +2; c2 +jbw +� �� +, +(128) +where we have defined +cjb = +⃗b · ⃗pT,j +b m +. +(129) +The azimuthally averaged equivalent of Eq. (128) can be obtained from Eq. (117) +⟨I(1) +ij,jj(⃗b)⟩av. =π1−ϵ +�b2 +4 +�2ϵ Γ(−2ϵ) +Γ(1 + ϵ) +� ∞ +0 +dw +Rij(w) +(1 + w)2+ϵ +× {1 + Re {[2F1 (−2ϵ, −ϵ; 1 − ϵ; wB) − 1] (1 + i cot(πϵ))}} . +(130) +We obtain +⟨I(1) +ij,jj(⃗b)⟩av. =π1−ϵ +�b2 +4 +�2ϵ Γ(−2ϵ) +Γ(1 + ϵ) × +� +1 +ϵ2 + 1 +ϵ (2 ln(B + 1) − 1) ++ +� +2Li2 +� +− 1 +B +� ++ ln2(B) + ln2(B + 1) − 2 ln(B + 1) − ζ3 +2 + 13π2 +24 ++ 3 +2 +� ++ ϵ +� +− 3Li3(−B) + 2Li3 +� +1 +B + 1 +� +− Li3 +� +B +B + 1 +� ++ Li4 +� +− 1 +B +� +− Li4 +� +B +B + 1 +� +− 2Li4 +� +1 +B + 1 +� ++ 1 +2Li2 +� +− 1 +B +� � +ln2(B + 1) − 2 ln(B + 1) − 6 +� +− 2Li3 +� +1 +B + 1 +� +ln(B + 1) − Li3 +� +B +B + 1 +� +ln(B + 1) + 1 +24 ln4(B) + 3 +8 ln4(B + 1) +− 2 +3 ln3(B + 1) ln(B) + 1 +3 ln3(B + 1) + 1 +4 ln2(B + 1) ln2(B) − 1 +2 ln(B + 1) ln2(B) ++ 1 +12π2 ln2(B) − 3 ln2(B) +2 ++ ln2(B + 1) ln(B) − 1 +12π2 ln2(B + 1) +− 1 +2 ln2(B + 1) + 7 +12π2 ln(B + 1) + 3 ln(B + 1) − 7 +2 − ζ2 +4 + ζ3 +6 − 5ζ4 +� +24 + ++ O +� +ϵ2� +� +. +(131) +We finally consider I(1) +ij,i(12)(⃗b). To obtain a one-fold integral representation of this contribu- +tion we can again take advantage of the result for I(1) +ij,ij, identifying pj = p1+p2. This also means +setting ⃗b · ⃗pT,j = 0 due to the absence of a transverse component in p1 + p2. Nevertheless, the +identification between the results is not completely straightforward because of the additional +difference in the integrand +� +(pi · pj) +(pi · k)(pj · k) +�ϵ +−→ +� +(p1 · p2) +(p1 · k)(p2 · k) +�ϵ +. +(132) +However, we can notice that +� +(pi · pj) +(pi · k)(pj · k) +�ϵ += +� 2 +k2 +T +�ϵ +(1 + w)−ϵ +(133) +while +� +(p1 · p2) +(p1 · k)(p2 · k) +�ϵ += +� +(pi · pj) +(pi · k)((pj · k) − (pi · k)) +�ϵ += +� 2 +k2 +T +�ϵ +. +(134) +Thus we can take care of this additional difference by adding a factor (1 + w)ϵ to the one-fold +representation of I(1) +ij,ij. Therefore, we have +⟨I(1) +ij,i(12)(⃗b)⟩av. = +�p1 · p2 +2m4 +�λ +π1−ϵ +�b2 +4 +�2ϵ−λ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +� ∞ +0 +dw w−1+λ +(1 + w)R12(w) . +(135) +The expression for R12(w) can be obtained taking the massless limit of Eqs. (122)–(125), which +leads to the simplified expressions +R(−2) +12 += −1 +2 +(136) +R(−1) +12 += 0 +(137) +R(0) +12 = 5 +4ζ2 +(138) +R(1) +12 = 7 +6ζ3 , +(139) +and therefore straightforwardly +� ∞ +0 +dw w−1+λ +(1 + w) R12(w) = 1 +λ +� +− 1 +2ϵ2 + 5 +4ζ2 + ϵ7 +6ζ3 + O(ϵ2) +� ++ O(λ) . +(140) +By comparing with Eq. (126) we see that the λ → 0 singular terms cancel out as expected. By +25 + +using Eq. (114) we can write the final result for I(1) +ij (⃗b) as +⟨I(1) +ij (⃗b)⟩av. =π1−ϵ +�b2 +4 +�2ϵ Γ(−2ϵ) +2Γ(1 + ϵ) +� +1 +ϵ2 +� +1 − 2 ln +�2(pi · pj) +√sm +�� ++ 1 +ϵ +� +−1 − π2 +6 + 2 ln(B + 1) − ln2(B) − 2Li2 +� +− 1 +B +�� ++ +� +5 +6π2 ln +�2(pi · pj) +√sm +� +− 13π2 +12 +− 6Li2(−B) + 2Li3 +� +− 1 +B +� ++ 2Li3 +� +1 +B + 1 +� +− 1 +3 ln3(B) − 1 +3 ln3(B + 1) − ln(B + 1) ln2(B) − 2 ln2(B) + ln2(B + 1) ln(B) ++ ln2(B + 1) − 1 +3π2 ln(B) − 2 ln(B + 1) − 3ζ3 − 2Li2 +� +− 1 +B +� +(ln(B + 1) + 2) +� ++ ϵ +� +14 +3 ζ3 ln +�2(pi · pj) +√sm +� ++ 2ζ3 ln(B) − Li2 +2 +� +− 1 +B +� ++ 6Li2(−B) − 2Li3 +� +− 1 +B +� +− 6Li4 +� +− 1 +B +� ++ Li2 +� +− 1 +B +� � +− ln2(B) − ln2(B + 1) + 2 ln(B + 1) + π2 +2 + 6 +� ++ 2Li4 +� +1 +B + 1 +� ++ 2S2,2 +� +− 1 +B +� +− 2Li3 +� +− 1 +B +� +ln(B) + 2Li3 +� +1 +B + 1 +� +ln(B + 1) +− 1 +4 ln4(B) − 1 +4 ln4(B + 1) + 1 +3 ln3(B) + 2 +3 ln3(B + 1) ln(B) + 1 +12π2 ln2(B) +− 1 +2 ln2(B + 1) ln2(B) + ln(B + 1) ln2(B) + 3 ln2(B) − 2 ln2(B + 1) ++ 1 +3π2 ln(B) − 1 +2π2 ln(B + 1) − 13ζ3 +3 +− 29π4 +360 + π2 +12 + 4 +� ++ O +� +ϵ2� +� +. +(141) +3.3.2 +Massive-massive contribution: I(1) +34 +Let us now consider the purely massive contribution, i.e. I(1) +34 (⃗b) in Eq. (110), which can also +be written as +I(1) +34 = +� +dDk δ+(k2) +� +2(p3 · p4) +(p3 · k)(p4 · k) − +m2 +(p3 · k)2 − +m2 +(p4 · k)2 +� � +(p3 · p4) +2(p3 · k)(p4 · k) +�ϵ +R34ei⃗b·⃗kT , +(142) +where the functions R34 have been presented for the first time in Ref. [53], while in Ref. [54] a +simplified expression has been proposed. The coefficients R(n) +34 read +R(−2) +34 +=1 , +(143) +R(−1) +34 += ln(v+) − v− +v +� +ln +�α3 +v+ +� ++ ln +�α4 +v+ +�� +, +(144) +26 + +R(0) +34 = 1 +2 ln2(v+) + 1 +v +� +1 +(d3 + d4) +� +(α3v+ − α4v−) ln2 �α3 +v+ +� ++ +� +α4v+ − α3v− +� +ln2 �α4 +v+ +�� ++ +� +ln +�α3 +v+ +� ++ ln +�α4 +v+ +��� +v+ ln(v+) − ln(v) +� +− Li2 +�v− +v+ +�� ++ ζ2 +�7 +v − 19 +2 +� +, +(145) +R(1) +34 = +1 +d3 + d4 +� +� +1 − (d3 + d4) +� +� +ln +� +1 − α3 +v+ +� +ln2 �α3 +v+ +� ++ ln +� +1 − α4 +v+ +� +ln2 �α4 +v+ +� ++ 2 +� +ln +�α3 +v+ +� +Li2 +�α3 +v+ +� ++ ln +�α4 +v+ +� +Li2 +�α4 +v+ +�� +− Li2 +�v− +v+ +�� +ln +�α3 +v+ +� ++ ln +�α4 +v+ +�� ++ 2 +� +Li3 +�v− +v+ +� +− Li3 +�α3 +v+ +� +− Li3 +�α4 +v+ +� ++ ζ3 +�� +− 7ζ2 +� +ln +�α3 +v+ +� ++ ln +�α4 +v+ +�� ++ 1 +v +��� +α4v+ − α3v− +� +ln2 �α3 +v+ +� ++ +� +α3v+ − α4v− +� +ln2 �α4 +v+ +�� +ln(v+) ++ +� +α3 − α4 +�� +ln2 �α3 +v+ +� +− ln2 �α4 +v+ +�� +ln(v) +− +� +d3 ln +�α3 +v+ +� ++ d4 ln +�α4 +v+ +��� +Li2 +�v− +v+ +� +− 7ζ2 +� +�� ++ 1 +v +�� +ln(v+) +�3 + v +4 +ln(v+) − ln(v) +� +− 9v− +2 ζ2 +�� +ln +�α3 +v+ +� ++ ln +�α4 +v+ +�� +− v− +6 +� +ln3 �α3 +v+ +� ++ ln3 �α4 +v+ +�� ++ 2Li3 +� +1 − v− +v+ +� ++ Li3 +�v− +v+ +� +− +� +Li2 +�v− +v+ +� ++ ζ2 +� +5 + 19 +2 v +�� +ln(v+) + 12ζ2 ln(v) +� ++ 1 +6 ln3(v+) − +�7 +3 + 1 +v +� +ζ3 . (146) +In Eqs. (143)–(146) we used the same notation of Refs. [53, 54], introducing the variables +α3 = +m2(p4 · k) +(p3 · k)(p3 · p4) , +(147) +α4 = +m2(p3 · k) +(p4 · k)(p3 · p4) , +(148) +v± = 1 ± v +2 +, +(149) +d3 = 1 − 2α3 , +(150) +d4 = 1 − 2α4 , +(151) +with v defined as in Eq. (75). +We first discuss the contributions of R(−2) +34 +and R(−1) +34 . Both these coefficients are independent +of the gluon momentum k (note that α3α4 = m4/(p3 · p4)2), and, therefore, the corresponding +integrals can be evaluated with the same method. We start from the generalised Feynman +27 + +parametrisation +1 +AmBn = Γ(m + n) +Γ(m)Γ(n) +� 1 +0 +dx +xm−1(1 − x)n−1 +(xA + (1 − x)B)m+n , +(152) +to write the denominators in terms of a single scalar product. +For the term proportional to (p3 · p4)/(p3 · k p4 · k) in I(1) +34 , dropping overall constant terms, +the relevant integral is +� +dDk δ+(k2) +ei⃗b·⃗kT +(p3 · k)1+ϵ(p4 · k)1+ϵ = Γ(2 + 2ϵ) +Γ2(1 + ϵ) +� 1 +0 +dx +� +dDk +δ+(k2)(1 − x)ϵxϵ ei⃗b·⃗kT +((1 − x)(p4 · k) + x(p3 · k))2+2ϵ , +(153) +while when considering the term proportional to m2/(pj · k)2, with j = 3, 4 we need to evaluate +� +dDk δ+(k2) +ei⃗b·⃗kT +(pj · k)2+ϵ(pi · k)ϵ = +Γ(2 + 2ϵ) +Γ(2 + ϵ)Γ(ϵ) +� 1 +0 +dx +� +dDk +δ+(k2)(1 − x)−1+ϵx1+ϵ ei⃗b·⃗kT +((1 − x)(p4 · k) + x(p3 · k))2+2ϵ . +(154) +We see that both Eq. (153) and Eq. (154) depend on the same integral +I(1) +k (x) = +� +dDk δ+(k2) +ei⃗b·⃗kT +(p(x) · k)2+2ϵ , +(155) +with +pµ(x) = x pµ +3 + (1 − x) pµ +4 . +(156) +This integral can be evaluated with the techniques used in Sec. 3.2 and we find +⟨I(1) +k (x)⟩av. = −4−ϵb4ϵπ2−ϵ +sin(ϵπ) +Γ(−2ϵ) +Γ(−ϵ)Γ(2 + 2ϵ)(p2(x))−1−ϵ +2F1 +� +−2ϵ, 1 + ϵ, 1 − ϵ; −p2 +T(x) +p2(x) +� +. (157) +We are now left with the integration over the Feynman parameter x. It is convenient to expand +in ϵ the hypergeometric function +2F1 (−2ϵ, 1 + ϵ, 1 − ϵ; −X) =1 + 2 ln(1 + X) ϵ − 4Li2(−X) ϵ2 ++ 4 +3 +� +ln3(1 + X) + 3 ln(1 + X)Li2(−X) − 9Li3(−x) +− 6Li3 +� +X +1 + X +� � +ϵ3 + O(ϵ4) . +(158) +By substituting Eq. (157) in Eqs. (153), (154) we obtain a sum of integrals that in most cases can +be computed in terms of multiple polylogarithms. The remaining finite integrals are computed +numerically. +We now focus on the contribution of R(0) +34 . We can split R(0) +34 in a part independent on k, +28 + +R(0) +34;const, and one with an explicit k dependence, R(0) +34;k +R(0) +34 = R(0) +34;const + R(0) +34;k . +(159) +We define +R(0) +34;const =1 +v +�� +ln +�α3 +v+ +� ++ ln +�α4 +v+ +��� +v+ ln(v+) − ln(v) +� +− Li2 +�v− +v+ +�� ++ 1 +2 ln2(v+) ++ ζ2 +�7 +v − 19 +2 +� +. +(160) +and +R(0) +34;k = +1 +(d3 + d4)v +� +(α3v+ − α4v−) ln2 �α3 +v+ +� ++ +� +α4v+ − α3v− +� +ln2 �α4 +v+ +�� +≡ +1 +d3 + d4 +r(0) +34 . +(161) +The contribution of R(0) +34;const can be evaluated as those of R(−2) +34 +and R(−1) +34 . The singular part +of the contribution of R(0) +34;k can be computed by using the following identity +� +dDk (k2)−1−ϵf(k) ei⃗b·⃗kT = +� +dDk (k2)−1−ϵf(k)θ(µ − kT) + O(ϵ0) , +(162) +µ being an arbitrary mass scale and f(k) an arbitrary function of the momentum k. +We consider the integral of the k-dependent part of R(0) +34 +I(1,0) +34;k (⃗b) = +� +dDk δ+(k2) +1 +d3 + d4 +� +2(p3 · p4) +(p3 · k)(p4 · k) − +m2 +(p3 · k)2 − +m2 +(p4 · k)2 +� +(163) +× +� +(p3 · p4) +2(p3 · k)(p4 · k) +�ϵ +r(0) +34 ei⃗b·⃗kT , +up to order 1/ϵ. The reason to pull out the factor 1/(d3 + d4) is because it allows us to use the +identity +2(p3 · p4) +(p3 · k)(p4 · k) − +m2 +(p3 · k)2 − +m2 +(p4 · k)2 = +(p3 · p4) +(p3 · k)(p4 · k)(d3 + d4) , +(164) +which makes the integrand considerably simpler. If now we extract the pole structure of the +integral by using Eq. (162), we get +I(1,0) +34;k (⃗b) = +� +dDk δ+(k2) +� +p3 · p4 +(p3 · k)(p4 · k) +�1+ϵ +r(0) +34 θ(µ2 − k2 +0) + O(ϵ0) . +(165) +There is no singularity associated to the angular variables. We can thus safely set ϵ = 0 in the +29 + +angular integral, obtaining +I(1,0) +34;k (⃗b) = 2π +� ∞ +0 +dk0 +k1+4ϵ +0 +θ(µ2 − k2 +0) +� 1 +0 +dt +� 1 +−1 +d cos θ t2−2ϵδ(1 − t2) +1 +1 − v cos θr(0) +34 + O(ϵ0) , +(166) +with t = |⃗k|/k0. Now we can perform the integral over t by using the delta function, while the +(otherwise divergent) integration over k0 is regulated by the cutoff we inserted. We find for the +pole +I(1,0) +34;k (⃗b) +�� +pole = − π +4ϵ +� 1 +−1 +d cos θ +1 +1 − v cos θ r(0) +34 . +(167) +The integration of the pole part of Eq. (145) is thus finally reduced to a one-fold integral that +can be computed with standard methods. In order to write r(0) +34 in terms of v, cos θ the following +relations are useful +α3 = 1 − v cos θ +2 +, +(168) +α4 = 1 − v2 +2 +1 +1 − v cos θ . +(169) +The result for the pole part of this contribution reads +⟨I(1,0) +34;k (⃗b) +�� +pole⟩av. = π1−ϵ +�b2 +4 +�2ϵ Γ(−2ϵ) +Γ(ϵ + 1) +� +2 − (1 − β2)2 +2β2 +ln2 +�1 − β +β + 1 +�� +. +(170) +The finite part in ϵ of the contribution of R(0) +34 can be integrated numerically. +The last contribution to be computed is that from R(1) +34 . Since it comes with an overall ϵ +factor we can directly apply Eq. (162) to evaluate it. The analytic result is too lengthy to be +reported. +We conclude this subsection discussing the contribution of the one-loop heavy-quark vacuum +polarization. Such term can be inserted in the radiated soft-gluon line, thus leading to an +additional virtual contribution to the one-loop soft-gluon current. Then such contribution has +to be consistently taken into account through the renormalization procedure, which amounts +to the wave function renormalization of the soft-gluon line and the MS renormalization of αS +with nf + 1 quark flavours (the nf massless quarks and the heavy quark Q). +Finally, we +can apply the decoupling relation of the heavy quark [37] and introduce the running coupling +α +(nf) +S +(µ2 +R) that we use throughout this paper (see the comment at the beginning of Sect. 2.2). +To the purpose of computing the soft contributions at small qT, the final result of this entire +procedure is equivalent to avoiding the introduction of the heavy-quark vacuum polarization +and to directly renormalizing the QCD coupling with nf light-quark flavours as in Eq. (32). +30 + +3.4 +Light-quark pair production +We start the analysis of the double real contribution by focusing on the process in which a +massless soft quark-antiquark pair is radiated +c(p1)¯c(p2) +→ +Q(p3) ¯Q(p4) q(k1)¯q(k2) . +(171) +The corresponding factorisation formula for the squared matrix element is [59] +���M(0) +c¯c→Q ¯Qq¯q +��� +2 +∼ (g0µϵ +0)4⟨M(0) +c¯c→Q ¯Q|I(0) +q¯q (k1, k2)|M(0) +c¯c→Q ¯Q⟩ +(172) +where the singular contributions are controlled by the soft factor +I(0) +q¯q (k1, k2) = +� +J(0) +g,µ(k1 + k2) +�† Πµν(k1, k2) J(0) +g,ν(k1 + k2) + ... . +(173) +In Eq. (173) J(0) +g,µ is the tree-level soft current in Eq. (78) and we have defined the tensor Πµν +as: +Πµν(k1, k2) = +TR +(k1 · k2)2 (−gµνk1 · k2 + kµ +1kν +2 + kν +1kµ +2) . +(174) +The dots in Eq. (173) stand for gauge dependent contributions that are proportional to the total +charge of the hard partons, thereby vanishing when evaluated on the c¯c → Q ¯Q matrix element. +Our task is now to integrate Eq. (173) over the phase space of the q¯q pair after subtracting the +initial-state contribution, i.e., to evaluate the integral I(0) +q¯q (⃗b) in Eq. (59). +To perform this calculation, we first integrate over the light-quark momenta k1 and k2 while +keeping their total momentum k = k1 + k2 fixed. This procedure will leave us with expressions +similar to the ones for the NLO-like contribution already described in Sect. 3.2 and will be +useful in order to organise the final integration over k in a similar way. +To proceed in this direction, we rewrite the integration of the soft factor in Eq. (59) in the +following way: +� +dDk1 +(2π)D−1 +dDk2 +(2π)D−1δ+(k2 +1)δ+(k2 +2)I(0) +q¯q (k1, k2) ei⃗b·(⃗kT 1+⃗kT 2) = += +� +dDk +(2π)D−1J(0) +g,µ(k)J(0) +g,ν(k) F µν(k) ei⃗b·⃗kT , +(175) +obtained by inserting the identity +1 = +� +dDk δ(D)(k − k1 − k2) , +(176) +and by isolating the integral on the soft-quark momenta in the tensor F µν(k), defined as: +F µν(k) = +1 +(2π)D−1 +� +dDk1 +� +dDk2 Πµν(k1, k2)δ+(k1)δ+(k2)δ(D)(k − k1 − k2) . +(177) +31 + +We now continue with the computation of the tensor F µν. Since F µν is a symmetric tensor +fulfilling kµF µν = kνF µν = 0 it must take the form +F µν(k) = C +� +−gµν + kµkν +k2 +� +. +(178) +The normalisation factor C can be fixed by evaluating the quantity gµνF µν using Eq. (177) and +Eq. (178) and comparing the results. From Eq. (178) we immediately obtain +gµνF µν = −C(3 − 2ϵ) , +(179) +while from Eq. (177) +gµνF µν = −TR +2 − 2ϵ +(k2)1+ϵΓ( 3 +2 − ϵ)161−ϵπ +3 +2 −ϵ . +(180) +We can therefore write +C = +F(ϵ) +(k2)1+ϵ , +(181) +with +F(ϵ) = +TR(1 − ϵ) +Γ +� 5 +2 − ϵ +� +161−ϵπ +3 +2 −ϵ . +(182) +With the explicit expression for F µν just obtained, the right-hand side of Eq. (175) reads: +� +dDk +(2π)D−1J(0) +g,µ(k)J(0) +g,ν(k)F µν(k) ei⃗b·⃗kT = − +� +dDk +(2π)D−1 +F(ϵ) +(k2)1+ϵ +4 +� +i,j=1 +Ti · Tj +pi · pj +(pi · k)(pj · k) ei⃗b·⃗kT , +(183) +where the term in F µν proportional to kµkν gives no contribution because of colour conservation. +We observe that Eq. (183) has a similar structure to Eq. (57), the corresponding NLO +integral for single soft-gluon emission at tree-level, after the substitution +δ+(k2) +→ +F(ϵ) +(k2)1+ϵ , +(184) +which removes the on-shell constraint for the gluon. We can thus apply for the computation +a similar strategy as the one already employed in Sect. 3.2, when dealing with the NLO-like +contribution. +Before performing the final integration over k of the expression in Eq. (183), we need to +subtract the initial-state contribution from the soft current. We therefore write the integral +I(0) +q¯q (⃗b) in Eq. (59) as +I(0) +q¯q (⃗b) = − F(ϵ) +� +dDk +(2π)D−1 +1 +(k2)1+ϵ +� � +j=3,4 +� +m2 +(pj · k)2T2 +j + 2 +� +i=1,2 +�pi · pj +pj · k − +p1 · p2 +(p1 + p2)k +� Ti · Tj +pi · k +� ++ +2p3 · p4 +(p3 · k)(p4 · k)T3 · T4 +� +ei⃗b·⃗kT . +(185) +32 + +We can split Eq. (185) in different integrals according to the different colour factors +Iq¯q +jj (⃗b) = +� +dDk +(k2)1+ϵ +m2 +(pj · k)2 ei⃗b·⃗kT , +(186) +Iq¯q +ij (⃗b) = +� +dDk +(k2)1+ϵ +1 +pi · k +�pi · pj +pj · k − +p1 · p2 +(p1 + p2) · k +� +ei⃗b·⃗kT , +(187) +Iq¯q +34(⃗b) = +� +dDk +(k2)1+ϵ +p3 · p4 +(p3 · k)(p4 · k) ei⃗b·⃗kT . +(188) +In terms of these integrals, Eq. (185) reads +I(0) +q¯q (⃗b) = − +F(ϵ) +(2π)D−1 +� � +j=3,4 +� +Iq¯q +jj (⃗b) T2 +j + 2 +� +i=1,2 +Iq¯q +ij (⃗b) Ti · Tj +� ++ 2Iq¯q +34(⃗b) T3 · T4 +� +. +(189) +The integrals in Eqs. (186)–(188) can be evaluated with a similar strategy as to the one used +for the integrals in Eqs. (84)–(86). The azimuthally averaged results are +⟨Iq¯q +jj (⃗b)⟩av. =π1−ϵΓ(1 − ϵ)Γ(−2ϵ) +�b2 +4 +�2ϵ � +−1 +ϵ +2F1 (1, −2ϵ; 1 − ϵ; −B) +� +=π1−ϵΓ(1 − ϵ)Γ(−2ϵ) +�b2 +4 +�2ϵ � +− 1 +ϵ − 2 ln (1 + B) ++ ϵ +� +2 Li2 (−B) − ln2 (1 + B) +� ++ O(ϵ2) +� +, +(190) +⟨Iq¯q +ij (⃗b)⟩av. =1 +2π1−ϵΓ(1 − ϵ)Γ(−2ϵ) +�b2 +4 +�2ϵ Γ( λ +2 − 2ϵ)Γ( λ +2) +Γ(1 − 2ϵ) +× 2 +��pi · pj +m2 +�λ +2F1 +�λ +2, λ +2 − 2ϵ; 1ϵ; −B +� +− +�p1 · p2 +√s +�λ +2F1 +�λ +2, λ +2 − 2ϵ; 1 − ϵ; 0 +�� +=1 +2π1−ϵΓ(1 − ϵ)Γ(−2ϵ) +�b2 +4 +�2ϵ � +− 2 +ϵ ln +�2 pi · pj +m √s +� ++ 2Li2 (−B) ++ ϵ +3 +� +ln3 (1 + B) + 6 ln (1 + B) Li2 (−B) − 6 Li3 +� +B +B + 1 +�� ++ O(ϵ2) +� +, +(191) +⟨Iq¯q +34(⃗b)⟩av. =π1−ϵΓ(1 − ϵ)Γ(−2ϵ) +�b2 +4 +�2ϵ 1 + β2 +2β +� +−1 +ϵL0 − 2L1 + ϵ(2P2 − L2) + O(ϵ2) +� +. +(192) +The functions Ln and Pn have been defined in Eq. (98) and Eq. (99), respectively, while their +explicit expressions are reported in Eqs. (100)–(102). In the present case we also need the +33 + +function L2(β, θ), which reads +L2(β, θ) = 2(G(1, −1, − sec(θ), β) + G(1, −1, sec(θ), β) + G(1, 1, − sec(θ), β) ++ G(1, 1, sec(θ), β) + G(1, − sec(θ), −1, β) + G(1, − sec(θ), 1, β) − G(1, 1, 1, β) +− G(1, − sec(θ), − sec(θ), β) + G(1, sec(θ), −1, β) + G(1, sec(θ), 1, β) − G(1, 1, −1, β) +− G(1, sec(θ), − sec(θ), β) − G(1, sec(θ), sec(θ), β) − G(1, − sec(θ), sec(θ), β) +− G(1, −1, −1, β) − G(1, −1, 1, β)) . +(193) +Note that in Eq. (191), the expression for ⟨Iq¯q +ij (⃗b)⟩av. before the ϵ-expansion depends on the +regularisation parameter λ. As in Sec. 3.2 the integration in Eq. (187) needs to be carried out +separately for the two terms, by using the regulator factor in Eq. (92). We can then perform +the limit λ → 0 in the expanded result. +3.5 +Double gluon emission +We finally consider the contribution due to the emission of a soft-gluon pair, i.e. we consider +the process +c(p1)¯c(p2) +→ +Q(p3) ¯Q(p4)g(k1)g(k2) . +(194) +In the limit in which the two gluons become soft the singular behaviour is controlled by the +double-soft current J(0)µν +gg +(k1, k2) [59, 60]. The general expression of the squared soft current +reads +� +J(0)a1a2 +gg,µν (k1, k2) +�† dσµ(k1)dρν(k2)J(0)a1a2 +gg,σρ (k1, k2) = 1 +2 +� +J(0) +g +2(k1), J(0) +g +2(k2) +� ++ W(0) +gg (k1, k2) + ... , +(195) +where the purely non-abelian two-parton correlations are controlled by the function W(0) +gg (k1, k2), +which is defined as +W(0) +gg (k1, k2) = −CA +n +� +i,j=1 +Ti · Tj Sij(k1, k2) . +(196) +The dots in Eq. (195) stand for gauge-dependent terms proportional to the total colour charge +of the hard partons and, thus, give a vanishing contribution when evaluated on the c¯c → Q ¯Q +matrix element. The soft factor can be separated into massless and massive contributions +Sij(k1, k2) = Sm=0 +ij +(k1, k2) + +� +m2 +i Sm̸=0 +ij +(k1, k2) + m2 +j Sm̸=0 +ji +(k1, k2) +� +, +(197) +34 + +where mi(mj) = 0 for i(j) = 1, 2 and mi(mj) = m for i(j) = 3, 4. The massless contribution +reads [59] +Sm=0 +ij +(k1, k2) = (1 − ϵ) +(k1 · k2)2 +pi · k1 pj · k2 + pi · k2 pj · k1 +pi · (k1 + k2) pj · (k1 + k2) +− +(pi · pj)2 +2 pi · k1 pj · k2 pi · k2 pj · k1 +� +2 − pi · k1 pj · k2 + pi · k2 pj · k1 +pi · (k1 + k2) pj · (k1 + k2) +� ++ pi · pj +2 k1 · k2 +� +2 +pi · k1 pj · k2 ++ +2 +pj · k1 pi · k2 +− +1 +pi · (k1 + k2) pj · (k1 + k2) +× +� +4 + (pi · k1 pj · k2 + pi · k2 pj · k1)2 +pi · k1 pj · k2 pi · k2 pj · k1 +�� +, +(198) +pi, pj being the momenta of the emitters. The massive contribution is [60] +Sm̸=0 +ij +(k1, k2) = − +1 +4 k1 · k2 pi · k1 pi · k2 ++ +pi · pj pj · (k1 + k2) +2 pi · k1 pj · k2 pi · k2 pj · k1 pi · (k1 + k2) +− +1 +2 k1 · k2 pi · (k1 + k2) pj · (k1 + k2) +� +(pj · k1)2 +pi · k1 pj · k2 ++ +(pj · k2)2 +pi · k2 pj · k1 +� +. (199) +In the right-hand side of Eq. (195), W(0) +gg is the irreducible correlation component of double- +soft radiation, while the anticommutator term corresponds to the independent-emission compo- +nent. We have to evaluate the b-space contribution of the squared current in Eq. (195). Going +to b-space, the phase space for double-parton emission factorizes in terms of single-parton fac- +tors (see Eq. (63)). Therefore the b-space integral of the independent-emission component of +Eq. (195) is fully factorized and it leads to the straightforward exponentiation of the tree-level +single-emission contribution Fex,1 in Eq (61). Consequently the b-space contribution I(0) +gg (⃗b) of +double soft-gluon emission to Fex,2 in Eq. (62) is entirely due to the correlation component W(0) +gg +of Eq. (195). More precisely, we have to perform the integral in Eq. (60) where W(0) +gg (k1, k2) +�� +sub +is defined from W(0) +gg (k1, k2) in Eq. (196) after the proper subtraction of the contribution from +initial-state radiation. +Part of the contribution to I(0) +gg (⃗b) is similar to I(0) +q¯q (⃗b). The soft term in Eq. (173) involves +the factor +pµ +i pν +j Πµν (k1, k2) +pi · (k1 + k2) pj · (k1 + k2) = +TR +(k1 · k2)2 +−(pi · pj)(k1 · k2) + (pi · k1)(pj · k2) + (pi · k2)(pj · k1) +pi · (k1 + k2)pj · (k1 + k2) +. +(200) +In Eq. (198) we have some terms with a similar structure as the ones in Eq. (200). Those are +Sm=0 +ij +(k1, k2) +��� +12 = +4 +(k1 · k2) +(pi · pj) +(pi · k)(pj · k)− (1 − ϵ) +(k1 · k2)2 +(pi · k1) (pj · k2) + (pi · k2) (pj · k1) +pi · (k1 + k2) pj · (k1 + k2) +. (201) +35 + +Indeed by defining +�Πµν(k1, k2) = − +1 +(k1 · k2)2 (−4gµν(k1 · k2) + (1 − ϵ)kµ +1kν +2 + (1 − ϵ)kµ +2kν +1) +(202) +we can rewrite Eq. (201) as +Sm=0 +ij +(k1, k2) +��� +12 = +pµ +i +pi · (k1 + k2) +�Πµν(k1, k2) +pν +j +pj · (k1 + k2) . +(203) +Now the integration of Eq. (203) can be performed exactly in the same way as the integration +of Eq. (173) in the case of the emission of a soft q¯q pair in Sect. 3.4, leading to the same results +with an overall multiplicative factor. +By following the strategy of integrating over k1 and k2 at a fixed value of k = k1 + k2, +similarly to what was done in Eq. (177), we isolate the following integral +�F µν(k) = +1 +(2π)D−1 +� +dDk1 +� +dDk2 �Πµν(k1, k2)δ+(k2 +1)δ+(k2 +2)δ(D)(k − k1 − k2) . +(204) +The structure of �F µν(k) must be of the form +�F µν(k) = agµν + bkµkν +k2 +. +(205) +The coefficients a and b can be obtained by contracting Eq. (204) with gµν and kµkν. We find +�F µν = +1 +(k2)1+ϵ +1 +Γ +� 5 +2 − ϵ +� +161−ϵπ +3 +2 −ϵ +�11 − 7ϵ +2 +gµν + (1 − ϵ)kµkν +k2 +� +. +(206) +Because of current conservation, the second term will give no contribution to the integrals. +The first term, on the other hand, is exactly the same we obtained in the computation for the +soft-quark pair emission, but with an overall multiplicative factor: −(11 − 7ϵ)/(1 − ϵ). +Hence, to compute the integral of the contribution in Eq. (201), we can take the result for +the q¯q pair production and perform the formal substitution +nfTR −→ −CA +11 − 7ϵ +4(1 − ϵ) , +(207) +where we also included the Bose factor 1/2 of Eq. (60), which is due to the production of two +identical particles. +We can now define a new soft factor, in which we subtract the contribution that can be +computed as described above +�Sij(k1, k2) = �Sm=0 +ij +(k1, k2) + +� +m2 +i �Sm̸=0 +ij +(k1, k2) + m2 +j �Sm̸=0 +ji +(k1, k2) +� +, +(208) +36 + +where +�Sm=0 +ij +(k1, k2) = Sm=0 +ij +(k1, k2) − Sm=0 +ij +(k1, k2) +��� +12 , +(209) +�Sm̸=0 +ij +(k1, k2) = Sm̸=0 +ij +(k1, k2) . +(210) +We have +˜Sm=0 +ij +(k1, k2) = − +(pi · pj)2 +2(pi · k)(pj · k) +� +2 +(pi · k1)(pj · k1) + +1 +(pi · k1)(pj · k2) +� +− +(pi · pj) +2k2(pi · k)(pj · k) +((pi · k1)(pj · k2) − (pi · k2)(pj · k1))2 +(pi · k1)(pj · k2)(pi · k2)(pj · k1) ++ (pi · pj) +k2 +2 +(pi · k1)(pj · k2) + (1 ↔ 2) , +(211) +˜Sm̸=0 +ij +(k1, k2) = (pi · pj) +2(pi · k)2 +� +1 +(pi · k1)(pj · k1) + +1 +(pi · k1)(pj · k2) +� +− +1 +k2(pi · k) +1 +(pi · k1) +� +(pj · k1)2 +(pj · k)(pj · k2) − +(pi · k1)2 +(pi · k)(pi · k2) +� ++ (1 ↔ 2) , +(212) +where we have introduced k = k1 + k2. We now need to subtract the contribution from initial- +state radiation. We can use the same technique already used in the previous Sections. The +sum over the colour configurations can be organised as +4 +� +i,j=1 +˜Sij(k1, k2)Ti · Tj = +� +4 +� +i,j=1 +˜SijTi · Tj − +� +− ˜S12(T2 +1 + T2 +2) +�� ++ +� +− ˜S12(T2 +1 + T2 +2) +� +. +(213) +The second term on the r.h.s. is the same we would have for a colourless final state. The first +term is the new contribution to the subtracted current we have to compute and, by using colour +conservation, we can rewrite it as +� +j=3,4 +� +˜SjjT2 +j + +� +i=1,2 +� +2 ˜Sij − ˜S12 +� +Ti · Tj +� ++ 2 ˜S34T3 · T4 . +(214) +Hence we need now to compute +� +dDk1 dDk2 �Sij(k1, k2)δ+(k2 +1) δ+(k2 +2) , +(215) +for all the contributions involved in Eq. (214). This means we have to consider the following +combinations of emitters i and j: +• i and j being the two initial-state massless emitters; +• i being an initial-state massless emitter, j a final-state massive emitter; +• i = j being the same final-state massive emitter; +37 + +• i and j being the two final-state massive emitter. +It is convenient to integrate over the soft-gluon momenta k1 and k2 at fixed kµ = kµ +1 + kµ +2: +after that, we are left with only the integration over k. With this goal in mind, we define the +shorthand notation +� +(12) +f(k1, k2) ≡ Γ( 1 +2 − ϵ) +4ϵπ +1 +2 −ϵ +� +dDk1 dDk2 f(k1, k2) δ+(k2 +1) δ+(k2 +2) δ(D)(k − k1 − k2) , +(216) +and we apply it to the functions ˜Sm=0 +ij +and ˜Sm̸=0 +ij +. To perform this computation, we can first +integrate one of the soft-gluon momenta (e.g. k1) using the delta function δ(D)(k − k1 − k2). +Afterwards, we can go in the rest frame of k and integrate over the energy component and the +modulus of ⃗k2 by using the two remaining delta functions: this way only angular integrals are +left. +By following these steps, we obtain +� +(12) +˜Sm=0 +ij +=(k2)−1−ϵ(pi · pj) +(pi · k)(pj · k) +� +(1 + ⃗ni · ⃗nj) A+ +1,1 − 2 (1 − ⃗ni · ⃗nj) A− +1,1 + A1,0 + A0,1 +� +≡(k2)−1−ϵ(pi · pj) +(pi · k)(pj · k) fgg +ij (⃗ni · ⃗nj,⃗n2 +i ,⃗n2 +j) , +(217) +� +(12) +˜Sm̸=0 +ij +=(k2)−1−ϵ +(pj · k)2 +� +(1 − ⃗ni · ⃗nj) A− +1,1 − (1 + ⃗ni · ⃗nj) A+ +1,1 − 1 +2A+ +1,−1 + 3A1,0 − 1 +2A0,0 +� +≡(k2)−1−ϵ +(pj · k)2 ggg +ij (⃗ni · ⃗nj,⃗n2 +i ,⃗n2 +j) , +(218) +where we defined ⃗ni and ⃗nj as vectors in the centre-of-mass frame of k via pi = Ei(1,⃗ni) and +pj = Ej(1,⃗nj). In terms of invariants, we have +⃗n2 +i = 1 − +k2m2 +i +(pi · k)2 , +(219) +⃗n2 +j = 1 − +k2m2 +j +(pj · k)2 , +(220) +⃗ni · ⃗nj = 1 − +k2(pi · pj) +(pi · k)(pj · k) . +(221) +The functions fgg +ij , ggg +ij are defined as the sum of the angular integrals (with appropriate multi- +plicative factors) for the massless and massive case respectively. The angular integrals A± +k,l are +defined as +A± +k,l = +� π +0 +dθ +� π +0 +dφ +sinD−3 θ sinD−4 φ +(1 − ai cos θ)k(1 ± aj cos χ cos θ ± aj sin χ sin θ cos φ)l , +(222) +with: +ai = +� +⃗n2 +i +cos χ = ⃗ni · ⃗nj +� +⃗n2 +i⃗n2 +j +. +(223) +38 + +The expression of the angular integral in Eq. (222) in many cases of interest can be found in +Ref. [61]. Observe that A1,0 and A0,1 only depend on ai and aj respectively, and are independent +of the label ± in Eq. (222). +We now need to perform the integration over k (and, when needed, the explicit evaluation +of the angular integrals) of the expressions in Eqs. (217) and (218) for all the possible emitters. +3.5.1 +Massless-massless contribution: ˜S12 +We start by addressing the problem of the integration of Eq. (217) in the case in which both the +emitters are massless. The first step is to write explicitly the function fgg +ij for this configuration. +By using the results of Ref. [61] we find +� +(12) +˜Sm=0 +12 +(k1, k2) =π +2 +(p1 · p2) (k2)−1−ϵ +(p1 · k)(p2 · k) +� +− 8 +ϵ +� +1 − Γ(1 + ϵ)Γ(1 − ϵ) +�1 + ⃗n1 · ⃗n2 +1 − ⃗n1 · ⃗n2 +�ϵ� +− 4 +�1 − ⃗n1 · ⃗n2 +2 +� � 1 +0 +dt +1 − +� 1−⃗n1·⃗n2 +2 +� +t +� +(1 − t)−ϵ − 2(1 − t)ϵ� +� +, +(224) +where the integration over t in the last line is the integral representation of an hypergeometric +function. +Our task is now to compute +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm=0 +12 +(k1, k2) . +(225) +We observe that, while the first term on the right-hand side of Eq. (224) is singular in the limit +k2 → 0, the second term is regular since ⃗n1 · ⃗n2 → 1 as k2 → 0 (see Eq. (221)) and thus it can +be safely expanded in ϵ. +To proceed further, we need to regularise the additional collinear singularity due to the +presence of massless emitters, and we do it by partial fractioning +1 +(p1 · k)(p2 · k) = +1 +(p1 + p2) · k +� +1 +(p1 · k) + +1 +(p2 · k) +� +, +(226) +and by multiplying each singular contribution by the regulator already introduced in Eq. (92). +The next step is, after switching to light-cone coordinates, to add the integration over the delta +function of k2 +� +dK2 δ(k2 − K2) . +(227) +which is used to integrate over k+. Then we can introduce the dimensionless variables +x = K2 +k2 +T +y = k2 +− +k2 +T +, +(228) +obtaining expressions where the integrals over kT and the one over x and y are factorised. The +39 + +calculation can be completed with standard techniques: we find +⟨ +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm=0 +ij +�pi · k +m2 +�2λ +⟩av. = +�b2 +4 +�2ϵ−λ � √s +2m2 +�2λ +π2−ϵ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +× +� +1 +λ +� 4 +ϵ2 − 8ζ2 − 28ζ3ϵ + O(ϵ2) +� ++ +� +31ζ4ϵ + O(ϵ2) +� ++ O(λ) +� +. +(229) +3.5.2 +Massless-massive contribution: ˜Sij +We now consider the massless-massive contribution. In this case we have to integrate both +Eq. (217) and Eq. (218) for i = 1, 2 and j = 3, 4. +Mass-independent part +We start our analysis with the massless-like part of the soft factor. +After writing explicitly the function fgg +ij for this configuration by using the results of Ref. [61], +we split it into a regular and a singular part as done for the massless-massless contribution in +Sec. 3.5.1, immediately expanding in ϵ the regular part. Because of the ϵ-pole coming from +phase-space integration, the expansion of the integrand needs to be performed up to order ϵ. +We now describe the integration over k of the angular function fgg +ij . The collinear singularity +due to the presence of the massless momentum pi is regularised as before with the regulator +factor in Eq. (92). We then introduce a delta function δ(k2 − K2) as in Eq. (227) and we use +it to perform the integral over k+. +Unlike the massless-massless contribution to the double gluon soft current but similarly to +the single-gluon computation, the emitter has a non-zero transverse momentum and hence we +have a dependence on ⃗kT in (pj · k). As it is by now customary, we remove it with the shift +⃗kT → ⃗kT + k− +pj,− +⃗pT,j . +(230) +This way the only dependence left on the angular part of ⃗kT is in the exponential and the +angular integral can now be easily performed. +The integrals that are left are now the one over K2, over kT and over k−. We introduce the +dimensionless variables +u = K2 +k2 +T +w = k2 +− +k2 +T +m2 +p2 +j,− +, +(231) +in terms of which fgg +ij (⃗nj · ⃗ni, 1,⃗n2 +j) is now independent of k2 +T +fgg +ij (⃗ni · ⃗nj, 1,⃗n2 +j) ≡ f gg +ij (u, w) . +(232) +Because of this, the integral over kT factorises in the form +� ∞ +0 +dkT k−1−3ϵ+2λ +T +J−ϵ(bkT)ei ⃗b·⃗pT,j +√w kT /m , +(233) +40 + +and can be computed separately obtaining an hypergeometric function9. As for the integral +over the variables u and w we obtain, up to overall factors +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm=0 +ij +�pi · k +m2 +�2λ +∝ +� ∞ +0 +du dw u−1−ϵw−1+2ϵ +1 + u + w +2F1 +� +−2ϵ + λ, 1 +2 − 2ϵ + λ; 1 − ϵ; 1 +w +b2m2 +(⃗b · ⃗pT,j)2 +� +f gg +ij (u, w) . (234) +Notice that the collinear divergence, regulated by the parameter λ, is now described by the +limit w → 0. +It is now useful to perform some manipulation on Eq. (234) in order to move the dependence +on the regulator λ outside of the hypergeometric function. We use the following relation +2F1(a, b, c; z) =Γ(b − a)Γ(c) +Γ(b)Γ(c − a)(−z)−a +2F1 +� +a, a − c + 1, a − b + 1; 1 +z +� ++ Γ(a − b)Γ(c) +Γ(a)Γ(c − b)(−z)−b +2F1 +� +b, b − c + 1, b − a + 1, 1 +z +� +. +(235) +By applying it to Eq. (234) and by exploiting the w → 0 limits of the new hypergeometric +functions, the limit λ → 0 can be easily carried out and we obtain +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm=0 +ij +�pi · k +m2 +�2λ += (pi · pj)2λ +(m2)3λ +�b2 +4 +�2ϵ−λ +π1−ϵ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +× +� ∞ +0 +du dw f gg +ij (u, w) u−1−ϵw−1 +1 + u + w +� +wλ + +� +2F1 +� +−2ϵ, −ϵ; 1 +2; c2 +jbw +� +− 1 +− 4ϵicjb +√wΓ( 1 +2 − 2ϵ)Γ(1 + ϵ) +Γ(1 − 2ϵ)Γ( 1 +2 + ϵ) +2F1 +�1 +2 − 2ϵ, 1 +2 − ϵ; 3 +2; c2 +jbw +� �� +, +(236) +where cjb is defined in Eq. (129). By comparing Eq. (236) with the expression obtained in +the one loop case in Eq. (116), we can observe that they share the same structure, the only +differences being an overall multiplicative factor, an additional integration over u and the formal +substitution +u−1−ϵw−1 +1 + u + w → +1 +w (1 + w)1+ϵ . +(237) +By using this relation, the azimuthal average can be directly deduced from Eq. (117) +⟨ +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm=0 +ij +�pi · k +m2 +�2λ +⟩av. = (pi · pj)2λ +(m2)3λ +�b2 +4 +�2ϵ−λ +π1−ϵ Γ(−2ϵ + λ) +2Γ(1 + ϵ − λ) +× +� ∞ +0 +du dw f gg +ij (u, w) u−1−ϵw−1 +1 + u + w +� +wλ + Re +� +(2F1 (−2ϵ, −ϵ; 1 − ϵ; wB) − 1) +9 See formula 6.621 in Ref. [62]. +41 + +× (1 + i cot(πϵ)) +�� +. +(238) +The two-folded integral in Eq. (238) does not present significant complications and can be +computed with standard techniques. We obtain +⟨ +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm=0 +ij +�pi · k +m2 +�2λ +⟩av. = −(pi · pj)2λ +(m2)3λ +�b2 +4 +�2ϵ−λ +π2−ϵ +Γ(λ − 2ϵ) +2ϵΓ(ϵ − λ + 1) +× +� +1 +λ +� +−2 +ϵ + 4ζ2ϵ + 14ζ3ϵ2 + O(ϵ3) +� ++ +� +− +� +2 ln2 (B) + 4Li2 +� +− 1 +B +� ++ 2ζ2 +� ++ 2 +3ϵ +� +ln3 +� 1 +B + 1 +� ++ ln3 (B) + 3 ln (B) ln2 (B + 1) − 6 ln2 (B) ln (B + 1) +− 6 ln (B + 1) Li2 +� +− 1 +B +� +− 6Li3 +� +B +B + 1 +� +− 6 +� +2 ln +� 1 +B + 1 +� ++ 2 ln (B) +− ln (B + 1)) ζ2 +� ++ ϵ2 +� +7 +12 ln4 +� 1 +B + 1 +� ++ 14ζ2 ln2 +� 1 +B + 1 +� +− 2 +3 ln (B + 1) +× ln3 +� 1 +B + 1 +� ++ Li2 +� +B +B + 1 +� +ln2 +� 1 +B + 1 +� ++ 8 ln (B + 1) ζ2 ln +� 1 +B + 1 +� +− 31 +12 ln4 (B) + 11 +12 ln4 (B + 1) − 7 +3 ln (B) ln3 (B + 1) − 2Li2 +� +− 1 +B +� +2 ++ Li2 +� +B +B + 1 +� +2 + 10 +3 ln3 (B) ln (B + 1) − 4 ln (B + 1) Li3 +� +− 1 +B +� ++ 4 ln (B + 1) Li3 +� +B +B + 1 +� +− 12Li4 +� +− 1 +B +� ++ 6 ln (B + 1) Li3 +� +1 +B + 1 +� ++ 8Li4 +� +B +B + 1 +� ++ 10Li4 +� +1 +B + 1 +� +− 27 ln2 (B) ζ2 + 3Li2 +� +1 +B + 1 +� +× +� +ln2 (B + 1) − 6ζ2 +� +− 33 ln2 (B + 1) ζ2 + 60 ln (B) ln (B + 1) ζ2 +− 4Li2 +� +B +B + 1 +� +ζ2 − Li2 +� +− 1 +B +� � +ln2 +� 1 +B + 1 +� ++ 3 ln2 (B) − 2 ln2 (B + 1) ++2Li2 +� +B +B + 1 +� ++ 10ζ2 +� ++ 4 ln (B + 1) ζ3 + ζ4 +2 +� ++ O +� +ϵ2� +� ++ O (λ) +� +. +(239) +Combining the results in Eq. (239) and Eq. (229) to construct the second term in the square +bracket of Eq. (214) we see that the λ → 0 singularities cancel out, as expected. +Mass-dependent part +We now address the problem of the integration of the mass-dependent +part, thus considering Eq. (218). The structure of this integral is similar to the one we evalu- +ated for the mass-independent part, but with some differences that simplify the computation. +In particular, the overall factor multiplying the angular functions changes according to the +42 + +substitution +(k2)−1−ϵ(pi · pj) +(pi · k)(pj · k) +→ (k2)−1−ϵ +(pj · k)2 , +(240) +which in terms of the dimensionless variables introduced in Eq. (231), corresponds to: +u +1 + u + w → +2uw +(1 + u + w)2 . +(241) +This replacement removes the dependence on the massless momentum from the denominator +of the integrand: as a consequence, the integration of this term will not give rise to additional +collinear singularities and thus there is no need for the regulator we introduced in Eq. (92), +that we can safely drop. +Eq. (218) can thus be written in terms of two-folded integrals simply by applying the +substitution (241) in Eq. (238), setting λ = 0 and considering the function ggg +ij rather than fgg +ij . +By doing so, we obtain +⟨ +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm̸=0 +ij +⟩av. = +�b2 +4 +�2ϵ +π1−ϵ Γ(−2ϵ) +Γ(1 + ϵ) +� ∞ +0 +du dw +u−1−ϵ +(1 + u + w)2 +× +� +1 + Re {[2F1 (−2ϵ, −ϵ; 1 − ϵ; wB) − 1] (1 + i cot(πϵ))} +� +ggg +ij (u, w) , +(242) +where we have defined ggg +ij (⃗ni · ⃗nj, 1,⃗n2 +j) ≡ ggg +ij (u, w). The computation of this integral does not +present any particular additional challenge that cannot be solved with standard techniques. The +expression of the angular function ggg +ij can be obtained from Ref. [61] and it is again convenient +to identify and immediately expand its contribution which is regular in ϵ. To simplify the +expression of the integrand, we also isolated a part that, after integration, would have been +independent of any kinematical variables: we evaluated numerically this contribution to obtain +its constant result c0, finding c0 = −37.73041235261383. The final result reads +⟨ +� +dDk ei⃗b·⃗kT +� +(12) +˜Sm̸=0 +ij +⟩av. = − +�b2 +4 +�2ϵ +π2−ϵ Γ(−2ϵ) +ϵ Γ(1 + ϵ) +� +1 + ϵ +� +π2 +6 − 6 − 2Li2 +� +1 +1 − r2 +� +− ln2(r2 − 1) + 2 ln(r)(2 ln(r) + 2) +� ++ ϵ2 +6 +� +c0 + 30Li3 +� 1 +r2 +� ++ 72Li3 +� +1 +1 − r2 +� +− 120 +� +Li3 +� +1 +1 − r +� +− Li3 +� +1 +r + 1 +�� ++ 12Li2 +� 1 +r2 +� +(ln(r) − 5) + 16π2 coth−1 � +1 − 2r2� ++ 8 ln3(r − 1) − 16 ln3(r) ++ 8 ln3(r + 1) − 36 ln(r + 1) ln2(r − 1) + 36 ln2(r) ln(r − 1) +− 36 ln2(r + 1) ln(r − 1) − 192 ln2(r) + 36 ln2(r) ln(r + 1) ++ 120 ln(r) ln(r − 1) + 4π2 ln(r) − 144 ln(r) + 120 ln(r) ln(r + 1) ++ 63ζ3 + 13π2 + 24 − 30π2 ln(2) +�� +, +(243) +43 + +where the variable r is defined in Eq. (74). +3.5.3 +Massive-massive contribution: ˜Sjj +We now examine the case in which only one massive final-state emitter is involved and we +consider ˜Sjj, with j = 3, 4. By plugging in Eq. (208) the condition pi = pj, we obtain +˜Sjj = ˜Sm=0 +jj ++ 2m2 ˜Sm̸=0 +jj += +m4 +(pj · k1)(pj · k2)(pj · k)2 + +4m2 +k2(pj · k1)(pj · k2) += 2 +� +m4 +(pj · k)3 + +4m2 +k2(pj · k) +� +1 +(pj · k1) , +(244) +where k = k1+k2. This more compact expression for ˜Sjj allows us to obtain a simplified version +of Eqs. (217), (218) +� +(12) +˜Sjj = 2 +� +m4 +(pj · k)3 + +4m2 +k2(pj · k) +� (k2)−ϵ2−2+2ϵ +(pj · k) +A1,0 , +(245) +and, by using the result of the angular integral A1,0 from Ref. [61], we can write +� +(12) +˜Sjj = (k2)−1−ϵm2 +(pj · k)2 +2π +1 − 2ϵ +� m2k2 +(pj · k)2 + 4 +� +2F1 +�1 +2, 1, 3 +2;⃗n 2 +j +� +. +(246) +The integration of Eq. (246) can be carried out in a similar way as the integration of ˜Sij has +been performed in Sect. 3.5.2. Also in this case it is convenient to split the integrand into its +singular and regular part, immediately expanding in ϵ the latter. The final result reads +⟨ +� +dDk ei⃗b·⃗kT +� +(12) +˜Sjj⟩av. = +�b2 +4 +�2ϵ +π2−ϵ Γ(−2ϵ) +Γ(ϵ + 1) +� +2 +ϵ2 + 4 +ϵ +� +ln +� +r2� +− 1 +� ++ 1 +3 +� +−24Li2 +� +1 − r2� +− 24 ln +� +r2� +− 5π2 + 30 +� ++ ϵ +� +32 ln +� 2r +r + 1 +� +Li2 +�1 +2 +� +1 + 1 +r +�� +− 8 +3 +� +7 + 12 ln +� +2 +r − 1 +�� +Li2 +�1 − r +2 +� ++ +�254 +3 ++ 64 ln(r + 1) +� +Li2(1 − r) + 2 +� +4 + 1 +r + 8 ln +� +r r + 1 +(r − 1)2 +�� +Li2 +� 1 +r2 +� ++ +� +−182 +3 +− 8 +r + 32 ln +� (r − 1)2 +2r2(r + 1)2 +�� +Li2 +�1 +r +� ++ 32 ln +� 2r +r − 1 +� +Li2 +�r − 1 +2r +� ++ 32 ln +�r − 1 +2r +� +Li2 +�r − 1 +r +� ++ 8 +� +−3 + 4 ln +� +2r2r + 1 +r − 1 +�� +Li2(−r) +− 64 ln(r − 1)Li2 +� +1 +r + 1 +� ++ 8 +3 +� +7 + 12 ln +� +2 +r + 1 +�� +Li2 +� +2 +r + 1 +� ++ 32 ln +� 2r +r + 1 +� +Li2 +� +r +r + 1 +� ++ 32Li3 +�1 +2 +� +1 + 1 +r +�� +− 32Li3 +�1 − r +2 +� +44 + +− 64Li3(1 − r) + 8Li3 +� 1 +r2 +� +− 64Li3 +�1 +r +� ++ 32Li3 +�r − 1 +2r +� +− 32Li3 +�r − 1 +r +� +− 64Li3(−r) − 64Li3 +� +1 +r + 1 +� +− 32Li3 +� +2 +r + 1 +� +− 64Li3 +�r − 1 +r + 1 +� +− 32Li3 +� +r +r + 1 +� ++ 64Li3 +�r + 1 +1 − r +� +− 8Li3 +� +1 − r2� ++ 1 +3r +� +− 32r ln3(r − 1) ++ 96r ln3(r) + r ln2(r)(77 + 96 ln(2) − 144 ln(r + 1)) + 96r ln2(r − 1) +× ln(r + 1) − 2 ln(r − 1)(4r +� +5π2 − 7 ln(2) +� ++ ln(r)(−6 + r(−67 ++ 48 ln(2)) + 72r ln(r)) + 4r(7 − 48 ln(r)) ln(r + 1) + 96r ln2(r + 1)) ++ 12 ln(r) +� +2r +� +9 + 8 ln2(2) +� +− ln(r + 1)(1 + 2r(5 + 8 ln(2)) − 8r ln(r + 1)) +� ++ r(−60 + π2(27 + 16 ln(2)) + 4 ln2(2)(−7 + 8 ln(2)) + 2 +� +−24 + π2� +ln +� +r2� +− 8 ln3 � +r2� ++ 4 ln(r + 1) +� +6 +� +π2 − 4 ln2(2) +� ++ (7 + 36 ln(2)) ln(r + 1) +� ++ 12 ln2 � +r2� � +−2 + ln +� +r2 − 1 +�� ++ 276ζ(3)) +��� +. +(247) +3.5.4 +Massive-massive contribution: ˜S34 +We finally analyse the case of the interference between the two final-state emitters. Our task +is to integrate both the mass-independent and mass-dependent expressions in Eq. (217) and +Eq. (218) for i = 3, j = 4. +Mass-independent part +Let us start our analysis with the mass-independent contribution. +We need to integrate the dimensionless angular function fgg +34(⃗n3 ·⃗n4,⃗n2 +3,⃗n2 +4) defined in Eq. (217). +To evaluate the angular integrals contained therein, we relate them to the imaginary part of +a massive box diagram via the optical theorem10. All order results for the box diagram with a +single mass, equivalent to fix ai = aj in Eq. (222), are presented in Ref. [64], while results with +two different masses but only at the lowest order in ϵ can be found in Ref. [65]. We extended +the latter expression to all orders in ϵ, obtaining the angular integral A± +1,1 +A± +1,1 = +� π +0 +dθ +� π +0 +dφ +sinD−3 θ sinD−4 φ +(1 − a3 cos θ)(1 ± a4 cos χ cos θ ± a4 sin χ sin θ cos φ) += +2π +2ϵ − 1 +⃗n2 +3 + ⃗n3 · ⃗n4 +� +1 − ⃗n2 +3 (⃗n2 +3 + ⃗n2 +4 + (⃗n3 · ⃗n4)2 + 2⃗n3 · ⃗n4 − ⃗n2 +3⃗n2 +4) +× +× F1 +�1 +2 − ϵ; 1, 1 +2; 3 +2 − ϵ; +⃗n3 · ⃗n2 +4 − ⃗n2 +3⃗n2 +4 +(⃗n3 · ⃗n4)2 + 2⃗n3 · ⃗n4 + ⃗n2 +3 − ⃗n2 +3⃗n2 +4 + ⃗n2 +4 +, 1 − +⃗n2 +3 +⃗n2 +3 − 1 +� ++ (3 ↔ 4) . +(248) +By following the same strategy applied in the previous sections, we isolate in the angular +function fgg +34(⃗n3 ·⃗n4,⃗n 2 +3 ,⃗n 2 +4 ) a term σm=0 +34 +that will give rise to singularities upon integration over +10 See e.g. the discussion in Appendix A of Ref. [63]. +45 + +k2 and a regular term ρm=0 +34 +that vanishes in the k2 → 0 limit and that can be directly expanded +in ϵ +fgg +34(⃗n3 · ⃗n4,⃗n 2 +3 ,⃗n 2 +4 ) = −π +ϵ +� +σm=0 +34 ++ ϵ ρm=0 +34 ++ (p3 ↔ p4) +� +. +(249) +By using Eq. (248), the k2-singular part can be written in the following way +σm=0 +34 += 2 − 2 h(ϵ) +�1 − ⃗n2 +3 +4 +�−ϵ � +1 − 2ϵ (1 − ⃗n3 · ⃗n4)χ34 +⃗n2 +3 − ⃗n3 · ⃗n4 +2F1(1, 1 +2 − ϵ; 3 +2; χ34) +� +, +(250) +where the function h(ϵ) is defined as +h(ϵ) = π−1/2 4−ϵ Γ( 1 +2 − ϵ) Γ(1 + ϵ) , +(251) +and the k2-independent coefficient χ34 is given by +χ34 = +(⃗n2 +3 − ⃗n3 · ⃗n4)2 +(⃗n2 +3 − ⃗n2 +4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 +3 ⃗n2 +4 +, +(252) +and fulfils 0 ≤ χ34 ≤ 1. We observe that the factor (1−⃗n3·⃗n4)/(⃗n2 +3−⃗n3·⃗n4) is also independent +on k2. +Let us start by considering the integration of the singular contribution. +By inserting +Eqs. (249), (250) in Eq. (217) we obtain a sum of integrals with the following structure +Igg +34[fα] = +� +dDk ei⃗b·⃗kT (k2)−1−ϵ(p3 · p4) +(p3 · k)(p4 · k) fα(⃗n3,⃗n4) , +(253) +with three possible functions fα(⃗n3,⃗n4) (α = 1, 2, 3) +f1(⃗n3,⃗n4) = 1 , +(254) +f2(⃗n3,⃗n4) = +�1 − ⃗n2 +3 +4 +�−ϵ +, +(255) +f3(⃗n3,⃗n4) = −4π h(ϵ) +�1 − ⃗n2 +3 +4 +�−ϵ (1 − ⃗n3 · ⃗n4)χ34 +⃗n2 +3 − ⃗n3 · ⃗n4 +2F1(1, 1 +2 − ϵ; 3 +2; χ34) . +(256) +With those definitions, we have +� +dDk ei⃗b·⃗kT (k2)−1−ϵ(p3 · p4) +(p3 · k)(p4 · k) +� +−π +ϵ σm=0 +34 +� += −2π +ϵ Igg +34[f1] + 2π +ϵ h(ϵ)Igg +34[f2] + Igg +34[f3] . +(257) +We start by considering Igg +34[f1] +Igg +34[f1] = +� +dDk +p3 · p4 +(p3 · k)(p4 · k) (k2)−1−ϵ ei⃗b·⃗kT . +(258) +This integral is exactly the same appearing in Eq. (188) for the case of soft q¯q emission, and +the result up to O(ϵ) was already presented in Eq. (192). In the present case, however, we need +46 + +it up to O(ϵ2), since in Eq. (257) Igg +34[f1] already appears with an overall factor 1/ϵ. We find +⟨Igg +34[f1]⟩av. =π1−ϵ Γ(1 − ϵ)Γ(−2ϵ) +�b2 +4 +�2ϵ 1 + β2 +2β +� +−1 +ϵL0 − 2L1 + ϵ(2P2 − L2) +(259) +−2 +3ϵ2 (L3 − 6P3 − 3Q3) + O(ϵ3) +� +. +The functions Ln, Pn have been defined in Eqs. (98)–(99). Here we also introduced the function +Qn, defined as +Qn(β) = +� β +−β +dz +1 − z2Lin +� +− z2 tan2(θ) +z2 − sec2(θ) +� +. +(260) +The explicit expressions of the functions L0, L1 and P2 are already provided in Eqs. (100)–(102) +while L2 can be found in Eq. (193). The functions L3, P3 and Q3, that can be obtained in a +similar way. +We can use a similar strategy to evaluate Igg +34[f2] +Igg +34[f2] = +�m2 +4 +�−ϵ � +dDk ei⃗b·⃗kT (k2)−1−2ϵ(p3 · p4) +(p3 · k)1−2ϵ(p4 · k) . +(261) +In this case, the generalisation of Feynman parametrisation needs to be applied +1 +AmBn = Γ(m + n) +Γ(m)Γ(n) +� 1 +0 +dx +xm−1(1 − x)n−1 +(xA + (1 − x)B)m+n , +(262) +thereby obtaining +Igg +34[f2] = (p3 · p4) +�m2 +4 +�−ϵ +(1 − 2ϵ) +� 1 +0 +dx x−2ϵ +� +dDk ei⃗b·⃗kT +(k2)−1−2ϵ +(p(x) · k)2−2ϵ , +(263) +with p(x) = xp3 + (1 − x)p4. The azimuthally averaged integral ⟨Igg +34[f2]⟩av. can be evaluated as +⟨Igg +34[f2]⟩av. = (p3 · p4) +�m2 +4 +�−ϵ +(1 − 2ϵ) +� 1 +0 +dx +(p2(x))1−ϵx−2ϵ ⟨Iaux +2 +(x)⟩av. , +(264) +with +⟨Iaux +2 +(x)⟩av. = ⟨ +� +dDk ei⃗b·⃗kT (k2)−1−ϵ +(p(x) · k)2 p2(x) +� k2p2(x) +(p(x) · k)2 +�−ϵ +⟩ +av. += +�b2 +4 +�2ϵ π1−ϵ 2−2−2ϵ +ϵ2(1 − 2ϵ) Γ(1 − 2ϵ)Γ(1 − ϵ) +� +1 + p2 +T(x) +p2(x) +�2ϵ +. +(265) +The leftover integral over the Feynman parameter can be performed in a standard way and the +solution expressed in terms of multiple polylogarithms. +47 + +The last integral we need for the evaluation of the singular part is +Igg +34[f3] = −4πh(ϵ) +� +dDk ei⃗b·⃗kT (k2)−1−ϵ(p3 · p4) +(p3 · k)(p4 · k) +× +�1 − ⃗n2 +i +4 +�−ϵ (1 − ⃗n3 · ⃗n4)χ34 +⃗n2 +3 − ⃗n3 · ⃗n4 +2F1(1, 1 +2 − ϵ; 3 +2; χ34) , +(266) +with h(ϵ) and χ34 defined in Eqs. (251) and (252), respectively. +In order to simplify the expression of the integrand we define the auxiliary momentum +ℓµ = 1 +v +� +pµ +3 − +m2 +(p3 · p4)pµ +4 +� +, +(267) +with v defined as in Eq. (75). By using the definition of ℓµ and an integral representation for +the hypergeometric function, we can rewrite Igg +34[f3] as +Igg +34[f3] = − +� +dDk ei⃗b·⃗kT 2π(p3 · p4) +v +(k2)−1−2ϵ +(p4 · k) +� +m2 +(p3 · k)2 +�−ϵ +1 +ℓ · k +× +� 1 +0 +dt (t)− 1 +2 −ϵ(1 − t)ϵ +χ34 +1 − t χ34 +. +=2π(m2)−ϵ(p3 · p4) +v2 +� 1 +0 +dt (t)− 1 +2 −ϵ(1 − t)ϵ +� +dDk(k2)−1−2ϵ +ei⃗b·⃗kT +(p3 · k)1−2ϵ +× +� � +1 +1 − +t +v2 +� � +− +1 +(p4 · k) +� ++ +m2 +2(p3 · p4) +� +σ=±1 +1 +1 + σ +√ +t +v +1 +ℓ(σ) · k +� +, +(268) +where for convenience we introduced +ℓµ +(±) = pµ +3 ± +√ +t ℓµ . +(269) +By analysing the dependence on t of the integrand, we observe that it is expressed as a sum +of functions, some of which feature a divergence for t = v2. This singularity has no physical +origin and will eventually cancel in the final result when summing together these divergent +contributions. In order to regularise it, we fix a small imaginary part for v and take its finite +limit to zero at the end of the computation. The dependence of the integrand on k is via a +factor +Igg +34[f3] ∝ (k2)−1−2ϵ +(p3 · k)1−2ϵ +� +1 +(p4 · k); +1 +(ℓ(±) · k) +� +. +(270) +By applying the generalised Feynman parametrisation introduced in Eq. (262), we obtain a sin- +gle momentum integral equivalent to Iaux +2 +in Eq. (265), the only difference being the expression +of the auxiliary momentum, which now reads: +p(x) = xp3 + (1 − x)P , +(271) +48 + +with the three possibilities +P = p4, +P = ℓ(+) +P = ℓ(−) . +(272) +The integration over k can thus be performed by using the partial results already obtained. +The leftover integrals over the Feynman parameter x and the variable t we used for the integral +representation of the hypergeometric function can be performed with standard techniques, and +the result can be expressed in terms of multiple polylogarithms. +We now consider the contribution to fgg +34 that vanishes in the k2 → 0 limit, ρm=0 +34 +. It can be +written at all orders in ϵ in the following compact way +1 +π ρm=0 +34 +=Γ( 1 +2 − ϵ)Γ(ϵ) +√π +�1 − ⃗n2 +3 +⃗n2 +3 +�−ϵ 1 + D34 − 2 +� +⃗n2 +3 +� +⃗n2 +3 ++ 1 +ϵ +� +2 − (1 + D34)2F1( 1 +2, 1; 1 + ϵ, 1 − ⃗n2 +3) +� +− 2(1 − ⃗n3 · ⃗n4)χ34 +⃗n2 +3 − ⃗n3 · ⃗n4 +� 1 +0 +du(1 − u)− 1 +2 −ϵ +√1 − uχ34 +[(1 + uψ)ϵ − uϵ(1 + ψ)ϵ] ++ D34γ +� 1 +0 +du +(1 − u) +1 +2 −ϵ +� +1 − ⃗n2 +3u(1 − (1 − u)γ) +, +(273) +where the coefficient χ34 is defined in Eq. (252) and +D34 = +(1 + ⃗n3 · ⃗n4)(⃗n2 +3 + ⃗n3 · ⃗n4) +(⃗n2 +3 + ⃗n2 +4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 +3 ⃗n2 +4 +, +(274) +γ = +(⃗n3 · ⃗n4)2 − ⃗n2 +3⃗n2 +4 +(⃗n2 +3 + ⃗n2 +4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 +3 ⃗n2 +4 +, +(275) +ψ = − +(⃗n3 · ⃗n4)2 − ⃗n2 +3⃗n2 +4 +(⃗n2 +3 − ⃗n2 +4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 +3 ⃗n2 +4 +. +(276) +Because of its regular behaviour, ρm=0 +34 +can be safely expanded in ϵ +ρm=0 +34 += ρm=0, (0) +34 ++ ϵ ρm=0, (1) +34 ++ O(ϵ2) . +(277) +Due to the complexity of the functions ρm=0, (0) +34 +and ρm=0, (1) +34 +, we perform numerically the last +steps of their integration. The representation of the soft integrals in qT-space, rather than the +impact-parameter space used until now, is more convenient to this purpose, as it allows us to +trivially carry out the D − 2 dimensional integration of the transverse components of the soft +momentum k. The conversion to the representation in b-space can be obtained by applying to +the final result the relation in Eq. (70). To be specific, the integral that we will compute is +�� +dDk δ(D−2)(⃗kT − ⃗qT)(k2)−1−ϵ(p3 · p4) +(p3 · k)(p4 · k) (ρm=0, (0) +34 ++ ϵ ρm=0, (1) +34 +) +� +av. +. +(278) +To perform its azimuthal average, we can fix the azimuthal angle φ such that ⃗pT,3 · ⃗qT = +pT,3 qT cos φ. +With this choice the integral over the other angles becomes straightforward. +After integrating over dD−2kT, the remaining computation can be performed for instance by +49 + +introducing an integral over the virtuality of k. We obtain +(q2 +T)−1−ϵ +B( 1 +2, 1 +2 − ϵ) +� 1 +−1 +d cos φ +� ∞ +0 +dx dy 1 − ⃗n3 · ⃗n4 +2 x y +x−1−ϵ(1 − cos2 φ)−1/2−ϵ(ρm=0, (0) +34 ++ ϵ ρm=0, (1) +34 +) , +(279) +where we introduced the dimensionless integration variables +x = k2 +q2 +T +, +y = k− +qT +. +(280) +Given that the functions ρm=0, (0) +34 +and ρm=0, (1) +34 +vanish by construction in the limit x → 0, +the integrand in Eq. (279) can be safely expanded in ϵ and integrated numerically over x, y and +cos φ. The result is a function of the LO phase-space, which can be reduced to the dependence +on β and cos θ and can thus be provided in the form of a two-dimensional grid. +In order to obtain a more stable and fast numerical evaluation of Eq. (279) we isolate its +contributions that can be expressed only as a function of β. To be specific, we observe that in +the ϵ-expanded expression, +(q2 +T)−1−ϵ +B( 1 +2, 1 +2 − ϵ) +� 1 +−1 +d cos φ +� ∞ +0 +dx dy 1 − ⃗n3 · ⃗n4 +2 x2 y +(1 − cos2 φ)−1/2� +ρm=0, (0) +34 ++ ϵ ρm=0, (1) +34 +− ϵ ln +� +x(1 − cos2 φ) +� +ρm=0, (0) +34 +� +, +(281) +the integral of the first two terms in the square bracket is independent of cos θ, allowing us to +simply compute a one-dimensional grid. In addition, these terms can be integrated by using +the following identity, +1 +π +� 1 +−1 +d cos φ +� ∞ +0 +dx dy 1 − ⃗n3 · ⃗n4 +2 x2 y +x−1−ϵ(1 − cos2 φ)−1/2−ϵf(⃗n3 · ⃗n4,⃗n2 +3,⃗n2 +4) += +� 1 +0 +dt +� 1 +−1 +d cos φ +t2 +1 − t2 +1 +1 − vt cos φ f +� +1 − +1 − t2 +1 − vt cos φ, t2, 1 − (1 − v2) +1 − t2 +(1 − vt cos φ)2 +� +, +(282) +which is valid for a generic function f, and that can by proven by using the relation in Eq. (162). +The right hand side of the equation is only a two-fold integral, leading to a further simplification +of the corresponding numerical integration. +Mass-dependent part +We now consider the case of the contribution coming purely from +the massive case, ˜Sm̸=0 +34 +. +We can approach the computation in a way similar to that used +for the mass-independent part we just described, considering Eq. (218) rather than Eq. (217). +The factor multiplying the angular functions, however, is not anymore symmetric under the +exchange p3 ↔ p4. Unlike the mass-independent contribution, we thus cannot assume that the +final result respects such symmetry and we need to separately compute the integral of ˜Sm̸=0 +34 +and the one of ˜Sm̸=0 +43 +. +50 + +As it is by now customary, we write the dimensionless angular function ggg +34(⃗n3 · ⃗n4,⃗n 2 +3 ,⃗n 2 +4 ) +defined in Eq. (218) as the sum of a singular and a regular contribution +ggg +34(⃗n3 · ⃗n4,⃗n 2 +3 ,⃗n 2 +4 ) = −π +ϵ +� +σm̸=0 +34 ++ ϵ ρm̸=0 +34 ++ (p3 ↔ p4) +� +. +(283) +The singular part, which generates singularities after integration over k2, can be written as +σm̸=0 +34 += − (1 − ⃗n2 +3)−ϵΓ +� 1 +2 − ϵ +� +Γ(1 + ϵ) +√π +� +1 + ϵ 2χ34(1 − ⃗n3 · ⃗n4) 2F1 +� +1, 1 +2 − ϵ; 3 +2; χ34 +� +⃗n2 +3 − ⃗n3 · ⃗n4 +� ++ (1 − ⃗n2 +4)−ϵΓ +� 1 +2 − ϵ +� +Γ(1 + ϵ) +√π +� +1 + ϵ 2χ34(1 − ⃗n3 · ⃗n4) 2F1 +� +1, 1 +2 − ϵ; 3 +2; χ34 +� +⃗n3 · ⃗n4 − ⃗n2 +4 +� +, (284) +while the regular part can be directly expanded in ϵ and we can symbolically write +ρm̸=0 +34 += ρm̸=0 (0) +34 ++ ϵ ρm̸=0 (1) +34 ++ O(ϵ2) , +(285) +where higher order terms can be neglected in our computation. +Let us start with the integration of the singular part in Eq. (284). Its computation requires +the evaluation of integrals in the form +Igg +j [fα] = +� +dDk ei⃗b·⃗kT (k2)−1−ϵ +(pj · k)2 fα(⃗n3,⃗n4) , +(286) +with j = 3, 4 and where the possible functions fα(⃗n3,⃗n4) are the same already introduced in +the mass-independent case in Eq. (254). With this definition we have +� +j=3,4 +� +dDk ei⃗b·⃗kT (k2)−1−ϵ +(pj · k)2 +� +−π +ϵ (σm̸=0 +34 ++ σm̸=0 +43 +) +� += +� +− π +ϵ Igg +3 [f1] + π +ϵ h(ϵ)Igg +3 [f2] + 1 +2Igg +3 [f3] ++ π +ϵ Igg +4 [f1] − π +ϵ h(ϵ)Igg +4 [f2] + 1 +2Igg +4 [f3] +� ++ (p3 ↔ p4) . +(287) +We start from Igg +j [f1], +Igg +j [f1] = +� +dDk +1 +(k2)1+ϵ +ei⃗b·⃗kT +(pj · k)2 . +(288) +This integral has already been computed in Sect. 3.4 and we have m2Igg +j [f1] = Iq¯q +jj , where Iq¯q +jj is +defined in Eq. (186). The result for ⟨Iq¯q +jj (⃗b)⟩av. was reported in Eq. (190). +We now turn our attention to Igg +j [f2]. We first consider Igg +3 (f2). The integral to compute is +Igg +3 [f2] = +� +dDk ei⃗b·⃗kT (k2)−1−2ϵ +(p3 · k)2 +� +m2 +4(p3 · k)2 +�−ϵ +, +(289) +which has the same structure as Iaux +2 +, introduced in Eq. (265). We can thus take the result of +51 + +⟨Igg +3 [f2]⟩av. from Eq. (265) with the replacement p(x) → p3 +⟨Igg +3 [f2]⟩av. = 1 +m2 +�b2 +4 +�2ϵ π1−ϵ 2−2−2ϵ +ϵ2(1 − 2ϵ) Γ(1 − 2ϵ)Γ(1 − ϵ) +� +1 + p2 +3,T +m2 +�2ϵ +. +(290) +The integral Igg +4 [f2], on the other hand, is not proportional to Iaux +2 +(x) +Igg +4 [f2] = +� +dDk ei⃗b·⃗kT (k2)−1−2ϵ +(p4 · k)2 +� +m2 +4(p3 · k)2 +�−ϵ +, +(291) +but the structure of Iaux +2 +(x) can be recovered by applying the generalised Feynman parametri- +sation of Eq. (262): +⟨Igg +4 [f2]⟩av. = − 21+2ϵϵ (1 − 2ϵ) +� 1 +0 +dx x (1 − x)−1−2ϵ ⟨ +� +dDk ei⃗b·⃗kT +(k2)−1−2ϵ +(p(x) · k)2−2ϵ⟩ +av. += − 21+2ϵϵ (1 − 2ϵ) +� 1 +0 +dx +(p2(x))1−ϵx (1 − x)−1−2ϵ ⟨Iaux +2 +(x)⟩av. += − +1 +4 βm2 ϵπ1−ϵ +�b2 +4 +�2ϵ �τ +2 +�1−ϵ +Γ(1 − 2ϵ)Γ(1 − ϵ) +× +� β +−β +dy +� +1 + y +β +� � +1 − y +β +�−1−2ϵ � +1 +1 − y2 +�1−ϵ � Bτ +1 − τ +y2 +1 − y2 + 1 +�2ϵ +, +(292) +where in the last step we used the known result of ⟨Iaux +2 +(x)⟩av. from Eq. (265). At this stage, +the integrand cannot yet be expanded in ϵ, since the integral does not converge in the limit +ϵ → 0 due to a singularity in y = β. In order to perform the expansion, we need to isolate the +singular behaviour and subtract it. We consider the following auxiliary integral: +� β +−β +dy +� +1 − y2�2ϵ−1 +� +1 − y +β +�−2ϵ−1 � +1 + y +β +�1−2ϵ += +=2√πΓ(−2ϵ) +β +1 − β2 +� +1 +Γ +� 1 +2 − 2ϵ +� 2F1 +�1 +2, −2ϵ, 1 +2 − 2ϵ, β2 +� ++ϵ +1 + β2 +Γ +� 3 +2 − 2ϵ +� 2F1 +�1 +2, 1 − 2ϵ, 3 +2 − 2ϵ, β2 +�� +. +(293) +We observe that the integrand in Eq. (293) has the same behaviour as I4[f2] in the y → β limit, +while having a simpler structure that allows for a straightforward analytic integration. We can +thus add the r.h.s. of Eq. (293) to Eq. (292), while subtracting the l.h.s. at the integrand level +in order to obtain a regular expression that can be safely expanded. The resulting integral can +be evaluated separately at each order in ϵ in terms of multiple polylogarithms. +Let us finally consider the contribution of the function f3. By following the same proce- +dure already applied for the evaluation of Igg +34[f3] in the mass-independent case, we define the +52 + +momentum ℓ as in Eq. (267) and by using Eq. (268) we have: +⟨Igg +3 [f3]⟩av. = −π +v ⟨ +� +dDk ei⃗b·⃗kT (k2)−1−2ϵ +(p3 · k) +� +m2 +(p3 · k)2 +�−ϵ +1 +ℓ · k +� 1 +0 +dt t− 1 +2 −ϵ(1 − t)ϵ +χ34 +1 − t χ34 +⟩ +av. +, +(294) +which, once we substitute in it the definition of χ34 as in Eq. (252), gives us an expression that +only depends explicitly on two momenta, p3 and ℓ. By applying partial fractioning and defining +ℓ± as in Eq. (269) we obtain: +⟨Igg +3 [f3]⟩av. = −π +v ⟨ +� +dDk ei⃗b·⃗kT (m2)−ϵ +2 +� 1 +0 +dt t−1−ϵ(1 − t)ϵ +� +(k2)−1−2ϵ +(p3 · k)1−2ϵ(ℓ− · k) +− +(k2)−1−2ϵ +(p3 · k)1−2ϵ(ℓ+ · k) +� +⟩av. . +(295) +By applying the generalisation of Feynman parametrisation introduced in Eq. (262) we can +reduce the dependence of the denominators of the integrand to a single momentum, and re- +trieve the structure of Iaux +2 +(x) as defined in Eq. (265). The leftover integral over the Feynman +parameter x and the variable t can be computed in terms of multiple polylogarithms with a +standard procedure. +The evaluation of Igg +4 [f3] can be performed by following the same steps, but the differences in +the integrand make the procedure of partial fractioning a bit more involved. After introducing +the momentum ℓ and applying partial fractioning for a first time, we obtain an expression +similar to Eq. (295): +⟨Igg +4 [f3]⟩av. = −π +v ⟨ +� +dDk ei⃗b·⃗kT +(m2)−ϵ +2(p4 · k)2 +� 1 +0 +dt t−1−ϵ(1 − t)ϵ +� +(k2)−1−2ϵ +(p3 · k)−1−2ϵ(ℓ− · k) − +(k2)−1−2ϵ +(p3 · k)−1−2ϵ(ℓ+ · k) +� +⟩av. , (296) +but, in this case, we can not yet apply Feynman parametrisation, since each denominator in- +volves three different products of the momenta. We can circumvent this problem by performing +an additional partial fractioning: +⟨Igg +4 [f3]⟩av. = − π +v2(m2)−ϵ⟨ +� +dDk ei⃗b·⃗kT +� 1 +0 +dt t− 1 +2 −ϵ (1 − t)ϵ +� +1 − +t +v2 +� +� +(k2)−1−2ϵ +(p3 · k)−2ϵ(p4 · k)2 ++ m2 +p3 · p4 +� +− +� +1 + +t +v2 +� +� +1 − +t +v2 +� +(k2)−1−2ϵ +(p3 · k)1−2ϵ(p4 · k) + +m2 +2p3 · p4 +√ +t +v +× +�� +1 + +t +v2 +� +� +1 − +t +v2 +� +(k2)−1−2ϵ +(p3 · k)1−2ϵ(ℓ− · k) − +� +1 − +t +v2 +� +� +1 + +t +v2 +� +(k2)−1−2ϵ +(p3 · k)1−2ϵ(ℓ+ · k) +��� +⟩av. . +(297) +It is now possible to apply Feynman parametrisation to Eq. (297), obtaining integrals over the +momentum k that can be written in terms of Iaux +2 +. By using the known result of ⟨Iaux +2 +⟩ provided +53 + +in Eq. (265) to perform the integration over k, we are left with the final two integrals over the +Feynman parameter and the variable t, that can be performed with a standard procedure. +Let us finally analyse the regular part of ggg +34(⃗n3 · ⃗n4,⃗n2 +3,⃗n2 +4), that can be safely be expanded +in ϵ. We need to evaluate the following integral: +⟨ +� +j=3,4 +� +dDk ei⃗b·⃗kT (k2)−1−ϵ +(pj · k)2 (ρm̸=0 +34 ++ ρm̸=0 +43 +)⟩av. . +(298) +The explicit all-orders expression of the integrand reads: +1 +π +� +j=3,4 +1 +pj · k +� +ρm̸=0 +34 ++ ρm̸=0 +43 +� += (1 − ⃗n2 +3) +� +− 2 − 5ϵ +ϵ(1 − 2ϵ) − +1 +1 − 2ϵ +⃗n3 · ⃗n4 +⃗n2 +3 +� ++ (2 − ⃗n2 +3 − ⃗n2 +4) +� +1 +ϵ + Γ(1 − 2ϵ)Γ(ϵ) +Γ(1 − ϵ) +�1 − ⃗n2 +3 +4 +�−ϵ � +D34 +n1/2−ϵ +3 +− 1 +�� ++ 1 +√πΓ +�1 +2 − ϵ +� +Γ (ϵ) (1 − ⃗n2 +3)1−ϵ +�⃗n2 +3 + ⃗n3 · ⃗n4 +2(⃗n2 +3)3/2−ϵ − +6⃗n2 +3 +2(⃗n2 +3)3/2−ϵ + 2 +� ++ 1 +ϵ +� +3(1 − ⃗n2 +3) + D34(2 − ⃗n2 +3 − ⃗n2 +4) +� +2F1 +�1 +2, 1, 1 + ϵ; 1 − ⃗n2 +3 +� +− 1 − +� +⃗n2 +3 +ϵ⃗n2 +3 +(⃗n2 +3 + ⃗n3 · ⃗n4)2F1 +� +1, 1 − ϵ, 1 + ϵ; +2 +1 + +� +n2 +3 +− 1 +� +− 1 − ⃗n3 · ⃗n4 +⃗n2 +3 − ⃗n3 · ⃗n4 +(2 − ⃗n2 +3 − ⃗n2 +4)χ34 +� 1 +0 +du (1 − u)− 1 +2 −ϵ +√1 − χ34u [(1 + uψ)ϵ − uϵ(1 + ψ)ϵ] ++ D34 +γ +1 − γ (2 − ⃗n2 +3 − ⃗n2 +4) +� 1 +0 +du (1 − u) +1 +2 −ϵ +� +1 − ⃗n2 +3u +� +1 + u +γ +1 − γ +�−1 +, +(299) +where the variable χ34 has been defined in Eq. (252) while D34, γ and ψ are given in Eqs. (274)– +(276). We introduce the following notation for the ϵ-expansion of the integrand: +ρm̸=0 +ij += ρm̸=0, (0) +ij ++ ϵ ρm̸=0, (1) +ij ++ O(ϵ2) . +(300) +As for the case of the mass-independent contribution, due to the complexity of the functions +involved in the integrand, we perform numerically the last steps of this computation. We follow +the same steps as for the integration of ρm=0 +34 +, by considering the qT-space representation of the +integral and by fixing the azimuthal angle φ such that ⃗pT,3 · ⃗qT = pT,3 qT cos φ. After switching +to the dimensionless variables x, y already introduced in Eq. (280) and inserting an integral +54 + +over the virtuality of the momentum, we obtain: +⟨ +� +j=3,4 +� +dDk δ(D−2)(⃗kT − ⃗qT)(k2)−1−ϵ +(pj · k)2 (ρm=0, (0) +34 ++ ϵ ρm=0, (1) +34 +)⟩av. = += +q−1−ϵ +T +B( 1 +2, 1 +2 − ϵ) +� 1 +−1 +d cos φ +� ∞ +0 +dx dy +1 +2x2y(1 − cos2 φ)−1/2 +× +� +j=3,4 +1 +pj · k +� +ρm̸=0, (0) +34 ++ ϵ ρm̸=0, (1) +34 +− ϵ ln +� +x(1 − cos2 φ) +� +ρm̸=0, (0) +34 +� +. +(301) +We can observe that also in this case the integral of the first two terms in the square bracket +only depends on β and can be thus evaluated on a one-dimensional grid. We can also reduce +the 3-fold integrals of Eq. (301) in 2-fold ones by replacing the exponential by a θ-function +in the corresponding b-space representation, in a similar fashion as it was done in Eq. (282). +Adapting it to the present functions we obtain: +1 +π +� 1 +−1 +d cos φ +� ∞ +0 +dx dy +1 +2 x2 yx−1−ϵ(1 − cos2 φ)−1/2−ϵρ(⃗n3 · ⃗n4,⃗n2 +3,⃗n2 +4) += +� 1 +0 +dt +� 1 +−1 +d cos φ +t2 +1 − t2 ρ +� +1 − +1 − t2 +1 − vt cos φ, t2, 1 − (1 − v2) +1 − t2 +(1 − vt cos φ)2 +� +. +(302) +We have now summarized in detail the technical aspects of our calculation, and presented +explicit partial results in the cases the expressions were obtained in a compact analytic form. +Our complete final results are collected and implemented in a numerical code, which is described +in the next Section. +4 +Numerical results +In Sect. 2 we described in detail the ingredients entering the transverse-momentum resummation +formalism for heavy-quark production. To the purpose of the application to the qT-subtraction +framework, the key role is played by the coefficient HQ ¯Q defined in Eq. (14), which depends on +the subtracted matrix element � +M via the master formula (15), while � +M can be obtained through +Eq. (54). All the ingredients entering in these equations are finite, since the cancellation of the +IR poles has been carried out at the operator level as described in Eq. (41). The cancellation +is guaranteed by the relation with the subtracted soft anomalous dimension Γsub in Eqs. (36)– +(38): we were able to verify analytically this cancellation for all the contributions, with the +exception of the nf-independent part of the term proportional to the colour factor T3 · T4. +This term depends only on the variable β. As described in Section 3.5.4, part of this term was +evaluated numerically, and, therefore, only a numerical check of the cancellation is possible. In +Figure 1 we compare the coefficient of the 1/ϵ pole as a function of β computed analytically with +Eq. (38) against our numerical result. The lower plot shows the relative difference between the +two: The relative difference is below the 0.0005% in all the regions of the phase-space, showing +a perfect agreement with the prediction and providing a strong cross-check of our computation. +55 + +-2 +-1 +0 +1 +2 +3 +4 +5 +〈Fex,2 +(-1)〉av. +nf =0, T3.T4 component +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.0004 +-0.0002 +0.0000 +0.0002 +0.0004 +β +relative difference (%) +Figure 1: Numerical results for the coefficient of the 1/ϵ pole of the contribution proportional +to T3 · T4 in Fex,2 (red points), compared to the expected analytical result (gray curve) from +Eq. (38). The lower plot shows the relative difference (in percentage) between the two. +Having discussed the cancellation of the IR singularities, we now consider the ingredients +needed for the implementation of the function HQ ¯Q. In the qT-subtraction formalism, a final +average over the azimuthal degree of freedom of ⃗b is required (see Eq. (14)), and since the +operator D is defined in such a way that ⟨D⟩av. = 1, it gives no contribution in our computation, +except when interfering with the azimuthally dependent part of the C coefficients in Eq. (14). +Therefore, as a new perturbative ingredient at NNLO we just need to evaluate the subtracted +amplitude � +M through Eq. (54) at second order. At this order the subtracted amplitude Z−1 |M⟩ +appearing in Eq. (54) is provided by the numerical grids in Ref. [37]. The operator eV fin +c +at the +same order is already known from the implementation of qT-subtraction for a colourless final +state at NNLO [38] (see Eq. (55)). +We are left with the coefficient h, whose perturbative expansion is given in Eq. (47). The +term involving the commutator produces contributions proportional to three-parton correlators, +which vanish when evaluated on the Born c¯c → Q ¯Q amplitude. Therefore we can simply write +h(αS) = 1 + αS +2π ⟨F(0) +ex,1⟩av. + +�αS +2π +�2 � +⟨(F(0) +ex,1)2⟩av. − 1 +2 +� +⟨F(0) +ex,1⟩av. +�2 ++ ⟨F(0) +ex,2⟩av. − 2πβ0 ⟨F(1) +ex,1⟩av. +� ++ O(α3 +S) . +(303) +The two last terms in the O(α2 +S) contribution involve the product of two colour charges; we have +chosen to write the results in the numerical implementation in terms of the colour structures +56 + +T3 · T4 and Ti · Tj with i = 1, 2 and j = 3, 4. +The results for the Ti · Tj structure are obtained in a fully analytical way, and the explicit +expression, which can be obtained from the results in the previous Sections, is implemented in +the numerical code. The results corresponding to the colour structure T3·T4 have contributions +from the integral of the soft correlators Sm=0 +34 +and Sm̸=0 +34 +of Eq. (198) and (199), which are +partially obtained numerically in the form of a two-dimensional grid. +The numerical integration is performed using the implementation of global adaptive strate- +gies available in Mathematica. For the terms independent of θ, the integral is evaluated for +a grid in the variable β from 0 to 1, in steps of 0.001 in the range (0; 0.8) and a smaller step of +0.0001 in the high-energy region (0.8; 1), in which the variation of the function is larger. For the +remaining term, which depends both on β and cos θ, the integral is evaluated for a total number +of 5000 phase-space points in the range β ∈ (0; 1), cos θ ∈ (0; 1) (the result is symmetric under +the exchange cos θ → − cos θ), which were obtained from the NNLO parton level generator +Matrix [66] after the optimisation for the integration of the LO t¯t cross section. +Given that a numerical interpolation of the grid is already needed, and also due to the fact +that the numerical evaluation of the analytical terms entering the T3 · T4 structure of ⟨F(0) +ex,2⟩av. +is computationally very expensive, we decided to encode all the contributions proportional to +T3 · T4 in the terms ⟨F(0) +ex,2⟩av. − 2πβ0 ⟨F(1) +ex,1⟩av. in a two-dimensional grid composed by the same +phase-space points used for the numerical integration of the aforementioned pieces of Sm=0 +34 +and Sm̸=0 +34 +. The different pieces entering the final result are defined in three independent grids +and combined afterwards, in order to have a fully flexible implementation in the number of +light-quark flavours nf. The numerical evaluation of the multiple polylogarithms appearing in +some of our analytic expressions, needed for the construction of the grids, is performed using +GiNaC [67, 68]. +In addition to the contributions described above, the result for the azimuthal average of the +square of the NLO result, i.e. the term ⟨(F(0) +ex,1)2⟩av., is also obtained numerically, by simply +starting from the known result for F(0) +ex,1 and computing the azimuthal average of its square, +again in the same set of phase-space points used before. In this case, the results are grouped +in three different colour structures, (T3 · T4)2, CF T3 · T4 and C2 +F. +The grids described above are afterwards fitted using a spline approximation [69]. Given +that we do not expect our results for each phase-space point to have a large deviation from +the correct value, as the uncertainties of the numerical integration are at the per mille level, +the parameters of the spline fitting are chosen such that the fit is very close to the original +points. In addition, and in order to improve the quality of the fit, the grids are divided by +appropriate factors depending on β and cos θ before performing the fit, which were checked +to generate surfaces with smaller variations and therefore easier to fit. A concrete example +of this procedure is given by the way to handle the threshold region: all the grids had a +divergent logarithmic behaviour in the β → 1 limit. We thus divided all the points by a factor +(1 + log2(1 − β2)n), with the value of n chosen in order to get a regular grid in such limit, and +this factor was added back after the fitting. Also, in order to work with more evenly distributed +points, we worked with the variables β2 (instead of β) and cos θ. +57 + +We have studied the self-consistency of the fit by comparing the results obtained with it to +the original values on the grids used to construct it. We observed that, for the majority of the +points (93.9%), the difference is below the per mille level, while the points that agree better +than 1% almost cover the full phase-space (98.9%). The largest relative differences show up +in the grids corresponding to ⟨(F(0) +ex,1)2⟩av., in the regions in which simultaneously β and | cos θ| +are close to 1, the reason being the sudden variation of the fitted function in that area, and its +value being very close to zero. +In order to see if the error coming from the fitting of the grids has an impact in the +computation of a physical quantity, we checked the difference between the original grid and the +fit once combined with all the other ingredients entering the coefficient HQ ¯Q. This involves, +among other things, the evaluation of lower-order (colour-correlated) matrix elements, the finite +part of the two-loop amplitudes, plus all the soft contributions that were obtained and encoded +analytically. We performed this check for the specific case of top-quark pair production, using +OpenLoops [70] for the evaluation of tree-level and one-loop amplitudes, and the results from +Ref. [37] for the two-loop corrections. We observed that the point-wise difference is always +below 0.25%, and that, from the total of points, only a handful of them present a deviation +larger than 0.05%, indicating that the accuracy obtained through the fit is more than enough +to reproduce the original results. +The checks described above only tested the accuracy of the fit on the very same points used +to generate it: it is also important to perform some checks on the rest of the phase-space. To +this end, we reduced the number of points used to perform the fit by a factor of 2 and checked +how the accuracy of the final result is affected, finding results similar to the ones described +above, thereby confirming the reliability of our implementation. +We illustrate our final results in Figs. 2 and 3. As described in the text, we split our results +into the different colour structures appearing in h, specifically +h(αS) = 1 + αS +2π +� +h(1) +34 T3 · T4 + h(1) +33 CF +� ++ +�αS +2π +�2 � +h(2) +34 T3 · T4 + h(2) +13 T1 · T3 + h(2) +14 T1 · T4 + h(2) +23 T2 · T3 + h(2) +24 T2 · T4 ++ h(2) +3434 T3 · T4 T3 · T4 + h(2) +3433 T3 · T4 CF + h(2) +3333 C2 +F +� ++ O(α3 +S) . +(304) +We note that this particular choice of colour structures is not unique, and different choices +can be made which are related by colour conservation. Results for h(2) +34 and h(2) +13 are given in +Fig. 2, while h(2) +3434, h(2) +3433 and h(2) +3333 are presented in Fig. 3. In both cases, the results correspond +to nf = 0. The numerical code used to evaluate all the terms in Eq. (304) is included as +supplemental material of this paper, allowing for the evaluation of our final results for arbitrary +values of β, cos θ and nf. +58 + +0.001 +0.005 0.010 +0.050 0.100 +0.500 +1 +-30 +-20 +-10 +0 +10 +20 +30 +40 +1-β2 +h34 +(2) +nf=0, cosθ=0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-11.0 +-10.5 +-10.0 +-9.5 +cosθ +h34 +(2) +nf=0, β=0.5 +0.001 +0.005 0.010 +0.050 0.100 +0.500 +1 +0 +50 +100 +150 +1-β2 +h13 +(2) +nf=0, cosθ=0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +-14.5 +-14.0 +-13.5 +-13.0 +cosθ +h13 +(2) +nf=0, β=0.5 +Figure 2: Contributions to the second order coefficient of h proportional to T3 · T4 (upper +panels) and T1 · T3 (lower panels). The results correspond to nf = 0. The left panels show the +β dependence for a fixed value of cos θ = 0, while the cos θ dependence is shown in the right +panels for β = 0.5. +59 + +0.001 +0.005 0.010 +0.050 0.100 +0.500 +1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +1-β2 +h3434 +(2) +nf=0, cosθ=0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +cosθ +h3434 +(2) ×100 +nf=0, β=0.5 +0.001 +0.005 0.010 +0.050 0.100 +0.500 +1 +0 +10 +20 +30 +40 +50 +1-β2 +h3433 +(2) +nf=0, cosθ=0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +cosθ +h3433 +(2) ×100 +nf=0, β=0.5 +0.001 +0.005 0.010 +0.050 0.100 +0.500 +1 +0 +50 +100 +150 +200 +1-β2 +h3333 +(2) +nf=0, cosθ=0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +cosθ +h3333 +(2) ×100 +nf=0, β=0.5 +Figure 3: Same as in Fig. 2 for the contributions proportional to T3 ·T4 T3 ·T4 (upper panels), +CF T3 · T4 (middle panels) and C2 +F (lower panels). +60 + +5 +Summary +This paper has been devoted to the evaluation of the soft-parton contributions that are relevant +when a heavy-quark pair is produced at small transverse momenta in hadronic collisions. +When a colourless system (vector boson(s), Higgs boson(s) and so forth) is produced in +hadron collisions only soft and collinear radiation from the initial-state colliding partons plays +a role. When a heavy-quark pair is produced, the coloured heavy quarks can emit in turn soft +radiation (soft gluons and light quark-antiquark pairs), which gives an additional contribution +to the structure of the singular contributions at small transverse momenta. We have evaluated +such soft-parton contributions to NNLO in QCD perturbation theory. +Our computation has been carried out by using a semi-numerical approach, and evaluating +all the relevant integrals in impact parameter space. We have explicitly considered only the +contributions that are relevant to apply the qT subtraction formalism to this process. After +having introduced our framework in Sect. 2, in Sect. 3 we have provided the details of our +calculation, by first starting from the NLO in Sect. 3.2, which had already been obtained in +Ref. [27]. We then moved to the evaluation of the integrals from soft-gluon emission at one- +loop order in Sect. 3.3, soft light-quark pairs in Sect. 3.4, and finally double gluon emission +in Sect. 3.5. We have provided all the relevant details of the computation by highlighting the +difficulties that had to be overcome. At NNLO the most challenging contributions are those +from the double-real emission, and in particular, those from double gluon radiation. These +contributions need first to be integrated over the angles of the emitted partons by keeping their +total momentum k fixed. Then, the remaining integrals have been evaluated by splitting them +into a singular and a regular part as k2 → 0. For some of the contributions, the latter has +been evaluated numerically. After checking the cancellation of the ϵ poles, the complete results +for the final remainders are provided through a numerical code that is attached to the arXiv +distribution of the paper. +Together with the results already available in the literature, the soft-parton contributions +presented in this paper complete the evaluation at NNLO of the azimuthally-averaged transverse +momentum resummation formula for the production of heavy-quark pairs. In particular, the +results can straightforwardly be implemented to carry out fully differential NNLO calculations +for the production of a pair of heavy quarks with arbitrary mass by using the qT subtraction +formalism. +Acknowledgments +This work is supported in part by the Swiss National Science Foundation (SNF) under contract +200020 188464. +61 + +References +[1] P. Nason, S. Dawson, and R. K. Ellis, The Total Cross-Section for the Production of +Heavy Quarks in Hadronic Collisions, Nucl. Phys. B303 (1988) 607–633. +[2] W. Beenakker, H. 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C 79 (2019), no. 10 866, [arXiv:1907.13071]. +67 + diff --git a/_tFKT4oBgHgl3EQfVC14/content/tmp_files/load_file.txt b/_tFKT4oBgHgl3EQfVC14/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d1d5b532a8e4e41af0f113664073ccbd06a04a1 --- /dev/null +++ b/_tFKT4oBgHgl3EQfVC14/content/tmp_files/load_file.txt @@ -0,0 +1,3330 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf,len=3329 +page_content='TIF-UNIMI-2023-2 ZU-TH 04/23 PSI-PR-23-1 Soft-parton contributions to heavy-quark production at low transverse momentum Stefano Catani(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Simone Devoto(b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Massimiliano Grazzini(c) and Javier Mazzitelli(d) (a)INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Sezione di Firenze and Dipartimento di Fisica e Astronomia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Universit`a di Firenze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 50019 Sesto Fiorentino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Firenze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Italy (b)Dipartimento di Fisica “Aldo Pontremoli”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' University of Milano and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Sezione di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' I-20133 Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Italy (c)Physik Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Universit¨at Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 8057 Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Switzerland (d)Paul Scherrer Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' CH-5232 Villigen PSI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Switzerland Abstract We consider QCD radiative corrections to the production of a heavy-quark pair in hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We present the computation of the soft-parton contributions at low transverse momentum of the heavy-quark pair up to second order in the QCD coupling αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' These results complete the evaluation at the next-to-next-to- leading order (NNLO) of the transverse-momentum resummation formula for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Moreover, they give all the ingredients that are needed for the NNLO implementation of the qT subtraction formalism for heavy-quark production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We discuss the details of the computation and we provide a code that can be used to obtain the relevant results in numerical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' January 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='11786v1 [hep-ph] 27 Jan 2023 Contents 1 Introduction 1 2 Heavy-quark production at low transverse momentum 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 Resummation formalism for heavy-quark production .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 Soft contributions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 Double gluon emission .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 Massive-massive contribution: ˜Sjj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 Massive-massive contribution: ˜S34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 45 4 Numerical results 55 5 Summary 61 1 Introduction Heavy-quark pair production is one of the classic hard-scattering processes at hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For a sufficiently-heavy quark, the cross section is perturbatively computable as an expansion in the QCD coupling αS(µ2 R) where the renormalisation scale µR is of the order of the mass m of the heavy quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' A large variety of QCD studies of heavy-quark hadroproduction have been carried out over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In this context top-quark pair production plays a special role: being the heaviest particle in the Standard Model, the top quark couples strongly to the Higgs boson and is therefore particularly relevant for the mechanism of electroweak symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As such, top-quark pair production is especially relevant in searches for physics beyond the Standard Model, it constitutes a possible window on new physics and, at the same time, a crucial background in many analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Bottom and charm quark production have also been extensively studied at hadron colliders, and allow us to probe QCD at smaller energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For the above reasons, the study of the hadroproduction of a heavy-quark pair has attracted the attention and the efforts of the theoretical community for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Next-to-leading order 1 (NLO) QCD corrections to this process have been available since a long time, both for the total cross section and for differential distributions [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Nevertheless, because of the chal- lenging complications arising at the next order in the perturbative expansion, more than 20 years passed before next-to-next-to-leading order (NNLO) QCD corrections for top-quark pair production were also computed [6–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Further progress regards the combination of QCD and EW corrections [14] and the inclusion of top-quark decays [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Results by using the MS scheme for the renormalisation of the top-quark mass are also available [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' More recently, the NNLO calculation of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [12, 13] has been extended to bottom-quark pair production [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' One of the two available NNLO computations for heavy-quark production [12, 13, 17, 18] is based on the qT subtraction formalism [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The qT subtraction formalism is a method to handle and cancel the IR divergences in QCD computations at NLO, NNLO and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The method uses IR subtraction counterterms that are constructed by evaluating the qT distribution of the produced final-state system in the limit qT → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' If the produced final-state system is composed of colourless particles (such as vector bosons, Higgs bosons, and so forth), the behaviour of the qT distribution in the limit qT → 0 has a universal structure that is explicitly known to the next-to-next-to-next-to leading order (N3LO) through the formalism of transverse-momentum resummation [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The resummation formalism can be extended to the production of final states containing a heavy-quark pair [25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The heavy quarks do not lead to additional collinear singularities (which are absent because of the finite heavy-quark mass) but, being coloured, they lead to additional soft singularities that need to be properly taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The NNLO computations of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [12, 13, 17, 18] rely on the explicit evaluation of such soft- parton contributions due to the coloured massive quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The purpose of this paper is to report on the details of the computation of such soft- parton terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The final numerical results can be obtained by using the program attached to the arXiv submission of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 Our calculation is performed within the transverse- momentum resummation formalism of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' A similar computation, carried out within the framework of Soft Collinear Effective Theory (SCET) used in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [25, 26], has been presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that our formalism can be extended to the production of a heavy-quark pair accompanied by colourless particles [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Such extension has been recently applied to the evaluation of NNLO corrections to t¯tH [34] and Wb¯b [35] production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In this paper, however, we will limit ourselves to the case of heavy-quark production, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=', with no additional colourless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The soft-parton contributions relevant for the production of a heavy-quark pair and a colourless system will be documented elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 we review the resummation formalism for heavy- quark production and we discuss the soft-parton contributions we want to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 we illustrate our calculation, starting from single-gluon emission at tree level in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2, then going to single-gluon emission at one loop in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3, soft q¯q emission in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 and double- gluon emission in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Our numerical implementation and final results are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 5 we summarise our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 The soft-parton contributions evaluated in this work also enter the MiNNLOPS formalism for the matching of NNLO calculations to parton showers for heavy-quark production [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 2 Heavy-quark production at low transverse momentum 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 Resummation formalism for heavy-quark production We consider the inclusive hard-scattering process h1(P1) + h2(P2) → Q(p3) + ¯Q(p4) + X (1) where the collision of the two hadrons h1 and h2 with momenta P1 and P2 produces the heavy- quark pair Q ¯Q, and X denotes the accompanying final-state radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The heavy quarks have four-momenta p3 and p4, total momentum q = p3 + p4, invariant mass M 2 = q2 and total transverse momentum ⃗qT = ⃗p3,T + ⃗p4,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The rapidity of the Q ¯Q pair is y = 1/2 ln(q · P2/q · P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The knowledge of M, y and ⃗qT completely specifies the total momentum q of the heavy- quark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The kinematics of the observed heavy quarks is fully determined by q and two additional independent kinematical variables, that we denote by ⃗Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For example, we can choose ⃗Ω = {y3, φ3}, where y3 and φ3 are the rapidity and the azimuthal angle of the heavy quark Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The hadronic cross section corresponding to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (1) can be computed by convoluting par- tonic cross sections with parton distribution functions fa/h(x, µ2 F) (a = q, ¯q, g denotes the massless partons) of the colliding hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The partonic cross sections can be computed in QCD perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At the leading order (LO) only two partonic processes contribute: quark-antiquark annihilation q¯q → Q ¯Q and gluon fusion gg → Q ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For both processes the ⃗qT dependence of the cross section at LO is simply proportional to δ(2)(⃗qT), since no radiation is emitted at this perturbative order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At higher perturbative orders the partonic cross section in the limit qT → 0 receives large logarithmic contributions of the form αn+2 S 1 q2 T lnk(M 2/q2 T) (k ≤ 2n−1) that need be resummed to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The resummation is customarily carried out in impact parameter (⃗b) space, to factorise the kinematics of multiple parton emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The all-order structure of the logarithmically enhanced contributions can be written as [27] dσ(P1, P2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' ⃗qT, M, y, ⃗Ω) d2⃗qT dM 2 dy d⃗Ω = M 2 2P1 · P2 � c=q,¯q,g � dσ(0) c¯c � � d2⃗b (2π)2ei⃗b·⃗qT Sc(M, b) × � a1,a2 � 1 x1 dz1 z1 � 1 x2 dz2 z2 [(H∆)C1C2]c¯c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='a1a2fa1/h1(x1/z1, b2 0/b2)fa2/h2(x2/z2, b2 0/b2) , (2) where b0 = 2e−γE (γE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='. is the Euler number) and the kinematic variables x1 and x2 are defined as x1,2 = M √2P1 · P2 e±y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (3) The symbol � dσ(0) c¯c � is related to the LO cross section dˆσ(0) c¯c→Q ¯Q for the partonic process c(p1) + ¯c(p2) → Q(p3) + ¯Q(p4), c = q, ¯q, g (4) 3 with pi = xiPi (i = 1, 2), and we have � dσ(0) c¯c � = α2 S(M 2) dˆσ(0) c¯c→Q ¯Q M 2d⃗Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (5) We briefly recall the perturbative ingredients entering the resummation formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (2) (more details can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The formula contains process-dependent and process- independent contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The functions Ci include the contribution of radiation collinear to the initial-state partons at small momentum scales q ≲ 1/b, while the Sudakov form factor Sc accounts for soft and flavour-conserving collinear emissions at scales 1/b ≲ q ≲ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Since they are originated by the spin- and qT-dependent collinear splitting kernels, the functions Ci feature also a dependence on the azimuthal degree of freedom of ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' All the information on the process-dependent corrections is embodied in the term H∆, while the collinear functions Ci and the Sudakov form factor Sc are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The radiative factor ∆ is specific of heavy-quark pair production and is due to soft radiation from the Q ¯Q final state and from the initial-state and final-state interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It depends on the invariant mass M 2, on the kinematics of the partonic process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (4) and on the impact parameter ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The azimuthal dependence can be specified through the angle φ = φ3 −φb, where φ3 and φb are the azimuthal angles of ⃗p3,T and ⃗b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The hard-virtual term H, which embodies virtual contributions at scale q ∼ M, depends on the all-loop scattering amplitude Mc¯c→Q ¯Q for the partonic process c¯c → Q ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The explicit form of the term H∆ is (H∆)c¯c = ⟨ � Mc¯c→Q ¯Q|∆| � Mc¯c→Q ¯Q⟩ α2 S(M 2) ���M(0) c¯c→Q ¯Q ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (6) The symbol M(0) c¯c→Q ¯Q denotes the Born-level amplitude, while � Mc¯c→Q ¯Q represents the all-loop renormalised amplitude after subtraction of the IR singularities (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The amplitude | � Mc¯c→Q ¯Q⟩ is a vector in the colour space of {c, ¯c, Q, ¯Q} and ∆ is a colour-space operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the gluon fusion channel (c = g), the Lorentz (spin) indeces of � M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (6) are properly summed with the corresponding indeces of the gluon collinear functions Ci (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (11) and (13) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The action of the colour factor ∆ is expressed in terms of the operators2 D and V [27] ∆(⃗b, M) = V†(b, M)D(φ, αS(b2 0/b2))V(b, M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (7) The evolution factor V resums logarithmic terms αn S(M 2) lnk(M 2b2) (with k ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It is obtained by the exponentiation of the integral of the anomalous dimension matrix Γt, which is specific 2 Here and in the following, the additional dependence on the rapidity difference y34 = y3 − y4 is left under- stood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 4 of transverse-momentum resummation for QQ production V(b, M) = P q exp � − � M2 b2 0/b2 dq2 q2 Γt(αS(q2)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (8) The symbol P q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (8) denotes the anti path-ordering of the exponential matrix with respect to the integration variable q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The soft-parton factor D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (7) embodies the azimuthal correlations produced by the soft radiation and it is defined [27] in such a way that it gives a trivial contribution after integration over the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have ⟨D(φ, αS)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 1 , (9) where the symbol ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' denotes the average with respect to the azimuthal angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The explicit expressions of the factor H∆ up to O(αS) and of the anomalous dimension Γt up to O(α2 S) are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 In this paper we present a general discussion of the resummation factor H∆ and of its detailed origin and dependence on soft-parton contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Moreover we explicitly compute H∆ up to O(α2 S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This O(α2 S) result is also relevant in the context of the QCD computation of heavy-quark production at NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Indeed, as recalled below, it permits the NNLO implementation of the qT subtraction formalism for this production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Within the qT-subtraction formalism, the NNLO differential cross section dσQQ NNLO of the process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (1) is split into a part with qT = 0 and one with qT ̸= 0 dσQQ NNLO = dσQQ NNLO �� qT =0 + dσQQ NNLO �� qT ̸=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (10) Since at the Born level the final state QQ has qT = 0, the NNLO contributions at qT ̸= 0 are actually given by NLO contributions for the final state QQ+jets dσQQ NNLO �� qT ̸=0 = dσQQ+jets NLO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (11) At NNLO, we can hence handle the IR divergences of the qT ̸= 0 part with the available NLO techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By doing so, we are nevertheless left with additional singularities of purely NNLO origin connected to the limit qT → 0, for which we need an additional subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Following this strategy, we write the cross section as [19] dσQQ NNLO = HQQ NNLO ⊗ dσQQ LO + � dσQQ+jets NLO − dσCT NNLO � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (12) The cancellation of the extra singularities of NNLO type is performed by introducing the counterterm dσCT NNLO, while the coefficient HQQ NNLO embodies the information on the virtual 3 The explicit expressions of the corresponding resummation coefficients for the production of an arbitrary number of heavy quarks accompanied by a colourless system is reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Note that the expression of the first-order contribution D(1) to D therein is mistyped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The correct expression is obtained by replacing �b → −�b in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (25) and (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 5 corrections to the process and contains the qT = 0 contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The counterterm dσCT NNLO needs to capture the singular behaviour of the amplitude in the limit qT → 0 and it can been derived by using the knowledge on the low transverse-momentum spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In particular, it can be obtained from the NNLO perturbative expansion of the logarithmically-enhanced contributions of the resummation formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It depends [36] on the resummation coefficients that already appear in the case of a colourless final state, on the additional Q ¯Q resummation coefficients at O(αS) and on the anomalous dimension Γt at O(α2 S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The coefficient HQQ NNLO contains the virtual corrections to the process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (4) and contri- butions that compensate for the subtraction of the counterterm dσCT NNLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It is defined as the NNLO truncation of the following perturbative series HQQ = 1 + αS π HQQ(1) + �αS π �2 HQQ(2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (13) where HQQ can be expressed [12, 33, 36] in terms of the functions that we just introduced in the context of qT resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have HQQ = ⟨(HD)C1C2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (14) where the average is over the azimuthal angle φ, which appears [27] both in the factor D and through the functions Ci in the gluon channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Analogously to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (6), the explicit form of the term HD reads (HD)c¯c = ⟨ � Mc¯c→Q ¯Q|D| � Mc¯c→Q ¯Q⟩ α2 S(M 2) ���M(0) c¯c→Q ¯Q ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (15) The second-order coefficient HQQ(2) can be computed with the results presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 Soft contributions In our computation we regularise both ultraviolet and IR divergences by using conventional dimensional regularisation in D = 4 − 2ϵ space-time dimensions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The SU(Nc) QCD colour factors are CF = (N 2 c −1)/(2Nc), CA = Nc, TR = 1/2 and we use Cc = CF if c = q and Cc = CA if c = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We consider nf flavours of massless quarks in addition to the heavy quark Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The QCD running coupling αS(µ2 R) = α (nf) S (µ2 R) is introduced through MS renormalisation at the scale µR and decoupling of the heavy quark [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We start our discussion by considering the finite part � Mc¯c→Q ¯Q of the all-order virtual amplitude Mc¯c→Q ¯Q, which is defined through the relation [27] | � Mc¯c→Q ¯Q⟩ = � 1 − �Ic¯c→Q ¯Q � |Mc¯c→Q ¯Q⟩ , (16) 6 where the subtraction operator �Ic¯c→Q ¯Q in colour space has the following expansion �Ic¯c→Q ¯Q(αS(M 2), ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' {pi}) = ∞ � n=1 �αS(µ2 R) 2π �n �I(n) c¯c→Q ¯Q(ϵ, M 2/µ2 R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' {pi}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (17) It is useful to introduce the subtraction operator in the simpler case in which a colourless system F with invariant mass M is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In this case we can write [38] | � Mc¯c→F⟩ = � 1 − �Ic � |Mc¯c→F⟩ , (18) where the subtraction operator �Ic(αS(M 2), ϵ) is now a c-number, and it can be perturbatively expanded as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The explicit expression of the first two perturbative coefficients �I(1) c (ϵ, M 2/µ2 R) and �I(2) c (ϵ, M 2/µ2 R) can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that �Ic depends on the initial-state parton c, but it is completely independent of the produced colourless system F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For later convenience we also define Vc as follows Vc = ln(1 − �Ic) , (19) and we write its decomposition in IR divergent and IR finite components Vc = V sing c + V fin c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (20) The term V sing c , which includes the complete IR divergent contributions to Vc, is a perturbative series in powers of αS(M 2) and the corresponding perturbative coefficients are proportional to ϵ poles, with no additional ϵ dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The remaining ϵ dependence of Vc is entirely embodied in V fin c , which is finite in the limit ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The all-order virtual amplitude Mc¯c→F has IR divergent contributions that are cancelled by �Ic, and � Mc¯c→F in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (18) is IR finite in the limit ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Comparing the transverse-momentum resummation formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (2) with the correspond- ing formula for the production of a colourless system F [38], we recall [27] that the factor ⟨ � Mc¯c→Q ¯Q|∆| � Mc¯c→Q ¯Q⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (6) is analogous to the factor | � Mc¯c→F|2 for F production and, therefore, we can introduce the following master formula4 ⟨ � M|∆| � M⟩ = � ⟨M| eV ∗ c e2Fex(⃗b)eVc |M⟩ � ϵ=0 , (21) where we have written 1 − �Ic = eVc, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The term �Ic in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (18) is due to real emission contributions to the underlying partonic process c¯c → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' More precisely, �Ic is produced by radiation of final-state partons that are either soft or collinear to the colliding partons c and ¯c [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the case of Q ¯Q production, the underlying partonic process is c¯c → Q ¯Q, and the produced Q and ¯Q act as extra source of soft-parton radiation, while the accompanying initial-state collinear radiation is the same as for F production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The amount of extra soft 4 For convenience, here and in the following the amplitudes Mc¯c→Q ¯ Q and � Mc¯c→Q ¯ Q are denoted as M and � M, by removing the subscript c¯c → Q ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 7 radiation due to Q and ¯Q is embodied by the factor e2Fex in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This factor is the result of the integration of the soft-emission contributions after factorisation of the initial-state emission, which is taken into account by the factor eVc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that Fex is a colour space operator, which depends on the colour charges of the partons c, ¯c, Q, ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The real-emission factor eV ∗ c e2Fex(⃗b)eVc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (21) is IR divergent, and it cancels the IR divergences of the virtual amplitude M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This cancellation mechanism and the ensuing structure of the IR-finite terms ∆ and � M are discussed in the remaining part of this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The structure of the IR singular contributions in QCD amplitudes with massive partons is discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The IR singularities of the amplitude M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (21) are factorised in the IR divergent operator Z [43] that permits to write the IR-finite remainder Mfin of the amplitude as follows |Mfin(µIR)⟩ = Z−1(µIR) |M⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (22) Both Z(µIR) and Mfin(µIR) depend on the arbitrary subtraction scale µIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The operator Z(µIR) is a perturbative series in powers of αS(µ2 IR) and the corresponding perturbative coefficients are proportional to ϵ poles, with no additional ϵ dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Unless otherwise stated we will use µIR = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We write the operator Z as follows Z(M) = ZexZc(M) , (23) where the factor Zc(M) embodies the IR divergences due to the initial-state partons c and ¯c, while Zex includes the additional IR divergences due to soft wide-angle radiation from the colour-charged heavy quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore Zc(M) is the IR divergent operator of the amplitude Mc¯c→F in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (18), and we also have Zc(M) = e−V sing c , (24) since the real-emission factor eVc in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (18) cancels the virtual IR divergences of Mc¯c→F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The operator Zex can be obtained by exponentiation of the integral of the subtracted soft anomalous dimension Γsub introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have Zex(M) = P q exp � −1 2 � M2 0 dq2 q2 Γsub(αS(q2)) � , (25) where αS(q2) is the renormalised QCD coupling in D = 4 − 2ϵ dimensions and the perturbative expansion of Γsub is Γsub = αS 2πΓ(1) sub + �αS 2π �2 Γ(2) sub + O(α3 S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (26) 8 The explicit scale dependence of αS is αS(q2) = αS(µ2) �µ2 q2 �ϵ � 1 − β0 ϵ αS(µ2) � 1 − �µ2 q2 �ϵ� + O(α2 S) � (27) where β0 is the first coefficient of the QCD beta function 12πβ0 = 11CA − 2nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (28) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (27), the operator Zex in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (25) can be written as Zex(M) = e−Vex(M2) , (29) where the explicit expression of Vex up to O(α2 S) reads Vex(M 2) = αS(M 2) 2π � − 1 2ϵΓ(1) sub � + �αS(M 2) 2π �2 � 1 ϵ2 πβ0 2 Γ(1) sub − 1 ϵ 1 4Γ(2) sub � + O(α3 S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (30) We note that the anti-path ordered operator ¯Pq in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (25) is irrelevant to evaluate Zex up to O(α2 S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (22)–(24) in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (21) we see that the colourless subtraction operator Vc only cancels the IR singularities of M that originate from the initial-state emission factor Zc(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The virtual IR divergences in Zex are removed by the IR divergences in Fex, as discussed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The perturbative expansion of Fex(⃗b) can be written as Fex(⃗b) = α0 2πSϵ �b2µ2 0 b2 0 �ϵ Fex,1 (φ) + �α0 2πSϵ �2 �b2µ2 0 b2 0 �2ϵ Fex,2 (φ) + O(α3 0) , (31) where α0 denotes the unrenormalised QCD coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In our calculation of Fex(⃗b), the renor- malisation of the coupling constant is taken into account by using the MS scheme: the running coupling αS is related to the bare coupling α0 via the relation α0µ2ϵ 0 Sϵ = αS(µ2 R)µ2ϵ R � 1 − αS(µ2 R)β0 ϵ + O(α2 S) � , (32) where5 β0 is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (28) and Sϵ = (4π)ϵe−ϵγE (33) is the customary D-dimensional spherical factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that the operator Fex(⃗b) fulfils the relation F† ex(⃗b) = Fex(−⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We also point out that, while the function Fex(⃗b) depends on the vector ⃗b, the dependence on b in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (31) is fully embodied in the prefactors b2nϵ and, therefore, the perturbative coefficients Fex,n(φ) only depend on the azimuthal degree of freedom φ of ⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the following, this dependence is left understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Each perturbative coefficient can also be 5 At the end of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 we comment on the contribution to Fex(⃗b) of heavy-quark loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 9 expanded in ϵ as follows Fex,1 = 1 ϵ F(−1) ex,1 + F(0) ex,1 + ϵ F(1) ex,1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (34) Fex,2 = 1 ϵ2 F(−2) ex,2 + 1 ϵ F(−1) ex,2 + F(0) ex,2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (35) The poles in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (34) and (35) are due to the soft singularities of the real-emission contributions and, as previously mentioned, they have to cancel the virtual IR divergences due to the factor Zex in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The cancellation of IR divergences leads to relations between the coefficients F(k) ex,n in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (34), (35) and the coefficients Γ(n) sub of the ϵ pole contributions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (29),(30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We find the following relations F(−1) ex,1 = −1 4 � Γ(1) sub + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , (36) F(−2) ex,2 = πβ0F(−1) ex,1 + 1 8 �� Γ(1) sub − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , F(−1) ex,1 � , (37) F(−1) ex,2 = −1 8 � Γ(2) sub + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � + 2πβ0F(0) ex,1 + 1 4 �� Γ(1) sub − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , F(0) ex,1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (38) The explicit calculation of Fex is presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have verified that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (36)–(38) are fulfilled by our final result for Fex, which is an important cross-check of our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can now consider the master formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (21), implement the cancellation of the real and virtual IR divergences and derive the expressions of ∆ and � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We write ⟨ � M|∆| � M⟩ = � ⟨Mfin| e−V† ex(M2)e−V sing∗ c eV ∗ c e2Fex(⃗b)eVce−V sing c e−Vex(M2) |Mfin⟩ � ϵ=0 = � ⟨Mfin| eV fin∗ c e−V† ex(M2)e2Fex(⃗b)e−Vex(M2)eV fin c |Mfin⟩ � ϵ=0 = � ⟨Mfin| eV fin∗ c V† sub(b, M)e−V† ex(b2 0/b2)e2Fex(⃗b)e−Vex(b2 0/b2)Vsub(b, M)eV fin c |Mfin⟩ � ϵ=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (39) In the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (39) we have used Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (22), (23), (24) and (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the second line we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (20), and the fact that Vc is a c-number that commutes with the other operators in colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the third line we have introduced the evolution operator Vsub, defined by the following relation: e−Vex(M2) = e−Vex(b2 0/b2) ¯Pq exp � −1 2 � M2 b2 0/b2 dq2 q2 Γsub(αS(q2)) � ≡ e−Vex(b2 0/b2)Vsub(b, M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (40) The IR poles in the third line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (39) are fully contained in the individual factors of the operator e−V† ex(b2 0/b2)e2Fex(⃗b)e−Vex(b2 0/b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Their cancellation takes place at the operator level after combining the exponential functions together, and it is guaranteed by the relations between Fex and Γsub that are reported in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (36) and (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore we can safely perform the limit 10 ϵ → 0 and we obtain a finite reminder that, for later convenience, we define as follows lim ϵ→0 � e−V† ex(b2 0/b2)e2Fex(⃗b)e−Vex(b2 0/b2)� = K†(−⃗b)K(⃗b) , (41) where we also used the relation F† ex(⃗b) = Fex(−⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To recast Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (39) and (41) in the form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (6) and (7) we isolate the azimuthal dependence of K†(−⃗b)K(⃗b) in a factor with azimuthal average equal to unity, thus identifying the operator D(φ, αS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We write K†(−⃗b)K(⃗b) = h(αS(b2 0/b2))D(φ, αS)h(αS(b2 0/b2)) , (42) with h†(αS(b2 0/b2)) = h(αS(b2 0/b2)) , (43) ⟨D(φ, αS)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (44) The expressions for the colour operators h and D can be trivially obtained from K as follows (h(αS(b2 0/b2))2 = ⟨K†(−⃗b)K(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (45) D(φ, αS(b2 0/b2)) = h−1(αS(b2 0/b2)) � K†(−⃗b)K(⃗b) � h−1(αS(b2 0/b2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (46) In terms of Fex and Γsub they read h(αS) = 1 + αS 2π ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' + �αS 2π �2 � ⟨(F(0) ex,1)2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' − 1 2 � ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' �2 + ⟨F(0) ex,2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' − 2πβ0 ⟨F(1) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' − 1 4 �� Γ(1) sub − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , ⟨F(1) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � � + O(α3 S) , (47) D(φ, αS) = 1 + 2 αS 2π � F(0) ex,1 � cor + 2 �αS 2π �2 �� F(0) ex,2 − 2πβ0F(1) ex,1 − 1 4 �� Γ(1) sub − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , F(1) ex,1 �� cor + � (F(0) ex,1)2� cor − ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � F(0) ex,1 � cor − � F(0) ex,1 � cor ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � + O(α3 S) , (48) where, to keep the notation compact, we have defined the azimuthal correlation (f)cor of an operator f as (f)cor = f − ⟨f⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (49) In the operator h of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (42) the scale of αS is b2 0/b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The scale in h can be evolved up to the hard scale M 2 by using the operator Vsub of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (40) and by introducing the operator V of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (8) through the following relation V(b, M) = h(αS(b2 0/b2))Vsub(b, M)h−1(αS(M 2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (50) 11 From here we also obtain the relation between the anomalous dimensions Γt and Γsub in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (8) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Computing the logarithmic derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (50) with respect to M 2 we find Γt(αS) = 1 2h(αS)Γsub(αS)h−1(αS) + β(αS)dh(αS) d ln αS h−1(αS) , (51) where we have introduced the QCD β function β(αS(q2)) = d ln αS(q2) d ln q2 = − ∞ � k=1 βk−1αk S(q2) , (52) with β0 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At O(α2 S) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (51) is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (40) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27] with the identification F(1) t = 2 ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='. We can collect all the results of our discussion by inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (41), (42) and (50) in the third line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (39), and we obtain ⟨ � M| ∆ | � M⟩ = ⟨ � M| V†(b, M)D(φ, αS(b2 0/b2))V(b, M) | � M⟩ , (53) where the IR finite matrix element | � M⟩ can be expressed as | � M⟩ = lim ϵ→0 � h(αS(M 2))eV fin c Z−1 |M⟩ � , (54) and Z−1 |M⟩ = |Mfin⟩ can be obtained from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 For convenience, we report the explicit expression of V fin c in the limit ϵ → 0 V fin c =Cc � − π2 12 �αS(M 2) 2π � + �αS(M 2) 2π �2 � �607 162 − 67 144π2 + π4 72 − 77 36 � CA + � −41 81 + 5 72π2 + 7 18ζ3 � nf − iπ4 6 β0 � + O(α3 S) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (55) In this Section we have discussed how the factors ∆ and � M that appear in the transverse- momentum resummation formalism of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 are related to the soft-radiation contribution Fex(⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The first order resummation coefficients that were presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27] depend on the first-order term Fex,1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the following sections we illustrate the explicit computation of the first- and second-order terms Fex,1 and Fex,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In particular, Fex,2, controls the NNLO contribution to the operator h (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (47)) and, through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (54), it allows us to evaluate the NNLO subtracted amplitude � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 6 To be precise the numerical expression of the two-loop amplitude in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37] is presented by using µIR = m as IR subtraction scale, while in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (54) the operator Z is defined at the IR subtraction scale µIR = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore the implementation of the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (54) requires the evolution of the numerical result presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37] from the scale m to the scale M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We also note that a fully analytic result for the two-loop amplitude in the q¯q → Q ¯Q channel became available recently [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 12 3 Details of the calculation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 The subtracted integrals The evaluation of the operator Fex(⃗b) introduced in the previous section requires the integration of the soft-parton contributions after subtraction of the corresponding contribution of initial- state emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can write this symbolically as Fex(⃗b) = 1 2 � FQ ¯Q − Fcolourless � ≡ 1 2Fsub .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (56) At NLO we just have to consider the emission of one soft gluon, which can be described by the customary tree-level eikonal factorisation formula, after subtraction of initial-state emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The relevant contribution is I(0) g (⃗b) = − � dDk (2π)D−1δ+(k2) ��J(0) g (k) ��2 sub ei⃗b·⃗kT , (57) where the subtracted squared current ���J(0) g (k) ��� 2 sub is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At NNLO we need to consider contributions from: single-gluon emission at one-loop order (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' emission of a soft quark-antiquark pair (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' emission of two soft gluons (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The corresponding integrals read I(1) g (⃗b) = − � dDk (2π)D−1δ+(k2) � J(0)† g (k)J(1) g (k) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � sub ei⃗b·⃗kT , (58) I(0) q¯q (⃗b) = � dDk1 (2π)D−1 dDk2 (2π)D−1δ+(k2 1)δ+(k2 2)I(0) q¯q (k1, k2) �� subei⃗b·(⃗kT 1+kT 2) , (59) I(0) gg (⃗b) = 1 2 � dDk1 (2π)D−1 dDk2 (2π)D−1δ+(k2 1)δ+(k2 2)W(0) gg (k1, k2) �� subei⃗b·(⃗kT 1+kT 2) , (60) where the soft factors � J(0)† g (k)J(1) g (k) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , I(0) q¯q (k1, k2) and W(0) gg (k1, k2) are explicitly given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (104), (173) and (196), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57), the label “sub” in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (58)–(60) denotes the subtraction procedure that removes the initial-state emission contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Details of this procedure are given in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' All the integrals are computed by using dimensional regularisation with D = 4 − 2ϵ dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The relations with the perturbative 13 coefficients Fex,1 and Fex,2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (31) are 2 × Sϵ 8π2 �b2 b2 0 �ϵ Fex,1(φ) = I(0) g (⃗b) , (61) 2 × S2 ϵ (8π2)2 �b2 b2 0 �2ϵ Fex,2(φ) = I(1) g (⃗b) + I(0) q¯q (⃗b) + I(0) gg (⃗b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (62) We observe that the expression for Fex,2 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (35)) has up to double poles in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This is however not the case for the different contributions defined here: in particular terms 1/ϵ3 are separately present in I(1) g and I(0) gg , but they cancel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' All the integrals presented so far have been written in b-space, that is, in the space of the impact parameter b, connected to the ordinary space (qT-space) by a Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The transformation from a b-space integral in a qT-space one is hence obtained with the formal substitution δ(D−2) � ⃗qT + ⃗kT1 + ⃗kT2 � −→ ei⃗b·(⃗kT 1+⃗kT 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (63) For the computation of the function h in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (47), azimuthal averages are required, which are denoted as ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='. We compute the D-dimensional azimuthal average of a function F(φ) as ⟨F(φ)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 1 B � 1 2, 1 2 − ϵ � � 1 −1 d cos φ (1 − cos2 φ)− 1 2 −ϵF(φ) , (64) where B(x, y) is the Euler beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that, when considering the azimuthally averaged result, the step from b-space to qT-space is straightforward, and is determined by the overall dependence on b of the integral under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Given a b-space function I(⃗b) we introduce the corresponding qT-space transform as ˜I(⃗qT) = 1 (2π)D−2 � dD−2⃗b I(⃗b) e−i⃗b·⃗qT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (65) Performing the azimuthal average in qT space of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (65) we obtain ⟨˜I(⃗qT)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 1 (2π)D−2 � dD−2⃗b ⟨I(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' e−i⃗b·⃗qT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (66) If the b-space function has the factorised form I(⃗b) = f(b2)¯I(ˆb) , (67) where the function ¯I(ˆb) depends only on the azimuthal angle of ⃗b, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (66) gives ⟨˜I(⃗qT)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = ⟨¯I(ˆb)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 (2π)D−2 � dD−2⃗b f(b2) e−i⃗b·⃗qT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (68) By inspection of the structure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (31), we see that the soft integrals to be evaluated at 14 NnLO are of the form I(⃗b) = b2nϵ ¯I(ˆb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (69) We can then use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (68) with f(b2) = b2nϵ to obtain ⟨˜I(⃗qT)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 4nϵπ−1+ϵ � 1 q2 T �1+(n−1)ϵ Γ(1 + (n − 1)ϵ) Γ(−nϵ) ⟨¯I(ˆb)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (70) We conclude this section by specifying the kinematical variables for the Born level process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The polar angle θ is defined as the angle between the beam axis and the momentum of the final-state heavy quark in the centre-of-mass frame of the colliding partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The variable β is defined as β = √ 1 − τ , (71) with 0 < τ < 1 τ = 4m2 s , (72) where s = (p1 + p2)2 = (p3 + p4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We also introduce the following auxiliary variables, that will be useful in order to write our partial results in a more compact form B = p2 T,3 m2 = p2 T,4 m2 = β2 1 − β2 sin2 θ , (73) r = √ 1 + B , (74) v = � 1 − � 2m2 s − 2m2 �2 = 2β 1 + β2 , (75) c = 1 − β 1 + β , (76) cT = 1 − √ 1 − r2 τ 1 + √ 1 − r2 τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (77) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 Single gluon emission at tree level The evaluation of the soft-gluon contributions at NLO has already been performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the following, we describe the strategy adopted to carry out the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that, for the extension to NNLO, we need to obtain the NLO result up to O(ϵ) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (47)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The integral I(0) g (⃗b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57) is obtained from the subtracted current ���J(0) g (k) ��� 2 sub, which is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We start from the tree-level eikonal current J(0) g (k) describing the emission of a soft gluon with momentum k from the c(p1)¯c(p2) → Q(p3) ¯Q(p4) Born level amplitude J(0) g,µ(k) = 4 � i=1 Ti piµ (pi · k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (78) 15 The corresponding factorisation formula reads7 |M(0) c¯c→Q ¯Qg|2 ∼ (g0µϵ 0)2⟨M(0) c¯c→Q ¯Q|J(0)† g,µ (k) dµν(k) J(0) g,ν(k)|M(0) c¯c→Q ¯Q⟩ = −(g0µϵ 0)2⟨M(0) c¯c→Q ¯Q| ��J(0) g (k) ��2 |M(0) c¯c→Q ¯Q⟩ , (79) where g0 is the bare coupling (g2 0 = 4πα0), dµν(k) = −gµν + gauge terms (80) is the spin-polarisation tensor of the soft gluon and the gauge terms give vanishing contribution due to current conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The square of the current can be written in the form ��J(0) g (k) ��2 = � j=3,4 � p2 j (pj · k)2T2 j + � i=1,2 2pi · pj (pi · k)(pj · k)Ti · Tj � + 2p3 · p4 (p3 · k)(p4 · k)T3 · T4 + 2p1 · p2 (p1 · k)(p2 · k)T1 · T2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (81) From this expression we need to subtract the initial-state contribution, which is relevant for the production of a colourless system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It reads ��J(0) g (k) ��2 colourless = 2p1 · p2 (p1 · k)(p2 · k)T1 · T2 = − (p1 · p2) (p1 · k)(p2 · k) � T2 1 + T2 2 � = − � (p1 · p2) (p1 · k)(p1 + p2) · k + (p1 · p2) (p2 · k)(p1 + p2) · k � � T2 1 + T2 2 � , (82) where we used the colour conservation relation T1 + T2 = 0 for the corresponding production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The subtracted squared current that appears in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57) is defined as ��J(0) g (k) ��2 sub = ��J(0) g (k) ��2 − ��J(0) g (k) ��2 colourless = � j=3,4 � m2 (pj · k)2T2 j + 2 � i=1,2 �pi · pj pj · k − p1 · p2 (p1 + p2)k � Ti · Tj pi · k � + 2p3 · p4 (p3 · k)(p4 · k)T3 · T4 , (83) We emphasise that each of the three colour contributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (83) is separately collinear safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (83) the evaluation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57) is reduced to the computation of the following 7 Here and in the following the unrenormalised scattering amplitudes are denoted as Mu = α0µ2ϵ 0 � M(0) + M(1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='. � where M(0) is the tree-level contribution, M(1) is the one-loop virtual correction and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 16 integrals Ijj(⃗b) = � dDk δ+(k2) m2 (pj · k)2 ei⃗b·⃗kT , (84) Iij(⃗b) = � dDk δ+(k2) 1 pi · k �pi · pj pj · k − p1 · p2 (p1 + p2) · k � ei⃗b·⃗kT , (85) I34(⃗b) = � dDk δ+(k2) p3 · p4 (p3 · k)(p4 · k) ei⃗b·⃗kT , (86) where i = 1, 2 labels an initial-state parton, while j = 3, 4 labels one of the final-state massive particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In terms of these definitions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57) reads I(0) g (⃗b) = − 1 (2π)D−1 � � j=3,4 � Ijj(⃗b) T2 j + 2 � i=1,2 Iij(⃗b) Ti · Tj � + 2 I34(⃗b) T3 · T4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (87) The simplest contribution is the one where only a massive final-state particle is involved, Ijj defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To perform its computation, we introduce light-cone coordinates p± = p0 ± pz √ 2 , pµkµ = p+k− + p−k+ − ⃗pT · ⃗kT , (88) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (84) becomes Ijj(⃗b) = � dk+ dk− dD−2⃗kT δ(2k+k− − ⃗k2 T) m2 ei⃗b·⃗kT � pj,+k− + pj,−k+ − ⃗pj,T · ⃗kT �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (89) The delta function can now be used to perform the integral over k+ Ijj(⃗b) = � dk− dD−2⃗kT 2 m2 k− ei⃗b·⃗kT � pj,−k2 T + 2pj,+k2 − − 2k−⃗pj,T · ⃗kT �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (90) The leftover angular integral can be simplified by removing the angular dependence from the denominator with an appropriate shift of the ⃗kT variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We obtain Ijj(⃗b) = (2π)1−ϵbϵm2 � dk− 2k− p2 j,− e i k− pj,− ⃗b·⃗pj,T � dkT k1−ϵ T J−ϵ(bkT) � k2 T + m2 k2 − p2 j,− �2 , (91) where Jn(x) is the Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Including the azimuthal average, we are now left with a three-fold integral that can be performed via standard techniques to all orders in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' A similar strategy can be followed to compute Iij(⃗b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (85), but this requires some additional care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The term Iij(⃗b) involves the contribution of the initial-state emitter that is massless, and this may lead to a collinear singularity in the region pi · k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The collinear singularity is absent in the complete integrand of Iij(⃗b), but it is present in the two separate 17 contributions that correspond to the two terms in the square bracket of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To apply the same integration procedure used for Ijj(⃗b), the two contributions must be computed separately and, therefore, a regulator for the collinear singularity needs be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We thus multiply the integrand by the factor [45, 46] �pi · k m2 �2λ , (92) where λ is a small, positive coefficient and the mass scale m has been chosen equal to the heavy-quark mass, but it is in principle arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' With the inclusion of this additional factor, the collinear singularity is regularised, and after integration it leads to poles in λ, which cancel with each other once the results from the two contributions are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The divergence that appears in the intermediate steps of the evaluation of Iij(⃗b) is just an artifact of the approximation used to compute the small-qT behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Similar divergences arise in SCET computations and are usually called rapidity divergences [47–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' However, we point out that the term Iij(⃗b) and our entire soft contributions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57)–(62) have no collinear or rapidity divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In our computation the collinear singularities from initial- state emission can only appear due to technical reasons, since for practical purposes we split integrable integrands in several non-integrable terms that are evaluated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now focus on the final integral, I34(⃗b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It can be computed from Ijj(⃗b) by using Feynman parametrisation I34(⃗b) = � 1 0 dx � dDk δ+(k2) p3 · p4 (p(x) · k)2 ei⃗b·⃗kT = p3 · p4 m2 � 1 0 dx Ijj(⃗b) �� pj=p(x) , (93) where we introduced the auxiliary momentum pµ(x) = xpµ 3 + (1 − x)pµ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (94) The integration over the Feynman parameter can be easily performed in terms of multiple polylogarithms after the azimuthal average and an expansion to O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now present our results for the azimuthally averaged integrals ⟨Ijj(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=', ⟨Iij(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' and ⟨I34(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='. When the all order result is available, we show both the expression before and after the ϵ expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We find ⟨Ijj(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵ Γ(1 − ϵ) �b2 4 �ϵ � −1 ϵ 2F1 (1, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −B) � =π1−ϵ Γ(1 − ϵ) �b2 4 �ϵ � − 1 ϵ − ln (1 + B) + ϵ Li2 (−B) + O(ϵ2) � , (95) ⟨Iij(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = lim λ→0 1 2π1−ϵ �b2 4 �ϵ Γ( λ 2 − ϵ)Γ( λ 2) � �pi · pj m �λ 2F1 �λ 2, λ 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −B � − �p1 · p2 √s �λ 2F1 �λ 2, λ 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −1 s � � 18 =π1−ϵΓ(1 − ϵ) 2 �b2 4 �ϵ � − 2 ϵ ln �2 pi · pj √s m � + Li2 (−B) + ϵLi3 (−B) + O(ϵ2) � , (96) ⟨I34(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵ Γ(1 − ϵ) �b2 4 �ϵ 1 + β2 2β � −1 ϵL0(β) − L1(β, θ) + ϵ P2(β, θ) + O(ϵ2) � , (97) where the coefficient λ is the one introduced with the collinear regulator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (92) and the functions Ln(β, θ), Pn(β, θ) are defined as Ln(β, θ) = (p3 · p4) 2β 1 + β2 � 1 0 dx p(x)2 lnn � 1 + ⃗pT(x)2 p(x)2 � → � β −β dz 1 − z2 lnn �1 − z2 cos θ 1 − z2 � , (98) Pn(β) = (p3 · p4) 2β 1 + β2 � 1 0 dx p(x)2Lin � −⃗pT(x)2 p(x)2 � → � β −β dz 1 − z2Lin �z2 sin2 θ z2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (99) The momentum pµ(x) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (94), and in the last step in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (98), (99) we have used ⃗p3 = −⃗p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The explicit expressions of the functions L0(β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' L1(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' θ) and P2(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' θ) read L0(β) = ln �1 + β 1 − β � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (100) L1(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' θ) = ln �1 + β 1 − β � ln (1 + B) − Li2 � 4β (1 + β)2 � − 1 2 ln2 �1 + β 1 − β � + Li2(1 − c cT) + Li2 � 1 − c cT � + 1 2 ln2 cT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (101) P2(β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' θ) = G � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� + G � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� + G � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� + G � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − 2 ln(1 − β)G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − 2 ln(1 − β)G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� − ln �sin2(θ) 4 � G � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� + ln �sin2(θ) 4 � G � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' β − 1 2β ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' sin2 �θ 2 �� + 2 ln(1 − β) ln �sin2(θ) 4 � ln � cos2 �θ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (102) with c and cT defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (76) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (77) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The function P2(β, θ) is expressed in terms of multiple polylogarithmic functions G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We note that the same kind of integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (98), (99) will also appear at a later stage in the computation of the double gluon emission contribution (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5): in this case though we will need Ln and Pn up to n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 Single gluon emission at one loop We now focus on the emission of a soft gluon at one loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The corresponding factorisation formula reads [52, 53] ⟨M(0) c¯c→Q ¯Qg|M(1) c¯c→Q ¯Qg⟩ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' ≃ − (g0µϵ 0)2 � ⟨M(0) c¯c→Q ¯Q|J(0) g (k) · J(0) g (k)|M(1) c¯c→Q ¯Q⟩ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � + (g0µϵ 0)4 � ⟨M(0) c¯c→Q ¯Q|J(0)† g (k) · J(1) g (k)|M(0) c¯c→Q ¯Q⟩ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � , (103) where J(1) g (k) is the one-loop correction to the soft-gluon current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The first contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (103) factorises the tree-level squared current from the interference between the c¯c → Q ¯Q Born and one-loop amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Such term does not lead to new soft contributions to Fex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The product of the tree and loop soft currents can be written as J(0)† g (k) · J(1) g (k) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =2CA � i̸=j � (pi · pj) (pi · k)(pj · k) − m2 (pj · k)2 � Rij Ti · Tj − 4π � i,j,k ′ pi · pj (pi · k)(pj · k)Iikf abcT a i T b kT c j , (104) where �′ i,j,k denotes the sum over distinct indices (i ̸= j, j ̸= k, k ̸= i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The expansion in ϵ of the Rij, Iij functions can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [53] and, in the case of two massive emitters, a simplified expression has been presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Since we limit ourselves to considering heavy-quark production, we only need to evaluate the contribution proportional to Rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In fact Iij is proportional to the three-partons correlator f abcT a i T b kT c j that vanishes when acting on the tree-level amplitudes of the process8 c¯c → Q ¯Q [56– 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By using colour conservation, we can apply the same procedure employed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 to isolate the initial-state radiation and thus replace the first contribution on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (104) with � J(0) g (k)·J(1) g (k) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � sub ≡ =2CA � i=1,2 j=3,4 �� 2(pi · pj) (pi · k)(pj · k) − m2 (pj · k)2 � Rij − 2(pi · pj) (pi · k)(p1 + p2) · kR12 � Ti · Tj + 2CA � 2(p3 · p4) (p3 · k)(p4 · k) − m2 (p3 · k)2 − m2 (p4 · k)2 � R34 T3 · T4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='. (105) where the dots stand for the contributions proportional to Iij that will eventually vanish when evaluated onto tree-level amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now need to expand Rij in powers of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to 8 Note that this is not generally the case for processes in which the heavy-quark pair is accompanied by particles with complex couplings (see the Note Added in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 20 match the normalisation used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [53] we write Rij = � (pi · pj) 2(pi · k)(pj · k) �ϵ Rij , (106) where Rij = ∞ � n=−2 R(n) ij ϵn , (107) and R(n) ij with n ≤ 2 are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The integral of the one-loop squared current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (58) can be organised into a massless-massive and a massive-massive contribution based on their colour factor I(1) g (⃗b) = − 2CA (2π)D−1 � � � � � � j=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 i=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 I(1) ij (⃗b) Ti · Tj + I(1) 34 (⃗b) T3 · T4 � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (108) where the massless-massive contribution reads I(1) ij (⃗b) = � dDk δ+(k2) �� 2(pi · pj) (pi · k)(pj · k) − m2 (pj · k)2 � Rij − 2(pi · pj) (pi · k)(p1 + p2) · kR12 � ei⃗b·⃗kT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (109) while the massive-massive contribution is I(1) 34 (⃗b) = � dDk δ+(k2) � 2(p3 · p4) (p3 · k)(p4 · k) − m2 (p3 · k)2 − m2 (p4 · k)2 � R34 ei⃗b·⃗kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (110) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 Massive-massless contribution: I(1) ij By inspecting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (109) we can identify three different contributions proportional to (pi·pj) (pi·k)(pj·k), m2 (pj·k)2 and (pi·pj) (pi·k)(p1+p2)·k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We therefore define the three auxiliary integrals I(1) ij,ij(⃗b) = � dDk δ+(k2) (pi · pj) (pi · k)(pj · k) Rij � (pi · pj) 2(pi · k)(pj · k) �ϵ ei⃗b·⃗kT , (111) I(1) ij,jj(⃗b) = � dDk δ+(k2) m2 (pj · k)2 Rij � (pi · pj) 2(pi · k)(pj · k) �ϵ ei⃗b·⃗kT , (112) I(1) ij,i(12)(⃗b) = � dDk δ+(k2) (pi · pj) (pi · k)(p1 + p2) · k R12 � (p1 · p2) 2(p1 · k)(p2 · k) �ϵ ei⃗b·⃗kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (113) In terms of these auxiliary integrals, I(1) ij reads I(1) ij (⃗b) = 2 I(1) ij,ij(⃗b) − I(1) ij,jj(⃗b) − 2I(1) ij,i(12)(⃗b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (114) We start from I(1) ij,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In this case we have a collinear singularity associated with the radiation from the initial-state massless particle which is due to the factor (pi · k) in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 21 To take care of it, we can introduce a λ regulator similarly to what was done in the case of the NLO contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (92) �pi · k m2 �2λ , (115) with λ being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The collinear singularity will then be translated into poles in λ, which will cancel with analogous poles in the massless-massless contribution I(1) ij,i(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' From this stage, we can closely follow the procedure used in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 to perform the integral over the phase space of the emitted gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We obtain I(1) ij,ij(⃗b) =(m2)−3λπ1−ϵ(pi · pj)2λ �b2 4 �2ϵ−λ Γ(−2ϵ + λ) 2Γ(1 + ϵ − λ) � ∞ 0 dw Rij(w) w(1 + w)1+ϵ × � � �wλ + � �2F1 � �−2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' �⃗b · ⃗pT,j b m �2 w � � − 1 − 4ϵi⃗b · ⃗pT,j b m ×√w Γ( 1 2 − 2ϵ)Γ(1 + ϵ) Γ(1 − 2ϵ)Γ( 1 2 + ϵ) 2F1 � �1 2 − 2ϵ, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' �⃗b · ⃗pT,j b m �2 w � � � � � � � , (116) which after azimuthal average becomes ⟨I(1) ij,ij(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = (m2)−3λπ1−ϵ(pi · pj)2λ �b2 4 �2ϵ−λ Γ(−2ϵ + λ) 2Γ(1 + ϵ − λ) � ∞ 0 dw Rij(w) w(1 + w)1+ϵ × � wλ + Re {[2F1 (−2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' wB) − 1] (1 + i cot(πϵ))} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (117) Expanding in ϵ the expression in the curly bracket of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (117) we obtain Re {[2F1 (−2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' wB) − 1] (1 + i cot(πϵ))} = − 2ϵ π Im [Li2 (wB)] + O(ϵ2) =2ϵ ln (wB) θ (wB − 1) + O(ϵ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (118) In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (116) and (117) we have defined the adimensional variable w as w = m2 p2 j,− k2 − k2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (119) The kinematical invariants can be written in terms of w as (pi · k) = (pi · pj) m kT √w , (120) (pj · k) = m 2 kT � 1 √w + √w � , (121) while the explicit expression of the coefficients R(n) ij presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [53] in terms of w reads R(−2) ij = −1 2 , (122) 22 R(−1) ij = 0 , (123) R(0) ij = 1 24 � 5π2 − 6w ln2 � w w + 1 �� , (124) R(1) ij = 1 12 � 6(w − 1)Li3 � w w + 1 � + 6(w − 1)Li2 � 1 w + 1 � ln � w w + 1 � + 2(7 − 3w)ζ3 + ln � w w + 1 � � π2(6w + 1) − 3(w − 1) ln � w w + 1 � ln(w + 1) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (125) Our task is now to integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (117) with the expansion of R defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (122)–(125).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The final result reads ⟨I(1) ij,ij(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =(m2)−3λπ1−ϵ(pi · pj)2λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2ϵ−λ Γ(−2ϵ + λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2Γ(1 + ϵ − λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2ϵ2 + 5π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 + 7ζ3ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ln2(B) − π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 6Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 6Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) + ln3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln3(B) + 3 ln(B) ln2(B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 6 ln2(B) ln(B + 1) − 2π2 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2π2 ln(B) + π2 ln(B + 1) − 6ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ζ3 ln(B) + ζ3 ln(B + 1) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4π2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B) − Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) − 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) + S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ln4(B) + 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ln4(B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ln(B + 1) ln3(B) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ln3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) − 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ln2(B + 1) ln2(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 73 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24π2 ln2(B) + 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 π2 ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 π2 ln2(B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2B + 1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='12 ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2B + 1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 π2 ln(B + 1) ln(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3π2 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) − 23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='240π4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ O (λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (126) with B defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (73) and S2,2 being the Nielsen generalised polylogarithm function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 23 We now consider the integral I(1) ij,jj(⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The only difference with I(1) ij,ij(⃗b) consists in the replacement (pi·pj) (pi·k)(pj·k) → m2 (pj·k)2, which in terms of our integration variable w implies 2 k2 T(1 + w) −→ 2 k2 T(1 + w) 2w (1 + w) , (127) that is, we simply need to multiply the integrand by a factor 2w/(1 + w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In addition, the presence of only final-state emitters in the integrand implies that we can set λ = 0 throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can use this method to obtain from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (116) an expression for I(1) ij,jj as an integral over w I(1) ij,jj(⃗b) =π1−ϵ �b2 4 �2ϵ Γ(−2ϵ) Γ(1 + ϵ) � ∞ 0 Rij(w) (1 + w)2+ϵ �� 2F1 � −2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' c2 jbw � − 4ϵicjb √w Γ( 1 2 − 2ϵ)Γ(1 + ϵ) Γ(1 − 2ϵ)Γ( 1 2 + ϵ) 2F1 �1 2 − 2ϵ, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' c2 jbw � �� , (128) where we have defined cjb = ⃗b · ⃗pT,j b m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (129) The azimuthally averaged equivalent of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (128) can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (117) ⟨I(1) ij,jj(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵ �b2 4 �2ϵ Γ(−2ϵ) Γ(1 + ϵ) � ∞ 0 dw Rij(w) (1 + w)2+ϵ × {1 + Re {[2F1 (−2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' wB) − 1] (1 + i cot(πϵ))}} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (130) We obtain ⟨I(1) ij,jj(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2ϵ Γ(−2ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Γ(1 + ϵ) × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ (2 ln(B + 1) − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln2(B) + ln2(B + 1) − 2 ln(B + 1) − ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 + 13π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 3Li3(−B) + 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2(B + 1) − 2 ln(B + 1) − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) − Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(B + 1) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ln4(B) + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='8 ln4(B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ln3(B + 1) ln(B) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ln3(B + 1) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ln2(B + 1) ln2(B) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ln(B + 1) ln2(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='12π2 ln2(B) − 3 ln2(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln2(B + 1) ln(B) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='12π2 ln2(B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ln2(B + 1) + 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='12π2 ln(B + 1) + 3 ln(B + 1) − 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 − ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 + ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 − 5ζ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (131) We finally consider I(1) ij,i(12)(⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To obtain a one-fold integral representation of this contribu- tion we can again take advantage of the result for I(1) ij,ij, identifying pj = p1+p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This also means setting ⃗b · ⃗pT,j = 0 due to the absence of a transverse component in p1 + p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Nevertheless, the identification between the results is not completely straightforward because of the additional difference in the integrand � (pi · pj) (pi · k)(pj · k) �ϵ −→ � (p1 · p2) (p1 · k)(p2 · k) �ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (132) However, we can notice that � (pi · pj) (pi · k)(pj · k) �ϵ = � 2 k2 T �ϵ (1 + w)−ϵ (133) while � (p1 · p2) (p1 · k)(p2 · k) �ϵ = � (pi · pj) (pi · k)((pj · k) − (pi · k)) �ϵ = � 2 k2 T �ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (134) Thus we can take care of this additional difference by adding a factor (1 + w)ϵ to the one-fold representation of I(1) ij,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore, we have ⟨I(1) ij,i(12)(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = �p1 · p2 2m4 �λ π1−ϵ �b2 4 �2ϵ−λ Γ(−2ϵ + λ) 2Γ(1 + ϵ − λ) � ∞ 0 dw w−1+λ (1 + w)R12(w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (135) The expression for R12(w) can be obtained taking the massless limit of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (122)–(125), which leads to the simplified expressions R(−2) 12 = −1 2 (136) R(−1) 12 = 0 (137) R(0) 12 = 5 4ζ2 (138) R(1) 12 = 7 6ζ3 , (139) and therefore straightforwardly � ∞ 0 dw w−1+λ (1 + w) R12(w) = 1 λ � − 1 2ϵ2 + 5 4ζ2 + ϵ7 6ζ3 + O(ϵ2) � + O(λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (140) By comparing with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (126) we see that the λ → 0 singular terms cancel out as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By 25 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (114) we can write the final result for I(1) ij (⃗b) as ⟨I(1) ij (⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2ϵ Γ(−2ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2Γ(1 + ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − 2 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2(pi · pj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='√sm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−1 − π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 + 2 ln(B + 1) − ln2(B) − 2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6π2 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2(pi · pj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='√sm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 13π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 6Li2(−B) + 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ln3(B) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ln3(B + 1) − ln(B + 1) ln2(B) − 2 ln2(B) + ln2(B + 1) ln(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln2(B + 1) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3π2 ln(B) − 2 ln(B + 1) − 3ζ3 − 2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='(ln(B + 1) + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ζ3 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2(pi · pj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='√sm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2ζ3 ln(B) − Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 6Li2(−B) − 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 6Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− ln2(B) − ln2(B + 1) + 2 ln(B + 1) + π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2Li4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 � − 1 B � − 2Li3 � − 1 B � ln(B) + 2Li3 � 1 B + 1 � ln(B + 1) − 1 4 ln4(B) − 1 4 ln4(B + 1) + 1 3 ln3(B) + 2 3 ln3(B + 1) ln(B) + 1 12π2 ln2(B) − 1 2 ln2(B + 1) ln2(B) + ln(B + 1) ln2(B) + 3 ln2(B) − 2 ln2(B + 1) + 1 3π2 ln(B) − 1 2π2 ln(B + 1) − 13ζ3 3 − 29π4 360 + π2 12 + 4 � + O � ϵ2� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (141) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 Massive-massive contribution: I(1) 34 Let us now consider the purely massive contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' I(1) 34 (⃗b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (110), which can also be written as I(1) 34 = � dDk δ+(k2) � 2(p3 · p4) (p3 · k)(p4 · k) − m2 (p3 · k)2 − m2 (p4 · k)2 � � (p3 · p4) 2(p3 · k)(p4 · k) �ϵ R34ei⃗b·⃗kT , (142) where the functions R34 have been presented for the first time in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [53], while in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [54] a simplified expression has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The coefficients R(n) 34 read R(−2) 34 =1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (143) R(−1) 34 = ln(v+) − v− v � ln �α3 v+ � + ln �α4 v+ �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (144) 26 R(0) 34 = 1 2 ln2(v+) + 1 v � 1 (d3 + d4) � (α3v+ − α4v−) ln2 �α3 v+ � + � α4v+ − α3v− � ln2 �α4 v+ �� + � ln �α3 v+ � + ln �α4 v+ ��� v+ ln(v+) − ln(v) � − Li2 �v− v+ �� + ζ2 �7 v − 19 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='(145) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='R(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='34 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='d3 + d4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − (d3 + d4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 �α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 �α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ζ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 7ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='α4v+ − α3v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 �α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='α3v+ − α4v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 7ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(v+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�3 + v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(v+) − ln(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 9v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln3 �α3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln3 �α4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�v− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 + 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln(v+) + 12ζ2 ln(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 ln3(v+) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ζ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (146) In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (143)–(146) we used the same notation of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [53, 54], introducing the variables α3 = m2(p4 · k) (p3 · k)(p3 · p4) , (147) α4 = m2(p3 · k) (p4 · k)(p3 · p4) , (148) v± = 1 ± v 2 , (149) d3 = 1 − 2α3 , (150) d4 = 1 − 2α4 , (151) with v defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We first discuss the contributions of R(−2) 34 and R(−1) 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Both these coefficients are independent of the gluon momentum k (note that α3α4 = m4/(p3 · p4)2), and, therefore, the corresponding integrals can be evaluated with the same method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We start from the generalised Feynman 27 parametrisation 1 AmBn = Γ(m + n) Γ(m)Γ(n) � 1 0 dx xm−1(1 − x)n−1 (xA + (1 − x)B)m+n , (152) to write the denominators in terms of a single scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For the term proportional to (p3 · p4)/(p3 · k p4 · k) in I(1) 34 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' dropping overall constant terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' the relevant integral is � dDk δ+(k2) ei⃗b·⃗kT (p3 · k)1+ϵ(p4 · k)1+ϵ = Γ(2 + 2ϵ) Γ2(1 + ϵ) � 1 0 dx � dDk δ+(k2)(1 − x)ϵxϵ ei⃗b·⃗kT ((1 − x)(p4 · k) + x(p3 · k))2+2ϵ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (153) while when considering the term proportional to m2/(pj · k)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' with j = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 4 we need to evaluate � dDk δ+(k2) ei⃗b·⃗kT (pj · k)2+ϵ(pi · k)ϵ = Γ(2 + 2ϵ) Γ(2 + ϵ)Γ(ϵ) � 1 0 dx � dDk δ+(k2)(1 − x)−1+ϵx1+ϵ ei⃗b·⃗kT ((1 − x)(p4 · k) + x(p3 · k))2+2ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (154) We see that both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (153) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (154) depend on the same integral I(1) k (x) = � dDk δ+(k2) ei⃗b·⃗kT (p(x) · k)2+2ϵ , (155) with pµ(x) = x pµ 3 + (1 − x) pµ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (156) This integral can be evaluated with the techniques used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 and we find ⟨I(1) k (x)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = −4−ϵb4ϵπ2−ϵ sin(ϵπ) Γ(−2ϵ) Γ(−ϵ)Γ(2 + 2ϵ)(p2(x))−1−ϵ 2F1 � −2ϵ, 1 + ϵ, 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −p2 T(x) p2(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (157) We are now left with the integration over the Feynman parameter x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It is convenient to expand in ϵ the hypergeometric function 2F1 (−2ϵ, 1 + ϵ, 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −X) =1 + 2 ln(1 + X) ϵ − 4Li2(−X) ϵ2 + 4 3 � ln3(1 + X) + 3 ln(1 + X)Li2(−X) − 9Li3(−x) − 6Li3 � X 1 + X � � ϵ3 + O(ϵ4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (158) By substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (157) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (153), (154) we obtain a sum of integrals that in most cases can be computed in terms of multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The remaining finite integrals are computed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now focus on the contribution of R(0) 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can split R(0) 34 in a part independent on k, 28 R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='const, and one with an explicit k dependence, R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k R(0) 34 = R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='const + R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (159) We define R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='const =1 v �� ln �α3 v+ � + ln �α4 v+ ��� v+ ln(v+) − ln(v) � − Li2 �v− v+ �� + 1 2 ln2(v+) + ζ2 �7 v − 19 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (160) and R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k = 1 (d3 + d4)v � (α3v+ − α4v−) ln2 �α3 v+ � + � α4v+ − α3v− � ln2 �α4 v+ �� ≡ 1 d3 + d4 r(0) 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (161) The contribution of R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='const can be evaluated as those of R(−2) 34 and R(−1) 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The singular part of the contribution of R(0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k can be computed by using the following identity � dDk (k2)−1−ϵf(k) ei⃗b·⃗kT = � dDk (k2)−1−ϵf(k)θ(µ − kT) + O(ϵ0) , (162) µ being an arbitrary mass scale and f(k) an arbitrary function of the momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We consider the integral of the k-dependent part of R(0) 34 I(1,0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k (⃗b) = � dDk δ+(k2) 1 d3 + d4 � 2(p3 · p4) (p3 · k)(p4 · k) − m2 (p3 · k)2 − m2 (p4 · k)2 � (163) × � (p3 · p4) 2(p3 · k)(p4 · k) �ϵ r(0) 34 ei⃗b·⃗kT , up to order 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The reason to pull out the factor 1/(d3 + d4) is because it allows us to use the identity 2(p3 · p4) (p3 · k)(p4 · k) − m2 (p3 · k)2 − m2 (p4 · k)2 = (p3 · p4) (p3 · k)(p4 · k)(d3 + d4) , (164) which makes the integrand considerably simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' If now we extract the pole structure of the integral by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (162), we get I(1,0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k (⃗b) = � dDk δ+(k2) � p3 · p4 (p3 · k)(p4 · k) �1+ϵ r(0) 34 θ(µ2 − k2 0) + O(ϵ0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (165) There is no singularity associated to the angular variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can thus safely set ϵ = 0 in the 29 angular integral, obtaining I(1,0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k (⃗b) = 2π � ∞ 0 dk0 k1+4ϵ 0 θ(µ2 − k2 0) � 1 0 dt � 1 −1 d cos θ t2−2ϵδ(1 − t2) 1 1 − v cos θr(0) 34 + O(ϵ0) , (166) with t = |⃗k|/k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Now we can perform the integral over t by using the delta function, while the (otherwise divergent) integration over k0 is regulated by the cutoff we inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We find for the pole I(1,0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k (⃗b) �� pole = − π 4ϵ � 1 −1 d cos θ 1 1 − v cos θ r(0) 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (167) The integration of the pole part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (145) is thus finally reduced to a one-fold integral that can be computed with standard methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to write r(0) 34 in terms of v, cos θ the following relations are useful α3 = 1 − v cos θ 2 , (168) α4 = 1 − v2 2 1 1 − v cos θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (169) The result for the pole part of this contribution reads ⟨I(1,0) 34;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='k (⃗b) �� pole⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = π1−ϵ �b2 4 �2ϵ Γ(−2ϵ) Γ(ϵ + 1) � 2 − (1 − β2)2 2β2 ln2 �1 − β β + 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (170) The finite part in ϵ of the contribution of R(0) 34 can be integrated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The last contribution to be computed is that from R(1) 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Since it comes with an overall ϵ factor we can directly apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (162) to evaluate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The analytic result is too lengthy to be reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We conclude this subsection discussing the contribution of the one-loop heavy-quark vacuum polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Such term can be inserted in the radiated soft-gluon line, thus leading to an additional virtual contribution to the one-loop soft-gluon current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Then such contribution has to be consistently taken into account through the renormalization procedure, which amounts to the wave function renormalization of the soft-gluon line and the MS renormalization of αS with nf + 1 quark flavours (the nf massless quarks and the heavy quark Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Finally, we can apply the decoupling relation of the heavy quark [37] and introduce the running coupling α (nf) S (µ2 R) that we use throughout this paper (see the comment at the beginning of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To the purpose of computing the soft contributions at small qT, the final result of this entire procedure is equivalent to avoiding the introduction of the heavy-quark vacuum polarization and to directly renormalizing the QCD coupling with nf light-quark flavours as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 Light-quark pair production We start the analysis of the double real contribution by focusing on the process in which a massless soft quark-antiquark pair is radiated c(p1)¯c(p2) → Q(p3) ¯Q(p4) q(k1)¯q(k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (171) The corresponding factorisation formula for the squared matrix element is [59] ���M(0) c¯c→Q ¯Qq¯q ��� 2 ∼ (g0µϵ 0)4⟨M(0) c¯c→Q ¯Q|I(0) q¯q (k1, k2)|M(0) c¯c→Q ¯Q⟩ (172) where the singular contributions are controlled by the soft factor I(0) q¯q (k1, k2) = � J(0) g,µ(k1 + k2) �† Πµν(k1, k2) J(0) g,ν(k1 + k2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (173) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (173) J(0) g,µ is the tree-level soft current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (78) and we have defined the tensor Πµν as: Πµν(k1, k2) = TR (k1 · k2)2 (−gµνk1 · k2 + kµ 1kν 2 + kν 1kµ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (174) The dots in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (173) stand for gauge dependent contributions that are proportional to the total charge of the hard partons, thereby vanishing when evaluated on the c¯c → Q ¯Q matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Our task is now to integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (173) over the phase space of the q¯q pair after subtracting the initial-state contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=', to evaluate the integral I(0) q¯q (⃗b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To perform this calculation, we first integrate over the light-quark momenta k1 and k2 while keeping their total momentum k = k1 + k2 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This procedure will leave us with expressions similar to the ones for the NLO-like contribution already described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 and will be useful in order to organise the final integration over k in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To proceed in this direction, we rewrite the integration of the soft factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (59) in the following way: � dDk1 (2π)D−1 dDk2 (2π)D−1δ+(k2 1)δ+(k2 2)I(0) q¯q (k1, k2) ei⃗b·(⃗kT 1+⃗kT 2) = = � dDk (2π)D−1J(0) g,µ(k)J(0) g,ν(k) F µν(k) ei⃗b·⃗kT , (175) obtained by inserting the identity 1 = � dDk δ(D)(k − k1 − k2) , (176) and by isolating the integral on the soft-quark momenta in the tensor F µν(k), defined as: F µν(k) = 1 (2π)D−1 � dDk1 � dDk2 Πµν(k1, k2)δ+(k1)δ+(k2)δ(D)(k − k1 − k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (177) 31 We now continue with the computation of the tensor F µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Since F µν is a symmetric tensor fulfilling kµF µν = kνF µν = 0 it must take the form F µν(k) = C � −gµν + kµkν k2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (178) The normalisation factor C can be fixed by evaluating the quantity gµνF µν using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (177) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (178) and comparing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (178) we immediately obtain gµνF µν = −C(3 − 2ϵ) , (179) while from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (177) gµνF µν = −TR 2 − 2ϵ (k2)1+ϵΓ( 3 2 − ϵ)161−ϵπ 3 2 −ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (180) We can therefore write C = F(ϵ) (k2)1+ϵ , (181) with F(ϵ) = TR(1 − ϵ) Γ � 5 2 − ϵ � 161−ϵπ 3 2 −ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (182) With the explicit expression for F µν just obtained, the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (175) reads: � dDk (2π)D−1J(0) g,µ(k)J(0) g,ν(k)F µν(k) ei⃗b·⃗kT = − � dDk (2π)D−1 F(ϵ) (k2)1+ϵ 4 � i,j=1 Ti · Tj pi · pj (pi · k)(pj · k) ei⃗b·⃗kT , (183) where the term in F µν proportional to kµkν gives no contribution because of colour conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We observe that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (183) has a similar structure to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (57), the corresponding NLO integral for single soft-gluon emission at tree-level, after the substitution δ+(k2) → F(ϵ) (k2)1+ϵ , (184) which removes the on-shell constraint for the gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can thus apply for the computation a similar strategy as the one already employed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2, when dealing with the NLO-like contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Before performing the final integration over k of the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (183), we need to subtract the initial-state contribution from the soft current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We therefore write the integral I(0) q¯q (⃗b) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (59) as I(0) q¯q (⃗b) = − F(ϵ) � dDk (2π)D−1 1 (k2)1+ϵ � � j=3,4 � m2 (pj · k)2T2 j + 2 � i=1,2 �pi · pj pj · k − p1 · p2 (p1 + p2)k � Ti · Tj pi · k � + 2p3 · p4 (p3 · k)(p4 · k)T3 · T4 � ei⃗b·⃗kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (185) 32 We can split Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (185) in different integrals according to the different colour factors Iq¯q jj (⃗b) = � dDk (k2)1+ϵ m2 (pj · k)2 ei⃗b·⃗kT , (186) Iq¯q ij (⃗b) = � dDk (k2)1+ϵ 1 pi · k �pi · pj pj · k − p1 · p2 (p1 + p2) · k � ei⃗b·⃗kT , (187) Iq¯q 34(⃗b) = � dDk (k2)1+ϵ p3 · p4 (p3 · k)(p4 · k) ei⃗b·⃗kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (188) In terms of these integrals, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (185) reads I(0) q¯q (⃗b) = − F(ϵ) (2π)D−1 � � j=3,4 � Iq¯q jj (⃗b) T2 j + 2 � i=1,2 Iq¯q ij (⃗b) Ti · Tj � + 2Iq¯q 34(⃗b) T3 · T4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (189) The integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (186)–(188) can be evaluated with a similar strategy as to the one used for the integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (84)–(86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The azimuthally averaged results are ⟨Iq¯q jj (⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵΓ(1 − ϵ)Γ(−2ϵ) �b2 4 �2ϵ � −1 ϵ 2F1 (1, −2ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −B) � =π1−ϵΓ(1 − ϵ)Γ(−2ϵ) �b2 4 �2ϵ � − 1 ϵ − 2 ln (1 + B) + ϵ � 2 Li2 (−B) − ln2 (1 + B) � + O(ϵ2) � , (190) ⟨Iq¯q ij (⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =1 2π1−ϵΓ(1 − ϵ)Γ(−2ϵ) �b2 4 �2ϵ Γ( λ 2 − 2ϵ)Γ( λ 2) Γ(1 − 2ϵ) × 2 ��pi · pj m2 �λ 2F1 �λ 2, λ 2 − 2ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' −B � − �p1 · p2 √s �λ 2F1 �λ 2, λ 2 − 2ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0 �� =1 2π1−ϵΓ(1 − ϵ)Γ(−2ϵ) �b2 4 �2ϵ � − 2 ϵ ln �2 pi · pj m √s � + 2Li2 (−B) + ϵ 3 � ln3 (1 + B) + 6 ln (1 + B) Li2 (−B) − 6 Li3 � B B + 1 �� + O(ϵ2) � , (191) ⟨Iq¯q 34(⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵΓ(1 − ϵ)Γ(−2ϵ) �b2 4 �2ϵ 1 + β2 2β � −1 ϵL0 − 2L1 + ϵ(2P2 − L2) + O(ϵ2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (192) The functions Ln and Pn have been defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (98) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (99), respectively, while their explicit expressions are reported in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (100)–(102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the present case we also need the 33 function L2(β, θ), which reads L2(β, θ) = 2(G(1, −1, − sec(θ), β) + G(1, −1, sec(θ), β) + G(1, 1, − sec(θ), β) + G(1, 1, sec(θ), β) + G(1, − sec(θ), −1, β) + G(1, − sec(θ), 1, β) − G(1, 1, 1, β) − G(1, − sec(θ), − sec(θ), β) + G(1, sec(θ), −1, β) + G(1, sec(θ), 1, β) − G(1, 1, −1, β) − G(1, sec(θ), − sec(θ), β) − G(1, sec(θ), sec(θ), β) − G(1, − sec(θ), sec(θ), β) − G(1, −1, −1, β) − G(1, −1, 1, β)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (193) Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (191), the expression for ⟨Iq¯q ij (⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' before the ϵ-expansion depends on the regularisation parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 the integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (187) needs to be carried out separately for the two terms, by using the regulator factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can then perform the limit λ → 0 in the expanded result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 Double gluon emission We finally consider the contribution due to the emission of a soft-gluon pair, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' we consider the process c(p1)¯c(p2) → Q(p3) ¯Q(p4)g(k1)g(k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (194) In the limit in which the two gluons become soft the singular behaviour is controlled by the double-soft current J(0)µν gg (k1, k2) [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The general expression of the squared soft current reads � J(0)a1a2 gg,µν (k1, k2) �† dσµ(k1)dρν(k2)J(0)a1a2 gg,σρ (k1, k2) = 1 2 � J(0) g 2(k1), J(0) g 2(k2) � + W(0) gg (k1, k2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (195) where the purely non-abelian two-parton correlations are controlled by the function W(0) gg (k1, k2), which is defined as W(0) gg (k1, k2) = −CA n � i,j=1 Ti · Tj Sij(k1, k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (196) The dots in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (195) stand for gauge-dependent terms proportional to the total colour charge of the hard partons and, thus, give a vanishing contribution when evaluated on the c¯c → Q ¯Q matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The soft factor can be separated into massless and massive contributions Sij(k1, k2) = Sm=0 ij (k1, k2) + � m2 i Sm̸=0 ij (k1, k2) + m2 j Sm̸=0 ji (k1, k2) � , (197) 34 where mi(mj) = 0 for i(j) = 1, 2 and mi(mj) = m for i(j) = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The massless contribution reads [59] Sm=0 ij (k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' k2) = (1 − ϵ) (k1 · k2)2 pi · k1 pj · k2 + pi · k2 pj · k1 pi · (k1 + k2) pj · (k1 + k2) − (pi · pj)2 2 pi · k1 pj · k2 pi · k2 pj · k1 � 2 − pi · k1 pj · k2 + pi · k2 pj · k1 pi · (k1 + k2) pj · (k1 + k2) � + pi · pj 2 k1 · k2 � 2 pi · k1 pj · k2 + 2 pj · k1 pi · k2 − 1 pi · (k1 + k2) pj · (k1 + k2) × � 4 + (pi · k1 pj · k2 + pi · k2 pj · k1)2 pi · k1 pj · k2 pi · k2 pj · k1 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (198) pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' pj being the momenta of the emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The massive contribution is [60] Sm̸=0 ij (k1, k2) = − 1 4 k1 · k2 pi · k1 pi · k2 + pi · pj pj · (k1 + k2) 2 pi · k1 pj · k2 pi · k2 pj · k1 pi · (k1 + k2) − 1 2 k1 · k2 pi · (k1 + k2) pj · (k1 + k2) � (pj · k1)2 pi · k1 pj · k2 + (pj · k2)2 pi · k2 pj · k1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (199) In the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (195), W(0) gg is the irreducible correlation component of double- soft radiation, while the anticommutator term corresponds to the independent-emission compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have to evaluate the b-space contribution of the squared current in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (195).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Going to b-space, the phase space for double-parton emission factorizes in terms of single-parton fac- tors (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (63)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore the b-space integral of the independent-emission component of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (195) is fully factorized and it leads to the straightforward exponentiation of the tree-level single-emission contribution Fex,1 in Eq (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Consequently the b-space contribution I(0) gg (⃗b) of double soft-gluon emission to Fex,2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (62) is entirely due to the correlation component W(0) gg of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (195).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' More precisely, we have to perform the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (60) where W(0) gg (k1, k2) �� sub is defined from W(0) gg (k1, k2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (196) after the proper subtraction of the contribution from initial-state radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Part of the contribution to I(0) gg (⃗b) is similar to I(0) q¯q (⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The soft term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (173) involves the factor pµ i pν j Πµν (k1, k2) pi · (k1 + k2) pj · (k1 + k2) = TR (k1 · k2)2 −(pi · pj)(k1 · k2) + (pi · k1)(pj · k2) + (pi · k2)(pj · k1) pi · (k1 + k2)pj · (k1 + k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (200) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (198) we have some terms with a similar structure as the ones in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Those are Sm=0 ij (k1, k2) ��� 12 = 4 (k1 · k2) (pi · pj) (pi · k)(pj · k)− (1 − ϵ) (k1 · k2)2 (pi · k1) (pj · k2) + (pi · k2) (pj · k1) pi · (k1 + k2) pj · (k1 + k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (201) 35 Indeed by defining �Πµν(k1, k2) = − 1 (k1 · k2)2 (−4gµν(k1 · k2) + (1 − ϵ)kµ 1kν 2 + (1 − ϵ)kµ 2kν 1) (202) we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (201) as Sm=0 ij (k1, k2) ��� 12 = pµ i pi · (k1 + k2) �Πµν(k1, k2) pν j pj · (k1 + k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (203) Now the integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (203) can be performed exactly in the same way as the integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (173) in the case of the emission of a soft q¯q pair in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4, leading to the same results with an overall multiplicative factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By following the strategy of integrating over k1 and k2 at a fixed value of k = k1 + k2, similarly to what was done in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (177), we isolate the following integral �F µν(k) = 1 (2π)D−1 � dDk1 � dDk2 �Πµν(k1, k2)δ+(k2 1)δ+(k2 2)δ(D)(k − k1 − k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (204) The structure of �F µν(k) must be of the form �F µν(k) = agµν + bkµkν k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (205) The coefficients a and b can be obtained by contracting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (204) with gµν and kµkν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We find �F µν = 1 (k2)1+ϵ 1 Γ � 5 2 − ϵ � 161−ϵπ 3 2 −ϵ �11 − 7ϵ 2 gµν + (1 − ϵ)kµkν k2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (206) Because of current conservation, the second term will give no contribution to the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The first term, on the other hand, is exactly the same we obtained in the computation for the soft-quark pair emission, but with an overall multiplicative factor: −(11 − 7ϵ)/(1 − ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Hence, to compute the integral of the contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (201), we can take the result for the q¯q pair production and perform the formal substitution nfTR −→ −CA 11 − 7ϵ 4(1 − ϵ) , (207) where we also included the Bose factor 1/2 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (60), which is due to the production of two identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can now define a new soft factor, in which we subtract the contribution that can be computed as described above �Sij(k1, k2) = �Sm=0 ij (k1, k2) + � m2 i �Sm̸=0 ij (k1, k2) + m2 j �Sm̸=0 ji (k1, k2) � , (208) 36 where �Sm=0 ij (k1, k2) = Sm=0 ij (k1, k2) − Sm=0 ij (k1, k2) ��� 12 , (209) �Sm̸=0 ij (k1, k2) = Sm̸=0 ij (k1, k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (210) We have ˜Sm=0 ij (k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' k2) = − (pi · pj)2 2(pi · k)(pj · k) � 2 (pi · k1)(pj · k1) + 1 (pi · k1)(pj · k2) � − (pi · pj) 2k2(pi · k)(pj · k) ((pi · k1)(pj · k2) − (pi · k2)(pj · k1))2 (pi · k1)(pj · k2)(pi · k2)(pj · k1) + (pi · pj) k2 2 (pi · k1)(pj · k2) + (1 ↔ 2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (211) ˜Sm̸=0 ij (k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' k2) = (pi · pj) 2(pi · k)2 � 1 (pi · k1)(pj · k1) + 1 (pi · k1)(pj · k2) � − 1 k2(pi · k) 1 (pi · k1) � (pj · k1)2 (pj · k)(pj · k2) − (pi · k1)2 (pi · k)(pi · k2) � + (1 ↔ 2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (212) where we have introduced k = k1 + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now need to subtract the contribution from initial- state radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can use the same technique already used in the previous Sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The sum over the colour configurations can be organised as 4 � i,j=1 ˜Sij(k1, k2)Ti · Tj = � 4 � i,j=1 ˜SijTi · Tj − � − ˜S12(T2 1 + T2 2) �� + � − ˜S12(T2 1 + T2 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (213) The second term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' is the same we would have for a colourless final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The first term is the new contribution to the subtracted current we have to compute and, by using colour conservation, we can rewrite it as � j=3,4 � ˜SjjT2 j + � i=1,2 � 2 ˜Sij − ˜S12 � Ti · Tj � + 2 ˜S34T3 · T4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (214) Hence we need now to compute � dDk1 dDk2 �Sij(k1, k2)δ+(k2 1) δ+(k2 2) , (215) for all the contributions involved in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (214).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This means we have to consider the following combinations of emitters i and j: i and j being the two initial-state massless emitters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' i being an initial-state massless emitter, j a final-state massive emitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' i = j being the same final-state massive emitter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 37 i and j being the two final-state massive emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It is convenient to integrate over the soft-gluon momenta k1 and k2 at fixed kµ = kµ 1 + kµ 2: after that, we are left with only the integration over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' With this goal in mind, we define the shorthand notation � (12) f(k1, k2) ≡ Γ( 1 2 − ϵ) 4ϵπ 1 2 −ϵ � dDk1 dDk2 f(k1, k2) δ+(k2 1) δ+(k2 2) δ(D)(k − k1 − k2) , (216) and we apply it to the functions ˜Sm=0 ij and ˜Sm̸=0 ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To perform this computation, we can first integrate one of the soft-gluon momenta (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' k1) using the delta function δ(D)(k − k1 − k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Afterwards, we can go in the rest frame of k and integrate over the energy component and the modulus of ⃗k2 by using the two remaining delta functions: this way only angular integrals are left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By following these steps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' we obtain � (12) ˜Sm=0 ij =(k2)−1−ϵ(pi · pj) (pi · k)(pj · k) � (1 + ⃗ni · ⃗nj) A+ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − 2 (1 − ⃗ni · ⃗nj) A− 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 + A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 + A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 � ≡(k2)−1−ϵ(pi · pj) (pi · k)(pj · k) fgg ij (⃗ni · ⃗nj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗n2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗n2 j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217) � (12) ˜Sm̸=0 ij =(k2)−1−ϵ (pj · k)2 � (1 − ⃗ni · ⃗nj) A− 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − (1 + ⃗ni · ⃗nj) A+ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − 1 2A+ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−1 + 3A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 − 1 2A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 � ≡(k2)−1−ϵ (pj · k)2 ggg ij (⃗ni · ⃗nj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗n2 i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗n2 j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218) where we defined ⃗ni and ⃗nj as vectors in the centre-of-mass frame of k via pi = Ei(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗ni) and pj = Ej(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗nj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In terms of invariants, we have ⃗n2 i = 1 − k2m2 i (pi · k)2 , (219) ⃗n2 j = 1 − k2m2 j (pj · k)2 , (220) ⃗ni · ⃗nj = 1 − k2(pi · pj) (pi · k)(pj · k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (221) The functions fgg ij , ggg ij are defined as the sum of the angular integrals (with appropriate multi- plicative factors) for the massless and massive case respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The angular integrals A± k,l are defined as A± k,l = � π 0 dθ � π 0 dφ sinD−3 θ sinD−4 φ (1 − ai cos θ)k(1 ± aj cos χ cos θ ± aj sin χ sin θ cos φ)l , (222) with: ai = � ⃗n2 i cos χ = ⃗ni · ⃗nj � ⃗n2 i⃗n2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (223) 38 The expression of the angular integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (222) in many cases of interest can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Observe that A1,0 and A0,1 only depend on ai and aj respectively, and are independent of the label ± in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (222).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now need to perform the integration over k (and, when needed, the explicit evaluation of the angular integrals) of the expressions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217) and (218) for all the possible emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 Massless-massless contribution: ˜S12 We start by addressing the problem of the integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217) in the case in which both the emitters are massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The first step is to write explicitly the function fgg ij for this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By using the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [61] we find � (12) ˜Sm=0 12 (k1, k2) =π 2 (p1 · p2) (k2)−1−ϵ (p1 · k)(p2 · k) � − 8 ϵ � 1 − Γ(1 + ϵ)Γ(1 − ϵ) �1 + ⃗n1 · ⃗n2 1 − ⃗n1 · ⃗n2 �ϵ� − 4 �1 − ⃗n1 · ⃗n2 2 � � 1 0 dt 1 − � 1−⃗n1·⃗n2 2 � t � (1 − t)−ϵ − 2(1 − t)ϵ� � , (224) where the integration over t in the last line is the integral representation of an hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Our task is now to compute � dDk ei⃗b·⃗kT � (12) ˜Sm=0 12 (k1, k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (225) We observe that, while the first term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (224) is singular in the limit k2 → 0, the second term is regular since ⃗n1 · ⃗n2 → 1 as k2 → 0 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (221)) and thus it can be safely expanded in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To proceed further, we need to regularise the additional collinear singularity due to the presence of massless emitters, and we do it by partial fractioning 1 (p1 · k)(p2 · k) = 1 (p1 + p2) · k � 1 (p1 · k) + 1 (p2 · k) � , (226) and by multiplying each singular contribution by the regulator already introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The next step is, after switching to light-cone coordinates, to add the integration over the delta function of k2 � dK2 δ(k2 − K2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (227) which is used to integrate over k+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Then we can introduce the dimensionless variables x = K2 k2 T y = k2 − k2 T , (228) obtaining expressions where the integrals over kT and the one over x and y are factorised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The 39 calculation can be completed with standard techniques: we find ⟨ � dDk ei⃗b·⃗kT � (12) ˜Sm=0 ij �pi · k m2 �2λ ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = �b2 4 �2ϵ−λ � √s 2m2 �2λ π2−ϵ Γ(−2ϵ + λ) 2Γ(1 + ϵ − λ) × � 1 λ � 4 ϵ2 − 8ζ2 − 28ζ3ϵ + O(ϵ2) � + � 31ζ4ϵ + O(ϵ2) � + O(λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (229) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 Massless-massive contribution: ˜Sij We now consider the massless-massive contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In this case we have to integrate both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218) for i = 1, 2 and j = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Mass-independent part We start our analysis with the massless-like part of the soft factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' After writing explicitly the function fgg ij for this configuration by using the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [61], we split it into a regular and a singular part as done for the massless-massless contribution in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1, immediately expanding in ϵ the regular part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Because of the ϵ-pole coming from phase-space integration, the expansion of the integrand needs to be performed up to order ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now describe the integration over k of the angular function fgg ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The collinear singularity due to the presence of the massless momentum pi is regularised as before with the regulator factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We then introduce a delta function δ(k2 − K2) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (227) and we use it to perform the integral over k+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Unlike the massless-massless contribution to the double gluon soft current but similarly to the single-gluon computation, the emitter has a non-zero transverse momentum and hence we have a dependence on ⃗kT in (pj · k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As it is by now customary, we remove it with the shift ⃗kT → ⃗kT + k− pj,− ⃗pT,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (230) This way the only dependence left on the angular part of ⃗kT is in the exponential and the angular integral can now be easily performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The integrals that are left are now the one over K2, over kT and over k−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We introduce the dimensionless variables u = K2 k2 T w = k2 − k2 T m2 p2 j,− , (231) in terms of which fgg ij (⃗nj · ⃗ni, 1,⃗n2 j) is now independent of k2 T fgg ij (⃗ni · ⃗nj, 1,⃗n2 j) ≡ f gg ij (u, w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (232) Because of this, the integral over kT factorises in the form � ∞ 0 dkT k−1−3ϵ+2λ T J−ϵ(bkT)ei ⃗b·⃗pT,j √w kT /m , (233) 40 and can be computed separately obtaining an hypergeometric function9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As for the integral over the variables u and w we obtain, up to overall factors � dDk ei⃗b·⃗kT � (12) ˜Sm=0 ij �pi · k m2 �2λ ∝ � ∞ 0 du dw u−1−ϵw−1+2ϵ 1 + u + w 2F1 � −2ϵ + λ, 1 2 − 2ϵ + λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 w b2m2 (⃗b · ⃗pT,j)2 � f gg ij (u, w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (234) Notice that the collinear divergence, regulated by the parameter λ, is now described by the limit w → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It is now useful to perform some manipulation on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (234) in order to move the dependence on the regulator λ outside of the hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We use the following relation 2F1(a, b, c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' z) =Γ(b − a)Γ(c) Γ(b)Γ(c − a)(−z)−a 2F1 � a, a − c + 1, a − b + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 z � + Γ(a − b)Γ(c) Γ(a)Γ(c − b)(−z)−b 2F1 � b, b − c + 1, b − a + 1, 1 z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (235) By applying it to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (234) and by exploiting the w → 0 limits of the new hypergeometric functions, the limit λ → 0 can be easily carried out and we obtain � dDk ei⃗b·⃗kT � (12) ˜Sm=0 ij �pi · k m2 �2λ = (pi · pj)2λ (m2)3λ �b2 4 �2ϵ−λ π1−ϵ Γ(−2ϵ + λ) 2Γ(1 + ϵ − λ) × � ∞ 0 du dw f gg ij (u, w) u−1−ϵw−1 1 + u + w � wλ + � 2F1 � −2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' c2 jbw � − 1 − 4ϵicjb √wΓ( 1 2 − 2ϵ)Γ(1 + ϵ) Γ(1 − 2ϵ)Γ( 1 2 + ϵ) 2F1 �1 2 − 2ϵ, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' c2 jbw � �� , (236) where cjb is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (129).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (236) with the expression obtained in the one loop case in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (116), we can observe that they share the same structure, the only differences being an overall multiplicative factor, an additional integration over u and the formal substitution u−1−ϵw−1 1 + u + w → 1 w (1 + w)1+ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (237) By using this relation, the azimuthal average can be directly deduced from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (117) ⟨ � dDk ei⃗b·⃗kT � (12) ˜Sm=0 ij �pi · k m2 �2λ ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = (pi · pj)2λ (m2)3λ �b2 4 �2ϵ−λ π1−ϵ Γ(−2ϵ + λ) 2Γ(1 + ϵ − λ) × � ∞ 0 du dw f gg ij (u, w) u−1−ϵw−1 1 + u + w � wλ + Re � (2F1 (−2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' wB) − 1) 9 See formula 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='621 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 41 × (1 + i cot(πϵ)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (238) The two-folded integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (238) does not present significant complications and can be computed with standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We obtain ⟨ � dDk ei⃗b·⃗kT � (12) ˜Sm=0 ij �pi · k m2 �2λ ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = −(pi · pj)2λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='(m2)3λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2ϵ−λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='π2−ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Γ(λ − 2ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2ϵΓ(ϵ − λ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ + 4ζ2ϵ + 14ζ3ϵ2 + 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3ϵ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 (B + 1) − 6ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 33 ln2 (B + 1) ζ2 + 60 ln (B) ln (B + 1) ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 4Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ζ2 − Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 3 ln2 (B) − 2 ln2 (B + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='B + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 10ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 4 ln (B + 1) ζ3 + ζ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ O (λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (239) Combining the results in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (239) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (229) to construct the second term in the square bracket of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (214) we see that the λ → 0 singularities cancel out, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Mass-dependent part We now address the problem of the integration of the mass-dependent part, thus considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The structure of this integral is similar to the one we evalu- ated for the mass-independent part, but with some differences that simplify the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In particular, the overall factor multiplying the angular functions changes according to the 42 substitution (k2)−1−ϵ(pi · pj) (pi · k)(pj · k) → (k2)−1−ϵ (pj · k)2 , (240) which in terms of the dimensionless variables introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (231), corresponds to: u 1 + u + w → 2uw (1 + u + w)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (241) This replacement removes the dependence on the massless momentum from the denominator of the integrand: as a consequence, the integration of this term will not give rise to additional collinear singularities and thus there is no need for the regulator we introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (92), that we can safely drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218) can thus be written in terms of two-folded integrals simply by applying the substitution (241) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (238), setting λ = 0 and considering the function ggg ij rather than fgg ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By doing so, we obtain ⟨ � dDk ei⃗b·⃗kT � (12) ˜Sm̸=0 ij ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = �b2 4 �2ϵ π1−ϵ Γ(−2ϵ) Γ(1 + ϵ) � ∞ 0 du dw u−1−ϵ (1 + u + w)2 × � 1 + Re {[2F1 (−2ϵ, −ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' wB) − 1] (1 + i cot(πϵ))} � ggg ij (u, w) , (242) where we have defined ggg ij (⃗ni · ⃗nj, 1,⃗n2 j) ≡ ggg ij (u, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The computation of this integral does not present any particular additional challenge that cannot be solved with standard techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The expression of the angular function ggg ij can be obtained from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [61] and it is again convenient to identify and immediately expand its contribution which is regular in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To simplify the expression of the integrand, we also isolated a part that, after integration, would have been independent of any kinematical variables: we evaluated numerically this contribution to obtain its constant result c0, finding c0 = −37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='73041235261383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The final result reads ⟨ � dDk ei⃗b·⃗kT � (12) ˜Sm̸=0 ij ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='π2−ϵ Γ(−2ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ Γ(1 + ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 + ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 − 6 − 2Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− ln2(r2 − 1) + 2 ln(r)(2 ln(r) + 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='c0 + 30Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 72Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 12Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='(ln(r) − 5) + 16π2 coth−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − 2r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 8 ln3(r − 1) − 16 ln3(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 8 ln3(r + 1) − 36 ln(r + 1) ln2(r − 1) + 36 ln2(r) ln(r − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 36 ln2(r + 1) ln(r − 1) − 192 ln2(r) + 36 ln2(r) ln(r + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 120 ln(r) ln(r − 1) + 4π2 ln(r) − 144 ln(r) + 120 ln(r) ln(r + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 63ζ3 + 13π2 + 24 − 30π2 ln(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (243) 43 where the variable r is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 Massive-massive contribution: ˜Sjj We now examine the case in which only one massive final-state emitter is involved and we consider ˜Sjj, with j = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By plugging in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (208) the condition pi = pj, we obtain ˜Sjj = ˜Sm=0 jj + 2m2 ˜Sm̸=0 jj = m4 (pj · k1)(pj · k2)(pj · k)2 + 4m2 k2(pj · k1)(pj · k2) = 2 � m4 (pj · k)3 + 4m2 k2(pj · k) � 1 (pj · k1) , (244) where k = k1+k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This more compact expression for ˜Sjj allows us to obtain a simplified version of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217), (218) � (12) ˜Sjj = 2 � m4 (pj · k)3 + 4m2 k2(pj · k) � (k2)−ϵ2−2+2ϵ (pj · k) A1,0 , (245) and, by using the result of the angular integral A1,0 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [61], we can write � (12) ˜Sjj = (k2)−1−ϵm2 (pj · k)2 2π 1 − 2ϵ � m2k2 (pj · k)2 + 4 � 2F1 �1 2, 1, 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='⃗n 2 j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (246) The integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (246) can be carried out in a similar way as the integration of ˜Sij has been performed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Also in this case it is convenient to split the integrand into its singular and regular part, immediately expanding in ϵ the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The final result reads ⟨ � dDk ei⃗b·⃗kT � (12) ˜Sjj⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='π2−ϵ Γ(−2ϵ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Γ(ϵ + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ2 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−24Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 24 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 5π2 + 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='32 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 2r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='7 + 12 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r − 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 64 ln(r + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2(1 − r) + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 8 ln ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 32 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�r − 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2(−r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 64 ln(r − 1)Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='7 + 12 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 32 ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 2r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='Li2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 32Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 32Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�1 − r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 64Li3(1 − r) + 8Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 64Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 32Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 32Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 64Li3(−r) − 64Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 32Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 64Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�r − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 32Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 64Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�r + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 8Li3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='1 − r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 32r ln3(r − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 96r ln3(r) + r ln2(r)(77 + 96 ln(2) − 144 ln(r + 1)) + 96r ln2(r − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='× ln(r + 1) − 2 ln(r − 1)(4r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5π2 − 7 ln(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ ln(r)(−6 + r(−67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 48 ln(2)) + 72r ln(r)) + 4r(7 − 48 ln(r)) ln(r + 1) + 96r ln2(r + 1)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 12 ln(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='9 + 8 ln2(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− ln(r + 1)(1 + 2r(5 + 8 ln(2)) − 8r ln(r + 1)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ r(−60 + π2(27 + 16 ln(2)) + 4 ln2(2)(−7 + 8 ln(2)) + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−24 + π2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='− 8 ln3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 4 ln(r + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='π2 − 4 ln2(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ (7 + 36 ln(2)) ln(r + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 12 ln2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='−2 + ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='r2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='+ 276ζ(3)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (247) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 Massive-massive contribution: ˜S34 We finally analyse the case of the interference between the two final-state emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Our task is to integrate both the mass-independent and mass-dependent expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218) for i = 3, j = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Mass-independent part Let us start our analysis with the mass-independent contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We need to integrate the dimensionless angular function fgg 34(⃗n3 ·⃗n4,⃗n2 3,⃗n2 4) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To evaluate the angular integrals contained therein, we relate them to the imaginary part of a massive box diagram via the optical theorem10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' All order results for the box diagram with a single mass, equivalent to fix ai = aj in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (222), are presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [64], while results with two different masses but only at the lowest order in ϵ can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We extended the latter expression to all orders in ϵ, obtaining the angular integral A± 1,1 A± 1,1 = � π 0 dθ � π 0 dφ sinD−3 θ sinD−4 φ (1 − a3 cos θ)(1 ± a4 cos χ cos θ ± a4 sin χ sin θ cos φ) = 2π 2ϵ − 1 ⃗n2 3 + ⃗n3 · ⃗n4 � 1 − ⃗n2 3 (⃗n2 3 + ⃗n2 4 + (⃗n3 · ⃗n4)2 + 2⃗n3 · ⃗n4 − ⃗n2 3⃗n2 4) × × F1 �1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1, 1 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' ⃗n3 · ⃗n2 4 − ⃗n2 3⃗n2 4 (⃗n3 · ⃗n4)2 + 2⃗n3 · ⃗n4 + ⃗n2 3 − ⃗n2 3⃗n2 4 + ⃗n2 4 , 1 − ⃗n2 3 ⃗n2 3 − 1 � + (3 ↔ 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (248) By following the same strategy applied in the previous sections, we isolate in the angular function fgg 34(⃗n3 ·⃗n4,⃗n 2 3 ,⃗n 2 4 ) a term σm=0 34 that will give rise to singularities upon integration over 10 See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' the discussion in Appendix A of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 45 k2 and a regular term ρm=0 34 that vanishes in the k2 → 0 limit and that can be directly expanded in ϵ fgg 34(⃗n3 · ⃗n4,⃗n 2 3 ,⃗n 2 4 ) = −π ϵ � σm=0 34 + ϵ ρm=0 34 + (p3 ↔ p4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (249) By using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (248), the k2-singular part can be written in the following way σm=0 34 = 2 − 2 h(ϵ) �1 − ⃗n2 3 4 �−ϵ � 1 − 2ϵ (1 − ⃗n3 · ⃗n4)χ34 ⃗n2 3 − ⃗n3 · ⃗n4 2F1(1, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' χ34) � , (250) where the function h(ϵ) is defined as h(ϵ) = π−1/2 4−ϵ Γ( 1 2 − ϵ) Γ(1 + ϵ) , (251) and the k2-independent coefficient χ34 is given by χ34 = (⃗n2 3 − ⃗n3 · ⃗n4)2 (⃗n2 3 − ⃗n2 4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 3 ⃗n2 4 , (252) and fulfils 0 ≤ χ34 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We observe that the factor (1−⃗n3·⃗n4)/(⃗n2 3−⃗n3·⃗n4) is also independent on k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Let us start by considering the integration of the singular contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (249), (250) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217) we obtain a sum of integrals with the following structure Igg 34[fα] = � dDk ei⃗b·⃗kT (k2)−1−ϵ(p3 · p4) (p3 · k)(p4 · k) fα(⃗n3,⃗n4) , (253) with three possible functions fα(⃗n3,⃗n4) (α = 1, 2, 3) f1(⃗n3,⃗n4) = 1 , (254) f2(⃗n3,⃗n4) = �1 − ⃗n2 3 4 �−ϵ , (255) f3(⃗n3,⃗n4) = −4π h(ϵ) �1 − ⃗n2 3 4 �−ϵ (1 − ⃗n3 · ⃗n4)χ34 ⃗n2 3 − ⃗n3 · ⃗n4 2F1(1, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' χ34) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (256) With those definitions, we have � dDk ei⃗b·⃗kT (k2)−1−ϵ(p3 · p4) (p3 · k)(p4 · k) � −π ϵ σm=0 34 � = −2π ϵ Igg 34[f1] + 2π ϵ h(ϵ)Igg 34[f2] + Igg 34[f3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (257) We start by considering Igg 34[f1] Igg 34[f1] = � dDk p3 · p4 (p3 · k)(p4 · k) (k2)−1−ϵ ei⃗b·⃗kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (258) This integral is exactly the same appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (188) for the case of soft q¯q emission, and the result up to O(ϵ) was already presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (192).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the present case, however, we need 46 it up to O(ϵ2), since in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (257) Igg 34[f1] already appears with an overall factor 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We find ⟨Igg 34[f1]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =π1−ϵ Γ(1 − ϵ)Γ(−2ϵ) �b2 4 �2ϵ 1 + β2 2β � −1 ϵL0 − 2L1 + ϵ(2P2 − L2) (259) −2 3ϵ2 (L3 − 6P3 − 3Q3) + O(ϵ3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The functions Ln, Pn have been defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (98)–(99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Here we also introduced the function Qn, defined as Qn(β) = � β −β dz 1 − z2Lin � − z2 tan2(θ) z2 − sec2(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (260) The explicit expressions of the functions L0, L1 and P2 are already provided in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (100)–(102) while L2 can be found in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (193).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The functions L3, P3 and Q3, that can be obtained in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can use a similar strategy to evaluate Igg 34[f2] Igg 34[f2] = �m2 4 �−ϵ � dDk ei⃗b·⃗kT (k2)−1−2ϵ(p3 · p4) (p3 · k)1−2ϵ(p4 · k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (261) In this case, the generalisation of Feynman parametrisation needs to be applied 1 AmBn = Γ(m + n) Γ(m)Γ(n) � 1 0 dx xm−1(1 − x)n−1 (xA + (1 − x)B)m+n , (262) thereby obtaining Igg 34[f2] = (p3 · p4) �m2 4 �−ϵ (1 − 2ϵ) � 1 0 dx x−2ϵ � dDk ei⃗b·⃗kT (k2)−1−2ϵ (p(x) · k)2−2ϵ , (263) with p(x) = xp3 + (1 − x)p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The azimuthally averaged integral ⟨Igg 34[f2]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' can be evaluated as ⟨Igg 34[f2]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = (p3 · p4) �m2 4 �−ϵ (1 − 2ϵ) � 1 0 dx (p2(x))1−ϵx−2ϵ ⟨Iaux 2 (x)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (264) with ⟨Iaux 2 (x)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = ⟨ � dDk ei⃗b·⃗kT (k2)−1−ϵ (p(x) · k)2 p2(x) � k2p2(x) (p(x) · k)2 �−ϵ ⟩ av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = �b2 4 �2ϵ π1−ϵ 2−2−2ϵ ϵ2(1 − 2ϵ) Γ(1 − 2ϵ)Γ(1 − ϵ) � 1 + p2 T(x) p2(x) �2ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265) The leftover integral over the Feynman parameter can be performed in a standard way and the solution expressed in terms of multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 47 The last integral we need for the evaluation of the singular part is Igg 34[f3] = −4πh(ϵ) � dDk ei⃗b·⃗kT (k2)−1−ϵ(p3 · p4) (p3 · k)(p4 · k) × �1 − ⃗n2 i 4 �−ϵ (1 − ⃗n3 · ⃗n4)χ34 ⃗n2 3 − ⃗n3 · ⃗n4 2F1(1, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' χ34) , (266) with h(ϵ) and χ34 defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (251) and (252), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to simplify the expression of the integrand we define the auxiliary momentum ℓµ = 1 v � pµ 3 − m2 (p3 · p4)pµ 4 � , (267) with v defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By using the definition of ℓµ and an integral representation for the hypergeometric function, we can rewrite Igg 34[f3] as Igg 34[f3] = − � dDk ei⃗b·⃗kT 2π(p3 · p4) v (k2)−1−2ϵ (p4 · k) � m2 (p3 · k)2 �−ϵ 1 ℓ · k × � 1 0 dt (t)− 1 2 −ϵ(1 − t)ϵ χ34 1 − t χ34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' =2π(m2)−ϵ(p3 · p4) v2 � 1 0 dt (t)− 1 2 −ϵ(1 − t)ϵ � dDk(k2)−1−2ϵ ei⃗b·⃗kT (p3 · k)1−2ϵ × � � 1 1 − t v2 � � − 1 (p4 · k) � + m2 2(p3 · p4) � σ=±1 1 1 + σ √ t v 1 ℓ(σ) · k � , (268) where for convenience we introduced ℓµ (±) = pµ 3 ± √ t ℓµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (269) By analysing the dependence on t of the integrand, we observe that it is expressed as a sum of functions, some of which feature a divergence for t = v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This singularity has no physical origin and will eventually cancel in the final result when summing together these divergent contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to regularise it, we fix a small imaginary part for v and take its finite limit to zero at the end of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The dependence of the integrand on k is via a factor Igg 34[f3] ∝ (k2)−1−2ϵ (p3 · k)1−2ϵ � 1 (p4 · k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 (ℓ(±) · k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (270) By applying the generalised Feynman parametrisation introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (262), we obtain a sin- gle momentum integral equivalent to Iaux 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265), the only difference being the expression of the auxiliary momentum, which now reads: p(x) = xp3 + (1 − x)P , (271) 48 with the three possibilities P = p4, P = ℓ(+) P = ℓ(−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (272) The integration over k can thus be performed by using the partial results already obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The leftover integrals over the Feynman parameter x and the variable t we used for the integral representation of the hypergeometric function can be performed with standard techniques, and the result can be expressed in terms of multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now consider the contribution to fgg 34 that vanishes in the k2 → 0 limit, ρm=0 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' It can be written at all orders in ϵ in the following compact way 1 π ρm=0 34 =Γ( 1 2 − ϵ)Γ(ϵ) √π �1 − ⃗n2 3 ⃗n2 3 �−ϵ 1 + D34 − 2 � ⃗n2 3 � ⃗n2 3 + 1 ϵ � 2 − (1 + D34)2F1( 1 2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 + ϵ, 1 − ⃗n2 3) � − 2(1 − ⃗n3 · ⃗n4)χ34 ⃗n2 3 − ⃗n3 · ⃗n4 � 1 0 du(1 − u)− 1 2 −ϵ √1 − uχ34 [(1 + uψ)ϵ − uϵ(1 + ψ)ϵ] + D34γ � 1 0 du (1 − u) 1 2 −ϵ � 1 − ⃗n2 3u(1 − (1 − u)γ) , (273) where the coefficient χ34 is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (252) and D34 = (1 + ⃗n3 · ⃗n4)(⃗n2 3 + ⃗n3 · ⃗n4) (⃗n2 3 + ⃗n2 4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 3 ⃗n2 4 , (274) γ = (⃗n3 · ⃗n4)2 − ⃗n2 3⃗n2 4 (⃗n2 3 + ⃗n2 4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 3 ⃗n2 4 , (275) ψ = − (⃗n3 · ⃗n4)2 − ⃗n2 3⃗n2 4 (⃗n2 3 − ⃗n2 4)2 + (⃗n3 · ⃗n4)2 − ⃗n2 3 ⃗n2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (276) Because of its regular behaviour, ρm=0 34 can be safely expanded in ϵ ρm=0 34 = ρm=0, (0) 34 + ϵ ρm=0, (1) 34 + O(ϵ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (277) Due to the complexity of the functions ρm=0, (0) 34 and ρm=0, (1) 34 , we perform numerically the last steps of their integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The representation of the soft integrals in qT-space, rather than the impact-parameter space used until now, is more convenient to this purpose, as it allows us to trivially carry out the D − 2 dimensional integration of the transverse components of the soft momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The conversion to the representation in b-space can be obtained by applying to the final result the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To be specific, the integral that we will compute is �� dDk δ(D−2)(⃗kT − ⃗qT)(k2)−1−ϵ(p3 · p4) (p3 · k)(p4 · k) (ρm=0, (0) 34 + ϵ ρm=0, (1) 34 ) � av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (278) To perform its azimuthal average, we can fix the azimuthal angle φ such that ⃗pT,3 · ⃗qT = pT,3 qT cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' With this choice the integral over the other angles becomes straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' After integrating over dD−2kT, the remaining computation can be performed for instance by 49 introducing an integral over the virtuality of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We obtain (q2 T)−1−ϵ B( 1 2, 1 2 − ϵ) � 1 −1 d cos φ � ∞ 0 dx dy 1 − ⃗n3 · ⃗n4 2 x y x−1−ϵ(1 − cos2 φ)−1/2−ϵ(ρm=0, (0) 34 + ϵ ρm=0, (1) 34 ) , (279) where we introduced the dimensionless integration variables x = k2 q2 T , y = k− qT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (280) Given that the functions ρm=0, (0) 34 and ρm=0, (1) 34 vanish by construction in the limit x → 0, the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (279) can be safely expanded in ϵ and integrated numerically over x, y and cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The result is a function of the LO phase-space, which can be reduced to the dependence on β and cos θ and can thus be provided in the form of a two-dimensional grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to obtain a more stable and fast numerical evaluation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (279) we isolate its contributions that can be expressed only as a function of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To be specific, we observe that in the ϵ-expanded expression, (q2 T)−1−ϵ B( 1 2, 1 2 − ϵ) � 1 −1 d cos φ � ∞ 0 dx dy 1 − ⃗n3 · ⃗n4 2 x2 y (1 − cos2 φ)−1/2� ρm=0, (0) 34 + ϵ ρm=0, (1) 34 − ϵ ln � x(1 − cos2 φ) � ρm=0, (0) 34 � , (281) the integral of the first two terms in the square bracket is independent of cos θ, allowing us to simply compute a one-dimensional grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In addition, these terms can be integrated by using the following identity, 1 π � 1 −1 d cos φ � ∞ 0 dx dy 1 − ⃗n3 · ⃗n4 2 x2 y x−1−ϵ(1 − cos2 φ)−1/2−ϵf(⃗n3 · ⃗n4,⃗n2 3,⃗n2 4) = � 1 0 dt � 1 −1 d cos φ t2 1 − t2 1 1 − vt cos φ f � 1 − 1 − t2 1 − vt cos φ, t2, 1 − (1 − v2) 1 − t2 (1 − vt cos φ)2 � , (282) which is valid for a generic function f, and that can by proven by using the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (162).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The right hand side of the equation is only a two-fold integral, leading to a further simplification of the corresponding numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Mass-dependent part We now consider the case of the contribution coming purely from the massive case, ˜Sm̸=0 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can approach the computation in a way similar to that used for the mass-independent part we just described, considering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218) rather than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (217).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The factor multiplying the angular functions, however, is not anymore symmetric under the exchange p3 ↔ p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Unlike the mass-independent contribution, we thus cannot assume that the final result respects such symmetry and we need to separately compute the integral of ˜Sm̸=0 34 and the one of ˜Sm̸=0 43 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 50 As it is by now customary, we write the dimensionless angular function ggg 34(⃗n3 · ⃗n4,⃗n 2 3 ,⃗n 2 4 ) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (218) as the sum of a singular and a regular contribution ggg 34(⃗n3 · ⃗n4,⃗n 2 3 ,⃗n 2 4 ) = −π ϵ � σm̸=0 34 + ϵ ρm̸=0 34 + (p3 ↔ p4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (283) The singular part, which generates singularities after integration over k2, can be written as σm̸=0 34 = − (1 − ⃗n2 3)−ϵΓ � 1 2 − ϵ � Γ(1 + ϵ) √π � 1 + ϵ 2χ34(1 − ⃗n3 · ⃗n4) 2F1 � 1, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' χ34 � ⃗n2 3 − ⃗n3 · ⃗n4 � + (1 − ⃗n2 4)−ϵΓ � 1 2 − ϵ � Γ(1 + ϵ) √π � 1 + ϵ 2χ34(1 − ⃗n3 · ⃗n4) 2F1 � 1, 1 2 − ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' χ34 � ⃗n3 · ⃗n4 − ⃗n2 4 � , (284) while the regular part can be directly expanded in ϵ and we can symbolically write ρm̸=0 34 = ρm̸=0 (0) 34 + ϵ ρm̸=0 (1) 34 + O(ϵ2) , (285) where higher order terms can be neglected in our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Let us start with the integration of the singular part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (284).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Its computation requires the evaluation of integrals in the form Igg j [fα] = � dDk ei⃗b·⃗kT (k2)−1−ϵ (pj · k)2 fα(⃗n3,⃗n4) , (286) with j = 3, 4 and where the possible functions fα(⃗n3,⃗n4) are the same already introduced in the mass-independent case in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (254).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' With this definition we have � j=3,4 � dDk ei⃗b·⃗kT (k2)−1−ϵ (pj · k)2 � −π ϵ (σm̸=0 34 + σm̸=0 43 ) � = � − π ϵ Igg 3 [f1] + π ϵ h(ϵ)Igg 3 [f2] + 1 2Igg 3 [f3] + π ϵ Igg 4 [f1] − π ϵ h(ϵ)Igg 4 [f2] + 1 2Igg 4 [f3] � + (p3 ↔ p4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (287) We start from Igg j [f1], Igg j [f1] = � dDk 1 (k2)1+ϵ ei⃗b·⃗kT (pj · k)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (288) This integral has already been computed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 and we have m2Igg j [f1] = Iq¯q jj , where Iq¯q jj is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (186).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The result for ⟨Iq¯q jj (⃗b)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' was reported in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (190).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We now turn our attention to Igg j [f2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We first consider Igg 3 (f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The integral to compute is Igg 3 [f2] = � dDk ei⃗b·⃗kT (k2)−1−2ϵ (p3 · k)2 � m2 4(p3 · k)2 �−ϵ , (289) which has the same structure as Iaux 2 , introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can thus take the result of 51 ⟨Igg 3 [f2]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265) with the replacement p(x) → p3 ⟨Igg 3 [f2]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 1 m2 �b2 4 �2ϵ π1−ϵ 2−2−2ϵ ϵ2(1 − 2ϵ) Γ(1 − 2ϵ)Γ(1 − ϵ) � 1 + p2 3,T m2 �2ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (290) The integral Igg 4 [f2], on the other hand, is not proportional to Iaux 2 (x) Igg 4 [f2] = � dDk ei⃗b·⃗kT (k2)−1−2ϵ (p4 · k)2 � m2 4(p3 · k)2 �−ϵ , (291) but the structure of Iaux 2 (x) can be recovered by applying the generalised Feynman parametri- sation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (262): ⟨Igg 4 [f2]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = − 21+2ϵϵ (1 − 2ϵ) � 1 0 dx x (1 − x)−1−2ϵ ⟨ � dDk ei⃗b·⃗kT (k2)−1−2ϵ (p(x) · k)2−2ϵ⟩ av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = − 21+2ϵϵ (1 − 2ϵ) � 1 0 dx (p2(x))1−ϵx (1 − x)−1−2ϵ ⟨Iaux 2 (x)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = − 1 4 βm2 ϵπ1−ϵ �b2 4 �2ϵ �τ 2 �1−ϵ Γ(1 − 2ϵ)Γ(1 − ϵ) × � β −β dy � 1 + y β � � 1 − y β �−1−2ϵ � 1 1 − y2 �1−ϵ � Bτ 1 − τ y2 1 − y2 + 1 �2ϵ , (292) where in the last step we used the known result of ⟨Iaux 2 (x)⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At this stage, the integrand cannot yet be expanded in ϵ, since the integral does not converge in the limit ϵ → 0 due to a singularity in y = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to perform the expansion, we need to isolate the singular behaviour and subtract it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We consider the following auxiliary integral: � β −β dy � 1 − y2�2ϵ−1 � 1 − y β �−2ϵ−1 � 1 + y β �1−2ϵ = =2√πΓ(−2ϵ) β 1 − β2 � 1 Γ � 1 2 − 2ϵ � 2F1 �1 2, −2ϵ, 1 2 − 2ϵ, β2 � +ϵ 1 + β2 Γ � 3 2 − 2ϵ � 2F1 �1 2, 1 − 2ϵ, 3 2 − 2ϵ, β2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (293) We observe that the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (293) has the same behaviour as I4[f2] in the y → β limit, while having a simpler structure that allows for a straightforward analytic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can thus add the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (293) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (292), while subtracting the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' at the integrand level in order to obtain a regular expression that can be safely expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The resulting integral can be evaluated separately at each order in ϵ in terms of multiple polylogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Let us finally consider the contribution of the function f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By following the same proce- dure already applied for the evaluation of Igg 34[f3] in the mass-independent case, we define the 52 momentum ℓ as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (267) and by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (268) we have: ⟨Igg 3 [f3]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = −π v ⟨ � dDk ei⃗b·⃗kT (k2)−1−2ϵ (p3 · k) � m2 (p3 · k)2 �−ϵ 1 ℓ · k � 1 0 dt t− 1 2 −ϵ(1 − t)ϵ χ34 1 − t χ34 ⟩ av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (294) which, once we substitute in it the definition of χ34 as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (252), gives us an expression that only depends explicitly on two momenta, p3 and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By applying partial fractioning and defining ℓ± as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (269) we obtain: ⟨Igg 3 [f3]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = −π v ⟨ � dDk ei⃗b·⃗kT (m2)−ϵ 2 � 1 0 dt t−1−ϵ(1 − t)ϵ � (k2)−1−2ϵ (p3 · k)1−2ϵ(ℓ− · k) − (k2)−1−2ϵ (p3 · k)1−2ϵ(ℓ+ · k) � ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (295) By applying the generalisation of Feynman parametrisation introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (262) we can reduce the dependence of the denominators of the integrand to a single momentum, and re- trieve the structure of Iaux 2 (x) as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The leftover integral over the Feynman parameter x and the variable t can be computed in terms of multiple polylogarithms with a standard procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The evaluation of Igg 4 [f3] can be performed by following the same steps, but the differences in the integrand make the procedure of partial fractioning a bit more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' After introducing the momentum ℓ and applying partial fractioning for a first time, we obtain an expression similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (295): ⟨Igg 4 [f3]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = −π v ⟨ � dDk ei⃗b·⃗kT (m2)−ϵ 2(p4 · k)2 � 1 0 dt t−1−ϵ(1 − t)ϵ � (k2)−1−2ϵ (p3 · k)−1−2ϵ(ℓ− · k) − (k2)−1−2ϵ (p3 · k)−1−2ϵ(ℓ+ · k) � ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' , (296) but, in this case, we can not yet apply Feynman parametrisation, since each denominator in- volves three different products of the momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can circumvent this problem by performing an additional partial fractioning: ⟨Igg 4 [f3]⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = − π v2(m2)−ϵ⟨ � dDk ei⃗b·⃗kT � 1 0 dt t− 1 2 −ϵ (1 − t)ϵ � 1 − t v2 � � (k2)−1−2ϵ (p3 · k)−2ϵ(p4 · k)2 + m2 p3 · p4 � − � 1 + t v2 � � 1 − t v2 � (k2)−1−2ϵ (p3 · k)1−2ϵ(p4 · k) + m2 2p3 · p4 √ t v × �� 1 + t v2 � � 1 − t v2 � (k2)−1−2ϵ (p3 · k)1−2ϵ(ℓ− · k) − � 1 − t v2 � � 1 + t v2 � (k2)−1−2ϵ (p3 · k)1−2ϵ(ℓ+ · k) ��� ⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (297) It is now possible to apply Feynman parametrisation to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (297), obtaining integrals over the momentum k that can be written in terms of Iaux 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' By using the known result of ⟨Iaux 2 ⟩ provided 53 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (265) to perform the integration over k, we are left with the final two integrals over the Feynman parameter and the variable t, that can be performed with a standard procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Let us finally analyse the regular part of ggg 34(⃗n3 · ⃗n4,⃗n2 3,⃗n2 4), that can be safely be expanded in ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We need to evaluate the following integral: ⟨ � j=3,4 � dDk ei⃗b·⃗kT (k2)−1−ϵ (pj · k)2 (ρm̸=0 34 + ρm̸=0 43 )⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (298) The explicit all-orders expression of the integrand reads: 1 π � j=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 1 pj · k � ρm̸=0 34 + ρm̸=0 43 � = (1 − ⃗n2 3) � − 2 − 5ϵ ϵ(1 − 2ϵ) − 1 1 − 2ϵ ⃗n3 · ⃗n4 ⃗n2 3 � + (2 − ⃗n2 3 − ⃗n2 4) � 1 ϵ + Γ(1 − 2ϵ)Γ(ϵ) Γ(1 − ϵ) �1 − ⃗n2 3 4 �−ϵ � D34 n1/2−ϵ 3 − 1 �� + 1 √πΓ �1 2 − ϵ � Γ (ϵ) (1 − ⃗n2 3)1−ϵ �⃗n2 3 + ⃗n3 · ⃗n4 2(⃗n2 3)3/2−ϵ − 6⃗n2 3 2(⃗n2 3)3/2−ϵ + 2 � + 1 ϵ � 3(1 − ⃗n2 3) + D34(2 − ⃗n2 3 − ⃗n2 4) � 2F1 �1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 + ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1 − ⃗n2 3 � − 1 − � ⃗n2 3 ϵ⃗n2 3 (⃗n2 3 + ⃗n3 · ⃗n4)2F1 � 1, 1 − ϵ, 1 + ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 1 + � n2 3 − 1 � − 1 − ⃗n3 · ⃗n4 ⃗n2 3 − ⃗n3 · ⃗n4 (2 − ⃗n2 3 − ⃗n2 4)χ34 � 1 0 du (1 − u)− 1 2 −ϵ √1 − χ34u [(1 + uψ)ϵ − uϵ(1 + ψ)ϵ] + D34 γ 1 − γ (2 − ⃗n2 3 − ⃗n2 4) � 1 0 du (1 − u) 1 2 −ϵ � 1 − ⃗n2 3u � 1 + u γ 1 − γ �−1 , (299) where the variable χ34 has been defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (252) while D34, γ and ψ are given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (274)– (276).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We introduce the following notation for the ϵ-expansion of the integrand: ρm̸=0 ij = ρm̸=0, (0) ij + ϵ ρm̸=0, (1) ij + O(ϵ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (300) As for the case of the mass-independent contribution, due to the complexity of the functions involved in the integrand, we perform numerically the last steps of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We follow the same steps as for the integration of ρm=0 34 , by considering the qT-space representation of the integral and by fixing the azimuthal angle φ such that ⃗pT,3 · ⃗qT = pT,3 qT cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' After switching to the dimensionless variables x, y already introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (280) and inserting an integral 54 over the virtuality of the momentum, we obtain: ⟨ � j=3,4 � dDk δ(D−2)(⃗kT − ⃗qT)(k2)−1−ϵ (pj · k)2 (ρm=0, (0) 34 + ϵ ρm=0, (1) 34 )⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = = q−1−ϵ T B( 1 2, 1 2 − ϵ) � 1 −1 d cos φ � ∞ 0 dx dy 1 2x2y(1 − cos2 φ)−1/2 × � j=3,4 1 pj · k � ρm̸=0, (0) 34 + ϵ ρm̸=0, (1) 34 − ϵ ln � x(1 − cos2 φ) � ρm̸=0, (0) 34 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (301) We can observe that also in this case the integral of the first two terms in the square bracket only depends on β and can be thus evaluated on a one-dimensional grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We can also reduce the 3-fold integrals of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (301) in 2-fold ones by replacing the exponential by a θ-function in the corresponding b-space representation, in a similar fashion as it was done in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (282).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Adapting it to the present functions we obtain: 1 π � 1 −1 d cos φ � ∞ 0 dx dy 1 2 x2 yx−1−ϵ(1 − cos2 φ)−1/2−ϵρ(⃗n3 · ⃗n4,⃗n2 3,⃗n2 4) = � 1 0 dt � 1 −1 d cos φ t2 1 − t2 ρ � 1 − 1 − t2 1 − vt cos φ, t2, 1 − (1 − v2) 1 − t2 (1 − vt cos φ)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (302) We have now summarized in detail the technical aspects of our calculation, and presented explicit partial results in the cases the expressions were obtained in a compact analytic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Our complete final results are collected and implemented in a numerical code, which is described in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 4 Numerical results In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 we described in detail the ingredients entering the transverse-momentum resummation formalism for heavy-quark production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To the purpose of the application to the qT-subtraction framework, the key role is played by the coefficient HQ ¯Q defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (14), which depends on the subtracted matrix element � M via the master formula (15), while � M can be obtained through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' All the ingredients entering in these equations are finite, since the cancellation of the IR poles has been carried out at the operator level as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The cancellation is guaranteed by the relation with the subtracted soft anomalous dimension Γsub in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (36)– (38): we were able to verify analytically this cancellation for all the contributions, with the exception of the nf-independent part of the term proportional to the colour factor T3 · T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This term depends only on the variable β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4, part of this term was evaluated numerically, and, therefore, only a numerical check of the cancellation is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In Figure 1 we compare the coefficient of the 1/ϵ pole as a function of β computed analytically with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (38) against our numerical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The lower plot shows the relative difference between the two: The relative difference is below the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0005% in all the regions of the phase-space, showing a perfect agreement with the prediction and providing a strong cross-check of our computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 55 2 1 0 1 2 3 4 5 〈Fex,2 (-1)〉av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' nf =0, T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='T4 component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0004 β relative difference (%) Figure 1: Numerical results for the coefficient of the 1/ϵ pole of the contribution proportional to T3 · T4 in Fex,2 (red points), compared to the expected analytical result (gray curve) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The lower plot shows the relative difference (in percentage) between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Having discussed the cancellation of the IR singularities, we now consider the ingredients needed for the implementation of the function HQ ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In the qT-subtraction formalism, a final average over the azimuthal degree of freedom of ⃗b is required (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (14)), and since the operator D is defined in such a way that ⟨D⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' = 1, it gives no contribution in our computation, except when interfering with the azimuthally dependent part of the C coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore, as a new perturbative ingredient at NNLO we just need to evaluate the subtracted amplitude � M through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (54) at second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At this order the subtracted amplitude Z−1 |M⟩ appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (54) is provided by the numerical grids in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The operator eV fin c at the same order is already known from the implementation of qT-subtraction for a colourless final state at NNLO [38] (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (55)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We are left with the coefficient h, whose perturbative expansion is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The term involving the commutator produces contributions proportional to three-parton correlators, which vanish when evaluated on the Born c¯c → Q ¯Q amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Therefore we can simply write h(αS) = 1 + αS 2π ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' + �αS 2π �2 � ⟨(F(0) ex,1)2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' − 1 2 � ⟨F(0) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' �2 + ⟨F(0) ex,2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' − 2πβ0 ⟨F(1) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' � + O(α3 S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (303) The two last terms in the O(α2 S) contribution involve the product of two colour charges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' we have chosen to write the results in the numerical implementation in terms of the colour structures 56 T3 · T4 and Ti · Tj with i = 1, 2 and j = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The results for the Ti · Tj structure are obtained in a fully analytical way, and the explicit expression, which can be obtained from the results in the previous Sections, is implemented in the numerical code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The results corresponding to the colour structure T3·T4 have contributions from the integral of the soft correlators Sm=0 34 and Sm̸=0 34 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (198) and (199), which are partially obtained numerically in the form of a two-dimensional grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The numerical integration is performed using the implementation of global adaptive strate- gies available in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For the terms independent of θ, the integral is evaluated for a grid in the variable β from 0 to 1, in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='001 in the range (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='8) and a smaller step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0001 in the high-energy region (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1), in which the variation of the function is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For the remaining term, which depends both on β and cos θ, the integral is evaluated for a total number of 5000 phase-space points in the range β ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1), cos θ ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 1) (the result is symmetric under the exchange cos θ → − cos θ), which were obtained from the NNLO parton level generator Matrix [66] after the optimisation for the integration of the LO t¯t cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Given that a numerical interpolation of the grid is already needed, and also due to the fact that the numerical evaluation of the analytical terms entering the T3 · T4 structure of ⟨F(0) ex,2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' is computationally very expensive, we decided to encode all the contributions proportional to T3 · T4 in the terms ⟨F(0) ex,2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' − 2πβ0 ⟨F(1) ex,1⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' in a two-dimensional grid composed by the same phase-space points used for the numerical integration of the aforementioned pieces of Sm=0 34 and Sm̸=0 34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The different pieces entering the final result are defined in three independent grids and combined afterwards, in order to have a fully flexible implementation in the number of light-quark flavours nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The numerical evaluation of the multiple polylogarithms appearing in some of our analytic expressions, needed for the construction of the grids, is performed using GiNaC [67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In addition to the contributions described above, the result for the azimuthal average of the square of the NLO result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' the term ⟨(F(0) ex,1)2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=', is also obtained numerically, by simply starting from the known result for F(0) ex,1 and computing the azimuthal average of its square, again in the same set of phase-space points used before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In this case, the results are grouped in three different colour structures, (T3 · T4)2, CF T3 · T4 and C2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The grids described above are afterwards fitted using a spline approximation [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Given that we do not expect our results for each phase-space point to have a large deviation from the correct value, as the uncertainties of the numerical integration are at the per mille level, the parameters of the spline fitting are chosen such that the fit is very close to the original points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In addition, and in order to improve the quality of the fit, the grids are divided by appropriate factors depending on β and cos θ before performing the fit, which were checked to generate surfaces with smaller variations and therefore easier to fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' A concrete example of this procedure is given by the way to handle the threshold region: all the grids had a divergent logarithmic behaviour in the β → 1 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We thus divided all the points by a factor (1 + log2(1 − β2)n), with the value of n chosen in order to get a regular grid in such limit, and this factor was added back after the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Also, in order to work with more evenly distributed points, we worked with the variables β2 (instead of β) and cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 57 We have studied the self-consistency of the fit by comparing the results obtained with it to the original values on the grids used to construct it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We observed that, for the majority of the points (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='9%), the difference is below the per mille level, while the points that agree better than 1% almost cover the full phase-space (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The largest relative differences show up in the grids corresponding to ⟨(F(0) ex,1)2⟩av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=', in the regions in which simultaneously β and | cos θ| are close to 1, the reason being the sudden variation of the fitted function in that area, and its value being very close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In order to see if the error coming from the fitting of the grids has an impact in the computation of a physical quantity, we checked the difference between the original grid and the fit once combined with all the other ingredients entering the coefficient HQ ¯Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' This involves, among other things, the evaluation of lower-order (colour-correlated) matrix elements, the finite part of the two-loop amplitudes, plus all the soft contributions that were obtained and encoded analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We performed this check for the specific case of top-quark pair production, using OpenLoops [70] for the evaluation of tree-level and one-loop amplitudes, and the results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [37] for the two-loop corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We observed that the point-wise difference is always below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='25%, and that, from the total of points, only a handful of them present a deviation larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='05%, indicating that the accuracy obtained through the fit is more than enough to reproduce the original results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The checks described above only tested the accuracy of the fit on the very same points used to generate it: it is also important to perform some checks on the rest of the phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' To this end, we reduced the number of points used to perform the fit by a factor of 2 and checked how the accuracy of the final result is affected, finding results similar to the ones described above, thereby confirming the reliability of our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We illustrate our final results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' As described in the text, we split our results into the different colour structures appearing in h, specifically h(αS) = 1 + αS 2π � h(1) 34 T3 · T4 + h(1) 33 CF � + �αS 2π �2 � h(2) 34 T3 · T4 + h(2) 13 T1 · T3 + h(2) 14 T1 · T4 + h(2) 23 T2 · T3 + h(2) 24 T2 · T4 + h(2) 3434 T3 · T4 T3 · T4 + h(2) 3433 T3 · T4 CF + h(2) 3333 C2 F � + O(α3 S) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (304) We note that this particular choice of colour structures is not unique, and different choices can be made which are related by colour conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Results for h(2) 34 and h(2) 13 are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2, while h(2) 3434, h(2) 3433 and h(2) 3333 are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In both cases, the results correspond to nf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The numerical code used to evaluate all the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' (304) is included as supplemental material of this paper, allowing for the evaluation of our final results for arbitrary values of β, cos θ and nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='500 1 30 20 10 0 10 20 30 40 1-β2 h34 (2) nf=0, cosθ=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 cosθ h34 (2) nf=0, β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='500 1 0 50 100 150 1-β2 h13 (2) nf=0, cosθ=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 cosθ h13 (2) nf=0, β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 Figure 2: Contributions to the second order coefficient of h proportional to T3 · T4 (upper panels) and T1 · T3 (lower panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The results correspond to nf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' The left panels show the β dependence for a fixed value of cos θ = 0, while the cos θ dependence is shown in the right panels for β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='0 1.' metadata={'source': 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+page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='06 cosθ h3333 (2) ×100 nf=0, β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5 Figure 3: Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2 for the contributions proportional to T3 ·T4 T3 ·T4 (upper panels), CF T3 · T4 (middle panels) and C2 F (lower panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 60 5 Summary This paper has been devoted to the evaluation of the soft-parton contributions that are relevant when a heavy-quark pair is produced at small transverse momenta in hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' When a colourless system (vector boson(s), Higgs boson(s) and so forth) is produced in hadron collisions only soft and collinear radiation from the initial-state colliding partons plays a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' When a heavy-quark pair is produced, the coloured heavy quarks can emit in turn soft radiation (soft gluons and light quark-antiquark pairs), which gives an additional contribution to the structure of the singular contributions at small transverse momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have evaluated such soft-parton contributions to NNLO in QCD perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Our computation has been carried out by using a semi-numerical approach, and evaluating all the relevant integrals in impact parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have explicitly considered only the contributions that are relevant to apply the qT subtraction formalism to this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' After having introduced our framework in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 2, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3 we have provided the details of our calculation, by first starting from the NLO in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='2, which had already been obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We then moved to the evaluation of the integrals from soft-gluon emission at one- loop order in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='3, soft light-quark pairs in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='4, and finally double gluon emission in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' We have provided all the relevant details of the computation by highlighting the difficulties that had to be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' At NNLO the most challenging contributions are those from the double-real emission, and in particular, those from double gluon radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' These contributions need first to be integrated over the angles of the emitted partons by keeping their total momentum k fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Then, the remaining integrals have been evaluated by splitting them into a singular and a regular part as k2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' For some of the contributions, the latter has been evaluated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' After checking the cancellation of the ϵ poles, the complete results for the final remainders are provided through a numerical code that is attached to the arXiv distribution of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Together with the results already available in the literature, the soft-parton contributions presented in this paper complete the evaluation at NNLO of the azimuthally-averaged transverse momentum resummation formula for the production of heavy-quark pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' In particular, the results can straightforwardly be implemented to carry out fully differential NNLO calculations for the production of a pair of heavy quarks with arbitrary mass by using the qT subtraction formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' Acknowledgments This work is supported in part by the Swiss National Science Foundation (SNF) under contract 200020 188464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFKT4oBgHgl3EQfVC14/content/2301.11786v1.pdf'} +page_content=' 61 References [1] P.' 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a/bNE_T4oBgHgl3EQfzRxG/content/tmp_files/2301.08322v1.pdf.txt b/bNE_T4oBgHgl3EQfzRxG/content/tmp_files/2301.08322v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a01a18083457e901d579b466c961f004c2e4c0d --- /dev/null +++ b/bNE_T4oBgHgl3EQfzRxG/content/tmp_files/2301.08322v1.pdf.txt @@ -0,0 +1,720 @@ +arXiv:2301.08322v1 [gr-qc] 19 Jan 2023 +Gravitational Baryogenesis: Problems and Possible +Resolution +Presented at 6th International Conference on Particle Physics and Astrophysics (ICCPA-2022) +E. Arbuzovaa,b, A. Dolgovb,c, K. Duttad, R. Rangarajane +January 23, 2023 +aDepartment of Higher Mathematics, Dubna State University, Universitetskaya st. 19, +Dubna, Moscow region 141983, Russia; +bDepartment of Physics, Novosibirsk State University, +Pirogova st. 2, Novosibirsk, 630090 Russia +cBogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, +Joliot-Curie st. 6, Dubna, Moscow region, 141980 Russia +dDepartment of Physical Sciences, Indian Institute of Science Education and Research +Kolkata, Mohanpur 741246, India; +e Mathematical and Physical Sciences Division, School of Arts and Sciences, Ahmedabad +University, Navrangpura, Ahmedabad 380009, India. +Abstract +The coupling of baryonic current to the derivative of the curvature scalar, R, +inherent to gravitational baryogenesis (GBG), leads to a fourth order differential +equation of motion for R instead of the algebraic one of General Relativity (GR). +The fourth-order differential equation is generically unstable. We consider a possi- +ble mechanism of stabilization of GBG by modification of gravity, introducing an +R2-term into the canonical action of GR. It is shown that this mechanism allows for +stabilization of GBG with bosonic and fermionic baryon currents. We have estab- +lished the region of the model parameters leading to stabilization of R. Still, the +standard cosmology would be noticeably modified. + +1 +Introduction +An excess of matter over antimatter in our Universe is crucial for our very existence and +is well supported by various observations. The local Universe is clearly matter dominated. +The amount of antimatter is very small and it can be explained as the result of high energy +collisions in space. On the other hand, matter and antimatter seem to have similar prop- +erties, therefore we could expect a matter-antimatter symmetric universe. The existence +of large regions of antimatter in our neighbourhood would produce high energy radiation +created by matter-antimatter annihilation on the boundaries between matter and anti- +matter domains, which is not observed. A satisfactory model of our Universe should be +able to explain the origin of the matter-antimatter asymmetry. Any initial asymmetry at +inflation could not solve the problem of observed excess of matter over antimatter, because +the energy density associated with the observed non-zero baryonic number density would +not allow for sufficiently long inflation. +The term baryogenesis is used to indicate the generation of the excess of matter +(baryons) over antimatter (antibaryons) or vice versa. +In 1967 Andrey Sakharov formulated three conditions today know as Sakharov Prin- +ciples [1], necessary to produce a matter-antimatter asymmetry in the initially symmetric +universe. These conditions include: +1. Non-conservation of baryonic number; +2. Breaking of symmetry between particles and antiparticles; +3. Deviation from thermal equilibrium. +However, not all of three Sakharov Principles are strictly necessary. For example, sponta- +neous baryogenesis (SBG) and gravitational bayogenesis (GBG) do not demand an explicit +C and CP violation and can proceed in thermal equilibrium. Moreover, these mechanisms +are usually most efficient in thermal equilibrium. +The statement that the cosmological baryon asymmetry can be created by spontaneous +baryogenesis in thermal equilibrium was mentioned in the original paper by A. Cohen +and D. Kaplan in 1987 [2] and in the subsequent papers by A. Cohen, D. Kaplan, and +A. Nelson [3,4] (for a review see [5–8]). +The term ”spontaneous” is related to spontaneous breaking of underlying symmetry of +the theory, which ensures the conservation of the total baryonic number in the unbroken +phase. This symmetry is supposed to be spontaneously broken and in the broken phase +the Lagrangian density acquires the term +LSBG = (∂µθ)Jµ +B , +(1) +where θ is a (pseudo) Goldstone field, and Jµ +B is the baryonic current of matter fields, +which becomes non-conserved as a result of the symmetry breaking. +For a spatially homogeneous field, θ = θ(t), the Lagrangian is reduced to a simple form +LSBG = ˙θ nB , nB ≡ J0 +B. +(2) +1 + +Here nB is the baryonic number density, so it is tempting to identify ˙θ with the chemical +potential, µB, of the corresponding system. However, such an identification is questionable +[9, 10]. It depends upon the representation chosen for the fermionic fields and is heavily +based on the assumption ˙θ ≈ const. In Ref. [9] the assumption ˙θ ≈ const is relaxed. +Stimulated by spontaneous baryogenesis the idea of gravitational baryogenesis was put +forward [11]. The scenario of SBG was modified by the introduction of the coupling of the +baryonic current to the derivative of the curvature scalar R: +SGBG = − 1 +M2 +� +d4x√−g (∂µR)Jµ +B , +(3) +where g is the determinant of the space-time metric tensor and the mass parameter M +determines the energy scale of baryogenesis. There are a lot of articles on the subject, and +a partial list of references is included in Refs. [12–16]. According to these papers, the GBG +mechanism can successfully explain the magnitude of the cosmological baryon asymmetry +of the universe. +However, it was argued in Refs. [17,18], that the back reaction of the created non-zero +baryonic density on the space-time curvature leads to strong instability of the cosmological +evolution. In this paper we show that the problem of stability can be solved by adding +to the Hilbert-Einstein action the quadratic in curvature term generated by quantum +corrections [19,20]. The underlying gravitational action has the form: +SGrav = −M2 +P l +16π +� +d4x √−g +� +R − R2 +6M2 +R +� +, +(4) +where MP l = 1.22 · 1019 GeV is the Planck mass, and we use the metric signature +(+, −, −, −). As is known, the R2-term leads to excitation of the scalar degree of freedom, +named scalaron, and MR is the scalaron mass. In the very early universe the R2-term can +generate inflation [21], and density perturbations. The amplitude of the observed den- +sity perturbations demands that MR = 3 · 1013 GeV [22] if the scalaron is the inflaton. +Otherwise MR > 3 · 1013 GeV is allowed. +Below we presume that the scalaron is the +inflaton. +2 +Instability problem of gravitational baryogenesis +The essential ingredient of the spontaneous baryogenesis is the coupling of the baryonic +current the derivative of the curvature scalar ∂µR (3). Taken over canonical cosmological +Friedmann-Lemaitre-Robertson-Walker background, this interaction can successfully fulfil +the task of generating the proper value of the baryon asymmetry of the universe. +However, any curvature dependent term in the Lagrangian of the theory would mod- +ify the equations of the General Relativity (GR). The modified GR equations have been +analysed in Refs. [9, 18]. Since interaction (3) is not just linear in the curvature term +multiplied by a constant, it leads to higher order equations describing evolution of grav- +itational fields. Higher order equations of motion are typically unstable with respect to +small perturbations. According to the results of Refs. [9, 18], it indeed happens in the +2 + +frameworks of the SBG scenario and the characteristic time of the exponential instability +is much shorter than the cosmological time. It creates serious problem for realisation of +the SBG mechanism. +In this work we suggest to consider possible stabilisation of SBG and have proved that +it can be realised but the resulting cosmological model suffers from too large value of R, +much larger than that in the classical Friedmann cosmology. Possible ways to cure this +shortcoming are mentioned. +3 +Stabilisation of gravitational baryogenesis in modi- +fied gravity +3.1 +Bosonic case. +Let us first consider the case when baryonic number is carried by a complex scalar field +φ [17]. The total action has the form: +Stot[φ] = − +� +d4x √−g +�M2 +P l +16π +� +R − R2 +6M2 +R +� ++ +1 +M2(∂µR)Jµ +(φ) − gµν∂µφ ∂νφ∗ + U(φ, φ∗) +� ++Smatt ,(5) +where U(φ, φ∗) is the potential of field φ and Smatt is the matter action which does not +include the field φ. In Eq. (5) R(t) is the classical curvature field, while φ(⃗x, t) is the +quantum operator of light scalar particles. +We assume that the potential U(φ, φ∗) is not invariant with respect to phase transfor- +mation φ → exp (iqβ)φ and thus the corresponding current +Jµ +(φ) = iq gµν(φ∗∂νφ − φ∂νφ∗) +(6) +is not conserved. Here q is the baryonic number of field φ. The non-conservation of the +current is necessary for the proper performance of the model, otherwise SGBG in Eq. (3) +can be integrated away by parts. +Varying action (5) over gµν we come to the following equations: +M2 +P l +16π +� +Rµν − 1 +2gµνR − +1 +3M2 +R +� +Rµν − 1 +4gµνR + gµνD2 − DµDν +� +R +� +− 1 +M2 +�� +Rµν − (DµDν − gµνD2) +� +DαJα +(φ) + 1 +2gµνJα +(φ) DαR − 1 +2 +� +J(φ)νDµR + J(φ)µDνR +�� +−1 +2 (Dµφ Dνφ∗ + Dνφ Dµφ∗) + 1 +2gµν [Dαφ Dαφ∗ − U(φ)] −(Dµφ)(Dνφ∗) += 1 +2 T (matt) +µν +, +(7) +where Dµ is the covariant derivative in metric gµν (of course, for scalars Dµ = ∂µ) and +T (matt) +µν +is the energy-momentum tensor of matter obtained from action Smatt. +3 + +Taking the trace of equation (7) with respect to µ and ν and changing sign we obtain: +M2 +P l +16π +� +R + +1 +M2 +R +D2R +� ++ +1 +M2 +� +(R + 3D2)DαJα +(φ) + Jα +(φ) DαR +� +− Dαφ Dαφ∗ + 2U(φ) += −1 +2 T (matt)= 0 , (8) +where T (matt) = gµνT (matt) +µν +is the trace of the energy-momentum tensor of matter. For +the usual relativistic matter T (matt) = 0, while for scalar field φ the trace of the energy- +momentum tensor is nonzero: +T µ +µ (φ) = −2Dαφ Dαφ∗ + 4U(φ). +(9) +The equation of motion for field φ is: +D2φ + ∂U +∂φ∗ = − iq +M2 +� +2DµR Dµφ + φD2R +� +. +(10) +According to definition (6) and Eq. (10), the current divergence is: +DµJµ = 2q2 +M2 +� +DµR (φ∗Dµφ + φDµφ∗) + |φ|2D2R +� ++ iq +� +φ∂U +∂φ − φ∗ ∂U +∂φ∗ +� +. +(11) +For homogeneous curvature scalar R(t) in spatially flat FLRW-metric +ds2 = dt2 − a2(t)dr2 +(12) +Eq. (8) is reduced to: +M2 +P l +16π +� +R + +1 +M2 +R +(∂2 +t + 3H∂t)R +� ++ +1 +M2 +� +(R + 3∂2 +t + 9H∂t)DαJα +(φ) + ˙R J0 +(φ) +� ++2U(φ) − (Dαφ)(Dαφ∗) = 0. +(13) +where J0 +(φ) is the baryonic number density of the φ-field, H = ˙a/a is the Hubble parameter, +and the divergence of the current is given by the expression: +DαJα +(φ) = 2q2 +M2 +� +˙R (φ∗ ˙φ + φ ˙φ∗) + ( ¨R + 3H ˙R) φ∗φ +� ++ iq +� +φ∂U +∂φ − φ∗ ∂U +∂φ∗ +� +. +(14) +As we see in what follows, the last two terms in Eq. (13) do not have an essential impact +on the cosmological instability found in Ref. [17] and will be disregarded below. +Let us note that the statement of exponential instability of R(t) [17] does not de- +pend on the conservation or non-conservation of the current from the potential term +(φ∂U/∂φ − φ∗∂U/∂φ∗) in Eq. (14). However if the current from this term is conserved then +the baryon asymmetry is not generated. On the other hand the term in square brackets +in Eq. (14) does not lead to generation of the baryon asymmetry but leads to exponential +instability of R(t). Below we ignore the last term of Eq. (14). +4 + +Performing thermal averaging of the normal ordered bilinear products of field φ in the +high temperature limit (see Appendix of Ref. [17]) in accordance with equations: +⟨φ∗φ⟩ = T 2 +12 , +⟨φ∗ ˙φ + ˙φ∗φ⟩ = 0 , +(15) +and using Eq. (14) we obtain the fourth order differential equation: +M2 +P l +16π +� +R + +1 +M2 +R +D2R +� ++ +q2 +6M4 +� +R + 3∂2 +t + 9H∂t +� �� +¨R + 3H ˙R +� +T 2� ++ +1 +M2 ˙R ⟨J0 +(φ)⟩ += −2U(φ) + (Dαφ)(Dαφ∗). (16) +Here ⟨J0 +(φ)⟩ is the thermal average value of the baryonic number density of φ, which is +supposed to vanish initially, but created through the process of the gravitational baryoge- +nesis. This term can be neglected because the baryon asymmetry is normally quite small. +Even if it is not small it does not have considerable impact on the explosive rise of the +curvature scalar. As we see in what follows the evolution of R(t) proceeds much faster +than the cosmological evolution, that is ¨R/ ˙R ≫ H. Consequently, we neglect the terms +proportional to R with respect to the terms proportional to the second derivative of R, ¨R. +We also consider the terms of the type HR as small w.r.t. to dR/dt. We can check that +this presumption is true a posteriori with the obtained solution for R(t). +Keeping only the dominant terms we simplify the above equation to: +d4R +dt4 + κ4 +M2 +R +d2R +dt2 + κ4R = −T µ +µ (φ) M4 +q2T 2, +(17) +where +κ4 = M2 +P lM4 +8πq2T 2 . +(18) +While studying the instability of the solution we do not take into account the r.h.s. of +Eq. (17) which does not depend upon R. Looking for the solution of Eq. (17) in the form +R = Rin exp(λt), we obtain the characteristic equation: +λ4 + κ4 +M2 +R +λ2 + κ4 = 0 +(19) +with the eigenvalues λ defined by the expression: +λ2 = − κ4 +2M2 +R +± κ2 +� +κ4 +4M4 +R +− 1. +(20) +There is no instability, if λ2 < 0 and Eq. (17) has only oscillating solutions. It is +realised, if κ4 > 4M4 +R. Using the expression in Eq. (18) for κ4 and taking MR = 3 · 1013 +GeV we find the stability condition: +M > 3 · 104 GeV +� q T +GeV +�1/2 +, +(21) +5 + +which is fulfilled for all interesting values of M. +The value of λ depends upon the relation between κ and MR. If κ ∼ MR then the +frequency of the oscillations of curvature is of the order of MR and |λ| ∼ MR. If κ ≫ MR +then there are two possible solutions |λ| ∼ MR and +|λ| ∼ κ(κ/MR) ≫ MR. +High +frequency oscillations of R would lead to efficient gravitational particle production and, as +a result, to damping of the oscillations. +3.2 +Fermionic case +In this section we consider the case when baryonic number is carried by fermions. The +gravitational part of the action has the form as in Eq. (4), while the fermionic part of the +action is the same as in Refs. [10,18]: +L[Q, L] = i +2( ¯Qγµ∇µQ − ∇µ ¯Q γµQ) − mQ ¯Q Q ++ i +2(¯Lγµ∇µL − ∇µ ¯LγµL) − mL ¯L L ++ +g +m2 +X +� +( ¯Q Qc)( ¯QL) + ( ¯QcQ)(¯LQ) +� ++ d +M2(∂µR)Jµ + Lmatt , +(22) +where Q is the quark-like field with non-zero baryonic number BQ, Qc is the charged con- +jugated quark operator, L is another fermionic field (lepton),, and ∇µ is the covariant +derivative of the Dirac fermions in tetrad formalism. The quark current is Jµ = BQ ¯QγµQ +with γµ being the curved space gamma-matrices, and Lmatt describes all other forms of +matter. The four-fermion interaction between quarks and leptons is introduced to ensure +the necessary non-conservation of the baryon number with mX being a constant parameter +with dimension of mass and g being a dimensionless coupling constant. In the term, de- +scribing interaction of the baryonic current of fermions with the derivative of the curvature +scalar, M is a constant parameter with dimension of mass and d = ±1 is dimensionless +coupling constant which is introduced to allow for an arbitrary sign of the above expression. +Gravitational equations of motion with an account of R2/M2 +R-term in analogy with +Eq. (7) take the form: +M2 +P l +8π +� +Rµν − 1 +2gµνR − +1 +3M2 +R +� +Rµν − 1 +4gµνR + gµνD2 − DµDν +� +R +� += gµν +2 +g +m2 +X +� +( ¯Q Qc)( ¯QL) + ( ¯QcQ)(¯LQ) +� ++ i +4 +� ¯Q(γµ∇ν + γν∇µ)Q − (∇ν ¯Q γµ + ∇µ ¯Q γν)Q +� ++ i +4 +�¯L(γµ∇ν + γν∇µ)L − (∇ν ¯L γµ + ∇µ ¯L γν)L +� +− 2d +M2 +� +Rµν + gµνD2 − DµDν +� +DαJα + +d +2M2 (Jµ∂νR + Jν∂µR) + T matt +µν +. +(23) +6 + +Taking the trace of Eq. (23) with an account of fermion equations of motion we obtain: +−M2 +P l +8π +� +R + +1 +M2 +R +D2R +� += mQ ¯QQ + mL ¯LL + 2g +m2 +X +� +( ¯Q Qc)( ¯QL) + ( ¯QcQ)(¯LQ) +� +− 2d +M2(R + 3D2)DαJα + Tmatt , +(24) +where Tmatt is the trace of the energy momentum tensor of all other fields. In the early +universe when various species are relativistic, we can take Tmatt = 0. The average expec- +tation value of the quark-lepton interaction term proportional to g is also small, so the +contribution of all matter fields may be neglected and hence the only term which remains +in the r.h.s. of Eq. (24) is that proportional to DαJα. +A higher order differential equation for R is obtained after we substitute the current +divergence, DαJα, calculated from the kinetic equation in the external field R [18], into +Eq. (24). For the spatially homogeneous case +DαJα = (∂t + 3H)nB = Icoll +B , +(25) +where the collision integral, Icoll +B , in the lowest order of perturbation theory is equal to: +Icoll +B += −3Bq(2π)4 +� +dνq1,q2 dν¯q3,l4δ4(q1 + q2 − q3 − l4) +� +|A(q1 + q2 → ¯q3 + l4)|2fq1fq2 − |A(¯q3 + l4 → q1 + q2)|2f¯q3fl4 +� +. +(26) +Here A(a → b) is the amplitude of the transition from state a to state b, BQ is the baryonic +number of quark, fa is the phase space distribution (the occupation number), and +dνq1,q2 = +d3q1 +2Eq1(2π)3 +d3q2 +2Eq2(2π)3, +(27) +where Eq = +� +q2 + m2 is the energy of particle with three-momentum q and mass m. The +element of phase space of final particles, dν¯q3,l4, is defined analogously. +We choose such representation of the quark operator, Q, for which the interaction +of baryonic current with the derivative of the curvature scalar in Eq. (22) vanishes but +reappears in the quark-lepton interaction term: +2g +m2 +X +� +e−3idBQR/M2 ( ¯Q Qc)( ¯QL) + e3idBQR/M2 ( ¯QcQ)(¯LQ) +� +. +(28) +We make the simplifying assumption that the evolution of R can be approximately de- +scribed by the law +R(t) ≈ R(t0) + (t − t0) ˙R. +(29) +We assume that ˙R(t) slowly changes at the characteristic time scale of the reactions, which +contribute to the collision integral (26), and so we can approximately take ˙R ≈ const. +According to the rules of quantum field theory the reaction probability is given by the +square of the integral over space and time of the amplitude of the corresponding process. In +7 + +the case of time independent interaction it leads to the energy conservation, ΣEin = ΣEfin. +If the interaction depends upon time the energy evidently is non-conserved and in our case, +e.g. for the reaction q1 + q2 → ¯q3 + l4, the energy balance has the form: +E(q1) + E(q2) = E(q3) + E(l4) + 3dBQ ˙R/M2. +(30) +In kinetic equilibrium the phase space distribution of fermions has the form +f = +1 +e(E/T−ξ) + 1 ≈ e−E/T+ξ, +(31) +where ξ = µ/T is the dimensionless chemical potential, different for quarks, ξq, and leptons, +ξl. +In thermal equilibrium case the condition of conservation of chemical potentials is +fulfilled, that is Σ ξin = Σ ξfin. In particular it demands that chemical potentials of particles +and antiparticles are equal by magnitude and have opposite signs: ξ = −¯ξ, as follows +e.g. from the consideration of particle-antiparticle annihilation into +different numbers +of photons. If energy is not conserved, due to time-dependent R(t), the conservation of +chemical potentials is also broken, as we see in what follows. +We assume that ξ ≪ 1 and hence distribution (31) turns into: +f ≈ e−E/T(1 + ξ). +(32) +We also assume that 3d BQ ˙R/(M2 T) ≪ 1 and correspondingly the balance of chemical +potentials in equilibrium for the reactions q1 + q2 ↔ ¯q3 + l4 leads to: +3ξq − ξl − 3d BQ ˙R(t) +M2 T += 0. +(33) +Following Ref. [18], we express +nB ≈ gsBQ +6 +ξqT 3, +(34) +where gs is the number of quark spin states. Since we are studying instability of R whose +timescale is presumed to be much smaller than the expansion rate of the Universe, we +approximate +DαJα ≈ ˙nB ≈ gsBQ +6 +˙ξqT 3 +(35) +≈ gsBQ +6 +˙ξeq +q T 3, +(36) +ξeq +q is obtained from Eq. (33), using the conservation of the sum of baryonic and leptonic +numbers which implies ξl = −ξq/3. Then +ξeq +q = 9d BQ ˙R(t) +10M2 T +. +(37) +8 + +Substituting Eq. (37) in Eq. (36) and neglecting the ˙T-term, Eq. (24) gives the following +fourth order differential equation for the curvature scalar: +d4R +dt4 + κ4 +f +M2 +R +d2R +dt2 + κ4 +fR = 0, +(38) +where +κ4 +f = +5M2 +P lM4 +36πgsB2 +QT 2 . +(39) +Once again, we consider terms containing R as small with respect to the terms containing +¨R. The value of κf is only slightly numerically different from κ in Eq. (18) and has the +same dependence upon the essential parameters, so the solutions of Eqs. (17) and (38) +practically coincide. +4 +Discussion +We have shown that discovered in Refs. [17, 18] exponential instability of the curvature +scalar inherent to the mechanism of spontaneous baryogenesis can be successfully cured +in modified gravity. The special form of gravity modification by introduction of R2-term +into canonical Hilbert-Einstein action of General Relativity was explored as a workable +mechanism. +However, the stabilized asymptotic value of R is extremely large and together with +possibly successful baryogenesis would still strongly perturb canonical cosmology. Possible +ways out of this problem could either be a more complicated model of F(R) gravity or a +proper account of particle production created by high frequency oscillations of R(t). Both +options open interesting possibilities for future research. +References +[1] Sakharov A. D. Violation of CP Invariance, C asymmetry, and baryon asymmetry +of the universe. Pisma Zh. Eksp. Teor. Fiz. 1967, 5, 32–35. +[2] Cohen A. G. and Kaplan D. B. Thermodynamic Generation of the Baryon Asym- +metry. Phys. Lett. B 1987, 199, 251–258. +[3] Cohen A. G. and Kaplan D. B. Spontaneous Baryogenesis. Nucl. Phys. B 1988, +n308, 913–928. +[4] Cohen A. G., Kaplan D. B and Nelson A. E. Spontaneous baryogenesis at the weak +phase transition. Phys. Lett. B 1991, 263, 86–92. +[5] Dolgov A. D. NonGUT baryogenesis. Phys. Rept. 1992, 222 (1992), 309–386. +9 + +[6] Rubakov V. A. and Shaposhnikov M. E. Electroweak baryon number nonconservation +in the early universe and in high-energy collisions. Usp. Fiz. Nauk 1996, 166, 493– +537. +[7] Riotto A. and Trodden M. Recent progress in baryogenesis. Ann. Rev. Nucl. Part. +Sci. 1999, 49, 35–75. +[8] Dolgov A.D. Baryogenesis, 30 years after. Surveys in High Energy Physics 1998 13, +83. +[9] Arbuzova E. V., Dolgov A.D. and Novikov V. A. General properties and kinetics of +spontaneous baryogenesis. Phys. Rev. D 2016, 94, no.12, 123501. +[10] Dasgupta A., Jain R. K. and Rangarajan R., Effective chemical potential in sponta- +neous baryogenesis. Phys. Rev. D 2018 98, no.8, 083527. +[11] Davoudiasl H., R. Kitano R., Kribs G. D., H. Murayama H., Steinhardt P. J. Grav- +itational baryogenesis. Phys. Rev. Lett. 2004, 93, 201301. +[12] Lambiase G. and Scarpetta G. Baryogenesis in f(R): Theories of Gravity. Phys. Rev. +D 2006, 74, 087504. +[13] Sadjadi H. M. A Note on Gravitational Baryogenesis. Phys. Rev. D 2007, 76, 123507. +[14] Lambiase G., Mohanty S. and Pizza L. Consequences of f(R)-theories of gravity on +gravitational leptogenesis. Gen. Rel. Grav. 2013, 45, 1771–1785. +[15] Fukushima M., Mizuno S. and Maeda K. i. Gravitational Baryogenesis after +Anisotropic Inflation. Phys. Rev. D 2016 93, no.10, 103513. +[16] Odintsov +S. +D. +and +Oikonomou +V. +K. +Gauss–Bonnet +gravitational +baryo- +genesis. Phys. Lett. B 2016, 760, 259–262. doi:10.1016/j.physletb.2016.06.074 +[arXiv:1607.00545 [gr-qc]]. +[17] Arbuzova E. V. and Dolgov A.D. Intrinsic problems of the gravitational baryogenesis. +Phys. Lett. B 2017, 769, 171–175. +[18] Arbuzova E. V. and Dolgov A.D. Instability of gravitational baryogenesis with +fermions. JCAP 2017, 06, 001. +[19] Ginzburg V. L., Kirzhnits D. A. and Lyubushin A. A. The role of quantum fluctua- +tions of the gravitational fields in general relativity theory and cosmology. Zh. Eksp. +Teor. Fiz. 1971, 60, 451–459. +[20] Gurovich V. T. and Starobinsky A. A. Quantum effects and regular cosmological +models. Sov. Phys. JETP 1979, 50, 844–852. +[21] Starobinsky A. A. A New Type of Isotropic Cosmological Models Without Singular- +ity. Phys. Lett. B 1980, 91, 99–102. +10 + +[22] Gorbunov D. S. and Panin A. G. Free scalar dark matter candidates in R2-inflation: +the light, the heavy and the superheavy. Phys. Lett. B 2012 718, 15–20. +11 + diff --git a/bNE_T4oBgHgl3EQfzRxG/content/tmp_files/load_file.txt b/bNE_T4oBgHgl3EQfzRxG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e4f18e0579c8a5ab5a587321b56807f5dcfbf30 --- /dev/null +++ b/bNE_T4oBgHgl3EQfzRxG/content/tmp_files/load_file.txt @@ -0,0 +1,384 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf,len=383 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='08322v1 [gr-qc] 19 Jan 2023 Gravitational Baryogenesis: Problems and Possible Resolution Presented at 6th International Conference on Particle Physics and Astrophysics (ICCPA-2022) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Arbuzovaa,b, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Dolgovb,c, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Duttad, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Rangarajane January 23, 2023 aDepartment of Higher Mathematics, Dubna State University, Universitetskaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 19, Dubna, Moscow region 141983, Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' bDepartment of Physics, Novosibirsk State University, Pirogova st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 2, Novosibirsk, 630090 Russia cBogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 6, Dubna, Moscow region, 141980 Russia dDepartment of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' e Mathematical and Physical Sciences Division, School of Arts and Sciences, Ahmedabad University, Navrangpura, Ahmedabad 380009, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Abstract The coupling of baryonic current to the derivative of the curvature scalar, R, inherent to gravitational baryogenesis (GBG), leads to a fourth order differential equation of motion for R instead of the algebraic one of General Relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The fourth-order differential equation is generically unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We consider a possi- ble mechanism of stabilization of GBG by modification of gravity, introducing an R2-term into the canonical action of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' It is shown that this mechanism allows for stabilization of GBG with bosonic and fermionic baryon currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We have estab- lished the region of the model parameters leading to stabilization of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Still, the standard cosmology would be noticeably modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 1 Introduction An excess of matter over antimatter in our Universe is crucial for our very existence and is well supported by various observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The local Universe is clearly matter dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The amount of antimatter is very small and it can be explained as the result of high energy collisions in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' On the other hand, matter and antimatter seem to have similar prop- erties, therefore we could expect a matter-antimatter symmetric universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The existence of large regions of antimatter in our neighbourhood would produce high energy radiation created by matter-antimatter annihilation on the boundaries between matter and anti- matter domains, which is not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' A satisfactory model of our Universe should be able to explain the origin of the matter-antimatter asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Any initial asymmetry at inflation could not solve the problem of observed excess of matter over antimatter, because the energy density associated with the observed non-zero baryonic number density would not allow for sufficiently long inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The term baryogenesis is used to indicate the generation of the excess of matter (baryons) over antimatter (antibaryons) or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In 1967 Andrey Sakharov formulated three conditions today know as Sakharov Prin- ciples [1], necessary to produce a matter-antimatter asymmetry in the initially symmetric universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' These conditions include: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Non-conservation of baryonic number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Breaking of symmetry between particles and antiparticles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Deviation from thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' However, not all of three Sakharov Principles are strictly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' For example, sponta- neous baryogenesis (SBG) and gravitational bayogenesis (GBG) do not demand an explicit C and CP violation and can proceed in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Moreover, these mechanisms are usually most efficient in thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The statement that the cosmological baryon asymmetry can be created by spontaneous baryogenesis in thermal equilibrium was mentioned in the original paper by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Cohen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Kaplan in 1987 [2] and in the subsequent papers by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Cohen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Kaplan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Nelson [3,4] (for a review see [5–8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The term ”spontaneous” is related to spontaneous breaking of underlying symmetry of the theory, which ensures the conservation of the total baryonic number in the unbroken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' This symmetry is supposed to be spontaneously broken and in the broken phase the Lagrangian density acquires the term LSBG = (∂µθ)Jµ B , (1) where θ is a (pseudo) Goldstone field, and Jµ B is the baryonic current of matter fields, which becomes non-conserved as a result of the symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' For a spatially homogeneous field, θ = θ(t), the Lagrangian is reduced to a simple form LSBG = ˙θ nB , nB ≡ J0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (2) 1 Here nB is the baryonic number density, so it is tempting to identify ˙θ with the chemical potential, µB, of the corresponding system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' However, such an identification is questionable [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' It depends upon the representation chosen for the fermionic fields and is heavily based on the assumption ˙θ ≈ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [9] the assumption ˙θ ≈ const is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Stimulated by spontaneous baryogenesis the idea of gravitational baryogenesis was put forward [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The scenario of SBG was modified by the introduction of the coupling of the baryonic current to the derivative of the curvature scalar R: SGBG = − 1 M2 � d4x√−g (∂µR)Jµ B , (3) where g is the determinant of the space-time metric tensor and the mass parameter M determines the energy scale of baryogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' There are a lot of articles on the subject, and a partial list of references is included in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' According to these papers, the GBG mechanism can successfully explain the magnitude of the cosmological baryon asymmetry of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' However, it was argued in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [17,18], that the back reaction of the created non-zero baryonic density on the space-time curvature leads to strong instability of the cosmological evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In this paper we show that the problem of stability can be solved by adding to the Hilbert-Einstein action the quadratic in curvature term generated by quantum corrections [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The underlying gravitational action has the form: SGrav = −M2 P l 16π � d4x √−g � R − R2 6M2 R � , (4) where MP l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='22 · 1019 GeV is the Planck mass, and we use the metric signature (+, −, −, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' As is known, the R2-term leads to excitation of the scalar degree of freedom, named scalaron, and MR is the scalaron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In the very early universe the R2-term can generate inflation [21], and density perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The amplitude of the observed den- sity perturbations demands that MR = 3 · 1013 GeV [22] if the scalaron is the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Otherwise MR > 3 · 1013 GeV is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Below we presume that the scalaron is the inflaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 2 Instability problem of gravitational baryogenesis The essential ingredient of the spontaneous baryogenesis is the coupling of the baryonic current the derivative of the curvature scalar ∂µR (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Taken over canonical cosmological Friedmann-Lemaitre-Robertson-Walker background, this interaction can successfully fulfil the task of generating the proper value of the baryon asymmetry of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' However, any curvature dependent term in the Lagrangian of the theory would mod- ify the equations of the General Relativity (GR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The modified GR equations have been analysed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [9, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Since interaction (3) is not just linear in the curvature term multiplied by a constant, it leads to higher order equations describing evolution of grav- itational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Higher order equations of motion are typically unstable with respect to small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' According to the results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [9, 18], it indeed happens in the 2 frameworks of the SBG scenario and the characteristic time of the exponential instability is much shorter than the cosmological time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' It creates serious problem for realisation of the SBG mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In this work we suggest to consider possible stabilisation of SBG and have proved that it can be realised but the resulting cosmological model suffers from too large value of R, much larger than that in the classical Friedmann cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Possible ways to cure this shortcoming are mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 3 Stabilisation of gravitational baryogenesis in modi- fied gravity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='1 Bosonic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Let us first consider the case when baryonic number is carried by a complex scalar field φ [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The total action has the form: Stot[φ] = − � d4x √−g �M2 P l 16π � R − R2 6M2 R � + 1 M2(∂µR)Jµ (φ) − gµν∂µφ ∂νφ∗ + U(φ, φ∗) � +Smatt ,(5) where U(φ, φ∗) is the potential of field φ and Smatt is the matter action which does not include the field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (5) R(t) is the classical curvature field, while φ(⃗x, t) is the quantum operator of light scalar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We assume that the potential U(φ, φ∗) is not invariant with respect to phase transfor- mation φ → exp (iqβ)φ and thus the corresponding current Jµ (φ) = iq gµν(φ∗∂νφ − φ∂νφ∗) (6) is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Here q is the baryonic number of field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The non-conservation of the current is necessary for the proper performance of the model, otherwise SGBG in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (3) can be integrated away by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Varying action (5) over gµν we come to the following equations: M2 P l 16π � Rµν − 1 2gµνR − 1 3M2 R � Rµν − 1 4gµνR + gµνD2 − DµDν � R � − 1 M2 �� Rµν − (DµDν − gµνD2) � DαJα (φ) + 1 2gµνJα (φ) DαR − 1 2 � J(φ)νDµR + J(φ)µDνR �� −1 2 (Dµφ Dνφ∗ + Dνφ Dµφ∗) + 1 2gµν [Dαφ Dαφ∗ − U(φ)] −(Dµφ)(Dνφ∗) = 1 2 T (matt) µν ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (7) where Dµ is the covariant derivative in metric gµν (of course,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' for scalars Dµ = ∂µ) and T (matt) µν is the energy-momentum tensor of matter obtained from action Smatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 3 Taking the trace of equation (7) with respect to µ and ν and changing sign we obtain: M2 P l 16π � R + 1 M2 R D2R � + 1 M2 � (R + 3D2)DαJα (φ) + Jα (φ) DαR � − Dαφ Dαφ∗ + 2U(φ) = −1 2 T (matt)= 0 , (8) where T (matt) = gµνT (matt) µν is the trace of the energy-momentum tensor of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' For the usual relativistic matter T (matt) = 0, while for scalar field φ the trace of the energy- momentum tensor is nonzero: T µ µ (φ) = −2Dαφ Dαφ∗ + 4U(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (9) The equation of motion for field φ is: D2φ + ∂U ∂φ∗ = − iq M2 � 2DµR Dµφ + φD2R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (10) According to definition (6) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (10), the current divergence is: DµJµ = 2q2 M2 � DµR (φ∗Dµφ + φDµφ∗) + |φ|2D2R � + iq � φ∂U ∂φ − φ∗ ∂U ∂φ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (11) For homogeneous curvature scalar R(t) in spatially flat FLRW-metric ds2 = dt2 − a2(t)dr2 (12) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (8) is reduced to: M2 P l 16π � R + 1 M2 R (∂2 t + 3H∂t)R � + 1 M2 � (R + 3∂2 t + 9H∂t)DαJα (φ) + ˙R J0 (φ) � +2U(φ) − (Dαφ)(Dαφ∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (13) where J0 (φ) is the baryonic number density of the φ-field, H = ˙a/a is the Hubble parameter, and the divergence of the current is given by the expression: DαJα (φ) = 2q2 M2 � ˙R (φ∗ ˙φ + φ ˙φ∗) + ( ¨R + 3H ˙R) φ∗φ � + iq � φ∂U ∂φ − φ∗ ∂U ∂φ∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (14) As we see in what follows, the last two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (13) do not have an essential impact on the cosmological instability found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [17] and will be disregarded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Let us note that the statement of exponential instability of R(t) [17] does not de- pend on the conservation or non-conservation of the current from the potential term (φ∂U/∂φ − φ∗∂U/∂φ∗) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' However if the current from this term is conserved then the baryon asymmetry is not generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' On the other hand the term in square brackets in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (14) does not lead to generation of the baryon asymmetry but leads to exponential instability of R(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Below we ignore the last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 4 Performing thermal averaging of the normal ordered bilinear products of field φ in the high temperature limit (see Appendix of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [17]) in accordance with equations: ⟨φ∗φ⟩ = T 2 12 , ⟨φ∗ ˙φ + ˙φ∗φ⟩ = 0 , (15) and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (14) we obtain the fourth order differential equation: M2 P l 16π � R + 1 M2 R D2R � + q2 6M4 � R + 3∂2 t + 9H∂t � �� ¨R + 3H ˙R � T 2� + 1 M2 ˙R ⟨J0 (φ)⟩ = −2U(φ) + (Dαφ)(Dαφ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (16) Here ⟨J0 (φ)⟩ is the thermal average value of the baryonic number density of φ, which is supposed to vanish initially, but created through the process of the gravitational baryoge- nesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' This term can be neglected because the baryon asymmetry is normally quite small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Even if it is not small it does not have considerable impact on the explosive rise of the curvature scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' As we see in what follows the evolution of R(t) proceeds much faster than the cosmological evolution, that is ¨R/ ˙R ≫ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Consequently, we neglect the terms proportional to R with respect to the terms proportional to the second derivative of R, ¨R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We also consider the terms of the type HR as small w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' to dR/dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We can check that this presumption is true a posteriori with the obtained solution for R(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Keeping only the dominant terms we simplify the above equation to: d4R dt4 + κ4 M2 R d2R dt2 + κ4R = −T µ µ (φ) M4 q2T 2, (17) where κ4 = M2 P lM4 8πq2T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (18) While studying the instability of the solution we do not take into account the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (17) which does not depend upon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Looking for the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (17) in the form R = Rin exp(λt), we obtain the characteristic equation: λ4 + κ4 M2 R λ2 + κ4 = 0 (19) with the eigenvalues λ defined by the expression: λ2 = − κ4 2M2 R ± κ2 � κ4 4M4 R − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (20) There is no instability, if λ2 < 0 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (17) has only oscillating solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' It is realised, if κ4 > 4M4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Using the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (18) for κ4 and taking MR = 3 · 1013 GeV we find the stability condition: M > 3 · 104 GeV � q T GeV �1/2 , (21) 5 which is fulfilled for all interesting values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The value of λ depends upon the relation between κ and MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' If κ ∼ MR then the frequency of the oscillations of curvature is of the order of MR and |λ| ∼ MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' If κ ≫ MR then there are two possible solutions |λ| ∼ MR and |λ| ∼ κ(κ/MR) ≫ MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' High frequency oscillations of R would lead to efficient gravitational particle production and, as a result, to damping of the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='2 Fermionic case In this section we consider the case when baryonic number is carried by fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The gravitational part of the action has the form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (4), while the fermionic part of the action is the same as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [10,18]: L[Q, L] = i 2( ¯Qγµ∇µQ − ∇µ ¯Q γµQ) − mQ ¯Q Q + i 2(¯Lγµ∇µL − ∇µ ¯LγµL) − mL ¯L L + g m2 X � ( ¯Q Qc)( ¯QL) + ( ¯QcQ)(¯LQ) � + d M2(∂µR)Jµ + Lmatt , (22) where Q is the quark-like field with non-zero baryonic number BQ, Qc is the charged con- jugated quark operator, L is another fermionic field (lepton),, and ∇µ is the covariant derivative of the Dirac fermions in tetrad formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The quark current is Jµ = BQ ¯QγµQ with γµ being the curved space gamma-matrices, and Lmatt describes all other forms of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The four-fermion interaction between quarks and leptons is introduced to ensure the necessary non-conservation of the baryon number with mX being a constant parameter with dimension of mass and g being a dimensionless coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In the term, de- scribing interaction of the baryonic current of fermions with the derivative of the curvature scalar, M is a constant parameter with dimension of mass and d = ±1 is dimensionless coupling constant which is introduced to allow for an arbitrary sign of the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Gravitational equations of motion with an account of R2/M2 R-term in analogy with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (7) take the form: M2 P l 8π � Rµν − 1 2gµνR − 1 3M2 R � Rµν − 1 4gµνR + gµνD2 − DµDν � R � = gµν 2 g m2 X � ( ¯Q Qc)( ¯QL) + ( ¯QcQ)(¯LQ) � + i 4 � ¯Q(γµ∇ν + γν∇µ)Q − (∇ν ¯Q γµ + ∇µ ¯Q γν)Q � + i 4 �¯L(γµ∇ν + γν∇µ)L − (∇ν ¯L γµ + ∇µ ¯L γν)L � − 2d M2 � Rµν + gµνD2 − DµDν � DαJα + d 2M2 (Jµ∂νR + Jν∂µR) + T matt µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (23) 6 Taking the trace of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (23) with an account of fermion equations of motion we obtain: −M2 P l 8π � R + 1 M2 R D2R � = mQ ¯QQ + mL ¯LL + 2g m2 X � ( ¯Q Qc)( ¯QL) + ( ¯QcQ)(¯LQ) � − 2d M2(R + 3D2)DαJα + Tmatt , (24) where Tmatt is the trace of the energy momentum tensor of all other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In the early universe when various species are relativistic, we can take Tmatt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The average expec- tation value of the quark-lepton interaction term proportional to g is also small, so the contribution of all matter fields may be neglected and hence the only term which remains in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (24) is that proportional to DαJα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' A higher order differential equation for R is obtained after we substitute the current divergence, DαJα, calculated from the kinetic equation in the external field R [18], into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' For the spatially homogeneous case DαJα = (∂t + 3H)nB = Icoll B , (25) where the collision integral, Icoll B , in the lowest order of perturbation theory is equal to: Icoll B = −3Bq(2π)4 � dνq1,q2 dν¯q3,l4δ4(q1 + q2 − q3 − l4) � |A(q1 + q2 → ¯q3 + l4)|2fq1fq2 − |A(¯q3 + l4 → q1 + q2)|2f¯q3fl4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (26) Here A(a → b) is the amplitude of the transition from state a to state b, BQ is the baryonic number of quark, fa is the phase space distribution (the occupation number), and dνq1,q2 = d3q1 2Eq1(2π)3 d3q2 2Eq2(2π)3, (27) where Eq = � q2 + m2 is the energy of particle with three-momentum q and mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The element of phase space of final particles, dν¯q3,l4, is defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We choose such representation of the quark operator, Q, for which the interaction of baryonic current with the derivative of the curvature scalar in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (22) vanishes but reappears in the quark-lepton interaction term: 2g m2 X � e−3idBQR/M2 ( ¯Q Qc)( ¯QL) + e3idBQR/M2 ( ¯QcQ)(¯LQ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (28) We make the simplifying assumption that the evolution of R can be approximately de- scribed by the law R(t) ≈ R(t0) + (t − t0) ˙R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (29) We assume that ˙R(t) slowly changes at the characteristic time scale of the reactions, which contribute to the collision integral (26), and so we can approximately take ˙R ≈ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' According to the rules of quantum field theory the reaction probability is given by the square of the integral over space and time of the amplitude of the corresponding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In 7 the case of time independent interaction it leads to the energy conservation, ΣEin = ΣEfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' If the interaction depends upon time the energy evidently is non-conserved and in our case, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' for the reaction q1 + q2 → ¯q3 + l4, the energy balance has the form: E(q1) + E(q2) = E(q3) + E(l4) + 3dBQ ˙R/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (30) In kinetic equilibrium the phase space distribution of fermions has the form f = 1 e(E/T−ξ) + 1 ≈ e−E/T+ξ, (31) where ξ = µ/T is the dimensionless chemical potential, different for quarks, ξq, and leptons, ξl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In thermal equilibrium case the condition of conservation of chemical potentials is fulfilled, that is Σ ξin = Σ ξfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' In particular it demands that chemical potentials of particles and antiparticles are equal by magnitude and have opposite signs: ξ = −¯ξ, as follows e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' from the consideration of particle-antiparticle annihilation into different numbers of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' If energy is not conserved, due to time-dependent R(t), the conservation of chemical potentials is also broken, as we see in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' We assume that ξ ≪ 1 and hence distribution (31) turns into: f ≈ e−E/T(1 + ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (32) We also assume that 3d BQ ˙R/(M2 T) ≪ 1 and correspondingly the balance of chemical potentials in equilibrium for the reactions q1 + q2 ↔ ¯q3 + l4 leads to: 3ξq − ξl − 3d BQ ˙R(t) M2 T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (33) Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [18], we express nB ≈ gsBQ 6 ξqT 3, (34) where gs is the number of quark spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Since we are studying instability of R whose timescale is presumed to be much smaller than the expansion rate of the Universe, we approximate DαJα ≈ ˙nB ≈ gsBQ 6 ˙ξqT 3 (35) ≈ gsBQ 6 ˙ξeq q T 3, (36) ξeq q is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (33), using the conservation of the sum of baryonic and leptonic numbers which implies ξl = −ξq/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Then ξeq q = 9d BQ ˙R(t) 10M2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (37) 8 Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (37) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (36) and neglecting the ˙T-term, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (24) gives the following fourth order differential equation for the curvature scalar: d4R dt4 + κ4 f M2 R d2R dt2 + κ4 fR = 0, (38) where κ4 f = 5M2 P lM4 36πgsB2 QT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (39) Once again, we consider terms containing R as small with respect to the terms containing ¨R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The value of κf is only slightly numerically different from κ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (18) and has the same dependence upon the essential parameters, so the solutions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' (17) and (38) practically coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 4 Discussion We have shown that discovered in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [17, 18] exponential instability of the curvature scalar inherent to the mechanism of spontaneous baryogenesis can be successfully cured in modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' The special form of gravity modification by introduction of R2-term into canonical Hilbert-Einstein action of General Relativity was explored as a workable mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' However, the stabilized asymptotic value of R is extremely large and together with possibly successful baryogenesis would still strongly perturb canonical cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Possible ways out of this problem could either be a more complicated model of F(R) gravity or a proper account of particle production created by high frequency oscillations of R(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Both options open interesting possibilities for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' References [1] Sakharov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Pisma Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' 1967, 5, 32–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' [2] Cohen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' and Kaplan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE_T4oBgHgl3EQfzRxG/content/2301.08322v1.pdf'} +page_content=' B.' 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0000000000000000000000000000000000000000..af8668d6157e79c92b167d8f5bd06106312384fe --- /dev/null +++ b/btE5T4oBgHgl3EQfEQ7d/content/tmp_files/2301.05413v1.pdf.txt @@ -0,0 +1,1388 @@ +ICTS-USTC/PCFT-23-02 +Constraint preserving boundary conditions in Bondi-Sachs gauge: a numerical study of stability of +pure AdS spacetime +Li-Ming Caoa ,b∗, Liang-Bi Wub†, and Yu-Sen Zhoub‡ +aPeng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China and +b Interdisciplinary Center for Theoretical Study and Department of Modern Physics, +University of Science and Technology of China, Hefei, Anhui 230026, China +(Dated: January 16, 2023) +In the Bondi-Sachs gauge, the Einstein equations with a cosmological constant coupled to a scalar field in +spherical symmetry are cast into a first order strongly hyperbolic formulation in which the lapse and shift are the +fundamental variables. For this system of equations, the lapse and shift are ingoing characteristic fields, and the +scalar field has three modes: ingoing, outgoing and static, respectively. A constraint-preserving initial boundary +value problem is constructed by using Bianchi identity. Using this scheme, we find that any small perturbation +of the scalar field at the boundary far away enough can cause the collapse of the pure AdS spacetime, and we +provide the numerical evidence for the formation of apparent horizons. The numerical evolution is performed +with a standard method of lines, second order in space and time. The evolution is performed using the standard +second order Runge-Kutta method while the space discrete derivative is second order central difference with +fourth order artificial dissipation. +I. +INTRODUCTION +Motivated mainly by the AdS/CFT correspondence, a very basic question “Is AdS stable?” is raised [1]. In more detail, one +may ask “Under the background with a negative cosmological constant, whether the pure AdS solution will collapse to form +a black hole under small perturbations of the scalar field?” It is difficult to solve this problem analytically. However, one can +attack the problem by numerical relativity. Actually, it has been shown in [1] that pure (global) AdS spacetime is not stable +under the small variation of the initial data. Instead of the perturbation on the initial surface, by using the numerical relativity, +we give some evidence that any small perturbation of the scalar field located on the boundary far away enough can also give rise +the collapse of the pure AdS solution. Under the perturbation, an apparent horizon will emerge inside the computational domain, +and a black hole forms. +The main task of numerical relativity is to solve Einstein equations numerically under different gauges or coordinates. There +are two common schemes of solving Einstein equations. In one of the schemes, the spacetime is foliated by spacelike hypersur- +faces. Each hypersurface represents an instant of time. Proper initial data is described on one time slice. By using a version of the +Einstein equations, one gets the data on a later time slice. This scheme is the so-called Cauchy problem for the Einstein equations. +The ADM formulation is the original version of the Cauchy problem [2]. Moreover, in order to get a well-posed strongly hyper- +bolic system, different research groups have converted the ADM formulation to the the Baumgarte–Shapiro–Shibata–Nakamura +(BSSN) formulation which is very robust in practice [3, 4]. +For another scheme which is called the characteristic formulation, spacetime is foliated by null hypersurfaces. Appropriate +data are prescribed on an initial retarded time slice and possibly on another hypersurface transverse to the retarded slice. In +numerical relativity, the original version of the characteristic problem is the Bondi-Sachs formulation [5]. Rounded discussion +can be found in the review paper by Wincour [6]. +For an actual evolutionary process, there are two alternatives for solving Einstein equations. One is called a constrained +scheme. It is a time scheme for integrating the 3+1 Einstein system in which some or all of the four constraints (Hamiltonian +and momentum constraints) are used to compute some metric coefficients at each step of the numerical evolution. The other is +called a free evolution scheme. It is a time scheme for integrating 3+1 Einstein system in which the constraint equations are +solved only to get the initial data [7]. +A straightforward manipulation of the Bianchi identities demonstrates that either strategy produces the same solution at the +analytical level [8]. Hence, there is no need to deal with constrained evolution which usually requires solving elliptic equations. +Solving elliptic equations at every time step demands a significant computational overhead. For this reason constrained evolu- +tions have been, for the most part, avoided beyond the two dimensional case. The more direct approach of free evolution can be +safely employed. However, free evolution in numerical implementations display violation of the constraints where only using +constraints to get the initial data. Choptuik gives some insights on implementing unconstrained code [9]. +When the free evolution scheme is considered, one’s purpose is to provide proper boundary conditions so that there exits a +unique solution to the evolution equation, while the constraints are preserved in the entire computational domain. In a strongly +hyperbolic formulation of gravity, it is sufficient that giving boundary conditions only to the characteristic modes that enter +∗ e-mail address: caolm@ustc.edu.cn +† e-mail address: liangbi@mail.ustc.edu.cn +‡ e-mail address: zhou ys@mail.ustc.edu.cn +arXiv:2301.05413v1 [gr-qc] 13 Jan 2023 + +2 +the computational domain at any given boundary [10]. Several efforts have illustrated the advantages of providing boundary +conditions through the use of constraints in a real numerical simulation [11]. +Now, in our paper, we are dealing with a first-order, quasi-linear strongly hyperbolic formulation of Einstein equations with a +negative cosmological constant. Such a system can be written as +˙u = Mu′ + l.o. , +in the spherically symmetry case, where u is a list of variables, the dot and the prime indicate time and spatial derivatives, +respectively. The term l.o.stands for lower order terms which do not have any derivatives. M is the so-called principal part +which is a diagonalizable matrix that might depend on the variables themselves and the spacetime coordinates, but not depend +on the derivatives of u. The evolution of the constraints (auxiliary system) is given by +˙uc = Mcu′ +c + l.o. . +A unique solution to the auxiliary system is fixed by giving initial data to uc and boundary conditions to the ingoing modes of +the system. Note that characteristic fields (eigenvectors of the principal part M) with positive eigenvalues are travelling to the +left, on the contrary, characteristic fields (eigenvectors of the principal part M) with negative eigenvalues are travelling to the +right. Therefore, for the left boundary, the boundary conditions are needed for the negative characteristic field. Accordingly, +for the right boundary, the boundary conditions are needed for the positive characteristic field. Since in the case of Einstein +equations, the auxiliary system is usually supposed to be a homogeneous system. The identically zero solution is obtained by +providing zero as initial data (This is the process of solving the initial data.) and zero boundary conditions to the eigenmodes of +Mc entering the domain. More details can found in the Ref.[8, 12]. +As far as we know, there are many papers studying the constraint preserving boundary condition formulation. Based on +its hyperbolic structure, a set of constraint preserving boundary conditions are introduced for the BSSN formulation of the +Einstein evolution equations in spherical symmetry [13]. The case of the first-order version of the Z4 system is considered, and +constraint-preserving boundary conditions of the Sommerfeld type are provided in [14]. The Einstein–Christoffel formulation +of the Einstein equations in the case of spherical symmetry are discussed in [8, 15]. Constraint preserving boundary condition +formulation in the Eddington-Finkelstein coordinates can be found in [11]. Recently, the sufficient conditions required for the +PDEs of the system to guarantee the constraint preservation are studied in [16], more meticulously. +Under the restriction of spherical symmetry, a constraint preserving formulation of the ADM problem for the Einstein equa- +tions transformed from the Bondi-Sachs formulation has been provided in the Ref.[17, 18]. After adding a scalar field, the +Einstein equations coupled to the scalar field in spherical symmetry are cast into a symmetric-hyperbolic system of equations +in which the scalar field, lapse, and shift are fundamental variables. Without the spherical symmetry, the hyperbolicity of the +Bondi-like gauge is studied in the Ref.[19, 20]. +In our paper, we adhere to the assumption of spherical symmetry, and extend the above Bondi-like formulation to the case with +a cosmological constant so as to study the instability of AdS spacetime. We use the constraint preserving formulation to give a +numerical evidence that the pure AdS spacetime is unstable under slight perturbations. It should be emphasized that our slight +perturbations are located at the outer boundary in our formulation which is different from the one where the perturbation happens +in the initial data. The relationship between the perturbation and the position of the apparent horizon is given numerically. This +work reveals the instability of AdS spacetime from another perspective. +The paper is organized as follows. In Sec.II, we present the main evolution equations and propagation of the constraints +by using the Bianchi identity. Numerical simulations is displayed in Sec.III. We consider two different small perturbations +and get the similar conclusions. We perform the numerical convergence test in Sec.IV and show that our numerical scheme is +second order self convergent as we use the second order Runge-Kutta in time integrator and the second order special difference +algorithm. Sec.V is the conclusion and the discussion. +II. +MAIN EVOLUTION EQUATIONS AND PROPAGATION OF THE CONSTRAINTS +The system we are considering is the same as the one in [1]. The action is given by +S = +1 +16πG +� +d4x√−g +� +R − 2Λ − ∇aϕ∇aϕ +� +, +(2.1) +where R is the Ricci scalar, g is the determinant of the metric gab, G is the gravitational constant, ϕ is a scalar field, Λ is the +cosmological constant. Varying the action (2.1), we get the equations of motion which can be expressed as +Rab − 1 +2gabR + Λgab = 8πTab , +(2.2) + +3 +and +∇a∇aϕ = 0 , +(2.3) +where Tab is the stress-energy tensor, which has a form +Tab = ∇aϕ∇bϕ − 1 +2gab∇cϕ∇cϕ . +(2.4) +We report here that the spherically symmetric ADM formulation is motivated by the Bondi-Sachs formulation. The line element +in the coordinate system (v, ˆr, θ, φ) adapted to the null slicing takes the form +ds2 = −e2β V +ˆr dv2 + 2e2βdvdˆr + ˆr2(dθ2 + sin2 θdφ2) , +(2.5) +where β = β(v, ˆr) and V = V (v, ˆr) are two undetermined functions. For convenience, the field equations are denoted in a more +compact form +Eab ≡ Gab + Λgab − 8πTab = 0 . +(2.6) +Based on the Bianchi identity, the independent components out of Eab are discussed in Appendix A. Therefore, one gets the +equations of evolution +Eˆrˆr = 0 , +Evˆr = 0 , +∇a∇aϕ = 0 , +(2.7) +and one constraint +Ev +ˆr = 0 . +(2.8) +Substituting the metric (2.5) into the above equations, we get +β,ˆr − 2πˆr(ϕ,ˆr)2 = 0 , +V,ˆr − e2β(1 − Λˆr2) = 0 , +2ϕ,vˆr + V +ˆr ϕ,ˆrˆr + +�V,ˆr +ˆr + V +ˆr2 +� +ϕ,ˆr + 2 +ˆr ϕ,v = 0 , +(2.9) +where the second equation comes from +gˆrˆrEˆrˆr + 2gvˆrEvˆr = 0. +The constraint variable Ev ˆr has the form +Ev +ˆr ≡ Gv +ˆr + Λδv +ˆr − 8πTv +ˆr = e−2β +ˆr2 +� +2V β,v − V,v − 8πˆr2� +ϕ,vϕ,v + V +ˆr ϕ,vϕ,ˆr +�� +. +(2.10) +The essential difference between the Bondi-Sachs problem and the ADM problem is that one of them uses null slices, which is +unique in spherical symmetry, whereas the other one uses spacelike slices, which is not unique. At every point of the spacetime +there is one ingoing or outgoing null tangent vector, but there is a one-parameter family of spacelike vectors. The parameter +corresponds to the “slope” of the slice at that point. Without losing generality, the relevant ADM formulation is obtained by +making a coordinate transformation [17, 18] +ˆr = r , +v = t + f(r) . +(2.11) +Clearly, one gets a unique relationship between the Bondi-Sachs and ADM variables, if one fixes the function f(r). Under the +coordinate transformation (2.11), the Bondi-Sachs metric (2.5) has the form +ds2 = − +e2β +f ′(2 − f ′V/r)dt2 + e2βf ′(2 − f ′V/r) +� +dr + (1 − f ′V/r) +f ′(2 − f ′V/r)dt +�2 ++ r2(dθ2 + sin2 θdφ2) . +(2.12) +Comparing with the standard ADM metric, +ds2 = −α2dt2 + γrr(dr + βrdt)2 + γT r2(dθ2 + sin2 θdφ2) , +(2.13) + +4 +one has the identifications +α2 = +e2β +f ′(2 − f ′V/r) , +βr = +1 − f ′V/r +f ′(2 − f ′V/r) , +(2.14) +with +γT = 1 , +γrr = (f ′)2 +α2 +(1 − f ′βr)2 , +(2.15) +where α and βr are referred to as the lapse and shift, f ′ ≡ df(r)/dr. Eqs.(2.15) are called as the advanced Bondi-Sachs gauge. +In the followings, we will set f(r) = r. This slicing corresponds to what is usually referred to as Kerr-Schild coordinates [21]. +In order to obtain an initial value problem in a time slicing, we transform the coordinates from (v, ˆr) into (t, r) by using +Eq.(2.11) and the variables from (β, V ) to (α, βr) with the help of Eq.(2.14). After some calculation, the initial value problem +is found +˙α = α,r + α(1 − 2βr) +2r +− α3 +2r (1 − Λr2) − 2πrα(P − Q)2 , +˙βr = βr +,r − (1 − βr)(1 − 2βr) +r ++ 1 − βr +r +α2(1 − Λr2) , +˙P = 2βrP,r + (1 − 2βr)Q,r + 2(1 − βr) +r +P + +�1 − Λr2 +r +α2 + 1 − 2βr +r +� +(Q − P) , +˙Q = P,r , +˙ϕ = P , +(2.16) +where, to reduce the order of the equation of the scalar field ϕ, we define the derivatives of the field ϕ as new variables +P ≡ +˙ϕ , +Q ≡ ϕ,r . +(2.17) +Since our numerical simulation is performed on the region [rmin, rmax], in order to construct the initial boundary value problem, +the boundary conditions have to be considered. To all intents and purposes, the constraint is used to give appropriate boundary +conditions, and can be written as +Ev +ˆr ≡ C1 ≡ 2(1 − 2βr) +rα3 +α,r + +2 +rα2 βr +,r + α2 − 1 + 2βr +r2α2 +− Λ − 4π +α2 +� +P 2 + (1 − 2βr)Q2� += 0 . +(2.18) +This is a first-order differential constraint on the lapse and shift. From Appendix A[see Eq.(A6)], constraint variable C1 satisfies +the following equation +˙C1 − C1,r + · · · = 0 , +(2.19) +where “ · · · ” represent undifferentiated terms. Therefore, the constraint variable C1 propagates at the characteristic speed v = 1. +Because our numerical evolution is performed in a finite spatial region [rmin, rmax], the ingoing constraint C1 = 0 must be +imposed on the outer boundary rmax [8]. Note that the definition of Q, i.e., Eq.(2.17), provides another constraint which is +denoted as C2 = 0. But the characteristic speed of this constraint is zero, so the boundary conditions is not necessary. +To illustrate the requirement of the boundary conditions for the numerical simulations, diagonalization of principal part is +carried out. First, the system of equations (2.16) has a block-diagonal principal part which can be written as a matrix form +M = +� +���� +1 0 +0 +0 +0 +0 1 +0 +0 +0 +0 0 2βr 1 − 2βr 0 +0 0 +1 +0 +0 +0 0 +0 +0 +0 +� +���� . +(2.20) +The eigenvalues of this matrix are +1, +1 , +1, +−(1 − 2βr), +0. + +5 +It is not hard to find that the characteristic fields are +α, +βr, +U −, +U +, +ϕ, +where +U − = P + (1 − 2βr)Q , +U + = P − Q . +(2.21) +The speed of α, βr, U − is 1, the speed of U + is 2βr − 1 and the speed of ϕ is 0. Due to the fact that the characteristic fields +span the whole eigenspace. It means that the system (2.16) is strongly hyperbolic [22]. Hence, after diagonalization, in terms of +these characteristic fields, the system (2.16) reads +˙α = α,r + α(1 − 2βr) +2r +− α3 +2r (1 − Λr2) − 2πrα(U +)2 , +˙βr = βr +,r − (1 − βr)(1 − 2βr) +r ++ 1 − βr +r +α2(1 − Λr2) , +˙U + = −(1 − 2βr)U + +,r + U − +r +− 1 − Λr2 +r +α2U + , +˙U − = U − +,r + U − +r +− 1 − Λr2 +r +α2U + + (U + − U −)α2(1 − Λr2) − 1 + 2βr +r +, +˙ϕ = (1 − 2βr)U + + U − +2(1 − βr) +. +(2.22) +Two initial constraints (C1 = 0, C2 = 0) in terms of characteristic fields become +2(1 − 2βr) +rα3 +α,r + +2 +rα2 βr +,r + α2 − 1 + 2βr +r2α2 +− Λ − 2π +α2 +�(U −)2 + (1 − 2βr)(U +)2 +1 − βr +� += 0 , +(2.23) +and +ϕ,r − −U + + U − +2(1 − βr) += 0 , +(2.24) +respectively. On account of the positive characteristic speed of the constraint variable C1, C1 = 0 needs to be enforced at the outer +boundary rmax. However, since our numerical simulation is a free evolution, C1 = 0 should be transformed in order to achieve +the convenience of the numerical discretization. Trading spatial derivatives for time derivatives, Eq.(2.23) in turn becomes +˙βr + 1 − 2βr +α +˙α − πr(1 − 2βr)U + + U − +1 − βr +� +(2βr − 1)U + + U −� += 0 . +(2.25) +Since the characteristic fields α, βr, U − get into the computing domain at the outer boundary, so the values of these fields are +specified freely. However, they are restricted to +C1|rmax = 0 . +(2.26) +Actually, there are only two freely chosen variables among α, βr, U −. In order to perform the scalar field perturbation entering +the computational domain [rmin, rmax], we enforce the constraint-preserving boundary condition by solving for α, given βr, U +, +U − at the outer boundary. From the boundary condition (2.25), we have +˙α = − +α +1 − 2βr ˙βr + πrα(U −)2 − (1 − 2βr)2(U +)2 +(1 − 2βr)(1 − βr) +. +(2.27) +For the inner boundary, we know that the characteristic variable U + has characteristic speed 2βr − 1. When βr < 1/2, +or 2βr − 1 < 0, strictly speaking, we should add an inner boundary condition for the characteristic variable U +. But in our +simulation, extrapolation is always used at the inner boundary whenever U + is an ingoing mode or an outgoing mode. For actual +simulations, some reasonable assumption is supposed to be made, whose validity can only be verified after the fact [17, 18]. +Our simulation tracks whether the apparent horizon appears. In Bondi-Sachs gauge, the expansion Θ is [23] +Θ = +√ +2 +r√γT +� 1 +α∂t(√γT r) + +� +1 +√γrr +− βr +α +� +∂r(√γT r) +� += +√ +2(1 − 2βr) +rα +. +(2.28) + +6 +Therefore, the location of the apparent horizon is given by βr(rH) = 1/2. Numerical simulations will be shown in the next +section. Until now, we have constructed the initial boundary value problem which is suitable for numerical calculation. +III. +NUMERICAL SIMULATIONS +To implement the proposed boundary treatment, a straightforward second-order dissipative method of lines (MOL) is chosen. +Spatial derivatives are discretized with second-order centered differences plus fourth-order dissipation as discussed in [24], while +for the time integrator we use second order Runge-Kutta. Our uniform grid structure consists of points i = 1, · · · , N, with grid +spacing ∆r = L/(N − 1), where L = rmax − rmin. Spatial derivatives are applied according to standard formulas [8] +Mf ′ → MD0f − ϵi +∆t(∆r)4D+D+D−D−f , +(3.1) +where +(D0f)i = fi+1 − fi−1 +2∆r +, +(D+f)i = fi+1 − fi +∆r +, +(D−f)i = fi − fi−1 +∆r +, +(3.2) +and +− ϵi +∆t(∆r)4D+D+D−D−f = − ϵi +∆t(fi+2 − 4fi+1 + 6fi − 4fi−1 + fi−2) , +(3.3) +M is given by Eq.(2.20), and ϵi is the dissipative factor. In order to evaluate derivatives at the boundaries, ghost zones which +are artificial points beyond the boundaries where field values are defined via extrapolation (second order one-sided difference +is actually the same as second order central difference with a third order extrapolation). Spatial indicators of these ghost points +are i = 0 and i = N + 1. Field values at these ghost points are defined via third order extrapolations and the same derivative +operator is applied at i = 1, · · · , N. After using standard finite differences for the spatial derivatives, we can rewrite our original +differential equations (2.22) as a coupled system of ordinary differential equations of the form +du +dt = S(u, t) , +(3.4) +where u is a vector constructed from the values of the function u in the spatial grid points. The second order Runge-Kutta +algorithm used in our simulation takes the form +u∗ = un + ∆tS(un, tn)/2 , +un+1 = un + ∆tS(u∗, tn + ∆t/2) , +(3.5) +where un = u(tn) and tn = n∆t. +For our simulation, u is a 5N − 2 dimensional vector. They are +α1 , · · · , αN , βr +1 , · · · , βr +N−1 , U + +1 , · · · , U + +N , U − +1 , · · · , U − +N−1 , ϕ1 , · · · , ϕN , +which are variables of MOL. βr +N and U − +N are the free variables at the outer boundary. They do not iterate in the RK algorithm. +Since the time-integration method is explicit, the time-step is limited by the Courant-Friedrichs-Lewy (CFL) condition. The +CFL condition takes the form as follow +max|λa|∆t ≤ ∆r , +(3.6) +where λa is the characteristic speed. We choose ∆t/∆r = 0.25, it is stable enough for the scheme. It should be pointed out that +since four order derivative cannot be obtained at the boundary points i = 1, i = N, so there is no dissipation at these points. It +means that ϵ1 = ϵN = 0. For the dissipation factor ϵi, it is chosen based on a von Neumann stability analysis [24]. We choose + +7 +ϵi = 0.6/16. Now, we write the form of ordinary differential equations (3.4) definitely. For i = 2, · · · , N − 1, +˙αi = αi+1 − αi−1 +2∆r ++ +�α(1 − 2βr) +2r +− α3 +2r (1 − Λr2) − 2πrα(U +)2� +i +− ϵi +∆t(αi+2 − 4αi+1 + 6αi − 4αi−1 + αi−2) , +˙βr +i += βr +i+1 − βr +i−1 +2∆r ++ +� +− (1 − βr)(1 − 2βr) +r ++ 1 − βr +r +α2(1 − Λr2) +� +i +− ϵi +∆t(βr +i+2 − 4βr +i+1 + 6βr +i − 4βr +i−1 + βr +i−2) , +˙U + +i += −(1 − 2βr +i )U + +i+1 − U + +i−1 +2∆r ++ +�U − +r +− 1 − Λr2 +r +α2U +� +i +− ϵi +∆t(U + +i+2 − 4U + +i+1 + 6U + +i − 4U + +i−1 + U + +i−2) , +˙U − +i += U − +i+1 − U − +i−1 +2∆r ++ +�U − +r +− 1 − Λr2 +r +α2U + + (U + − U −)α2(1 − Λr2) − 1 + 2βr +r +� +i +− ϵi +∆t(U − +i+2 − 4U − +i+1 + 6U − +i − 4U − +i−1 + U − +i−2) , +˙ϕi = +�(1 − 2βr)U + + U − +2(1 − βr) +� +i − ϵi +∆t(ϕi+2 − 4ϕi+1 + 6ϕi − 4ϕi−1 + ϕi−2) . +(3.7) +We have pointed out that for the inner boundary, the values of ghost points are obtained by interpolation. Therefore, at the inner +boundary i = 1, +˙α1 = α2 − α0 +2∆r ++ +�α(1 − 2βr) +2r +− α3 +2r (1 − Λr2) − 2πrα(U +)2� +1 , +˙βr +1 = βr +2 − βr +0 +2∆r ++ +� +− (1 − βr)(1 − 2βr) +r ++ 1 − βr +r +α2(1 − Λr2) +� +1 , +˙U + +1 += −(1 − 2βr +1)U + +2 − U + +0 +2∆r ++ +�U − +r +− 1 − Λr2 +r +α2U +� +1 , +˙U − +1 += U − +2 − U − +0 +2∆r ++ +�U − +r +− 1 − Λr2 +r +α2U + + (U + − U −)α2(1 − Λr2) − 1 + 2βr +r +� +1 , +˙ϕ1 = +�(1 − 2βr)U + + U − +2(1 − βr) +� +1 , +(3.8) +where +α0 = 3α1 − 3α2 + α3 , +βr +0 = 3βr +1 − 3βr +2 + βr +3 , +U + +0 += 3U + +1 − 3U + +2 + U + +3 , +U − +0 += 3U − +1 − 3U − +2 + U − +3 , +ϕ0 = 3ϕ1 − 3ϕ2 + ϕ3 . +(3.9) +At the outer boundary, since only the characteristic field U + leaves out of the computing domain, we extrapolate U + at the ghost +zone point i = N + 1. That is +U + +N+1 = U + +N−2 − 3U + +N−1 + 3U + +N . +(3.10) +So at i = N, +˙U + +N = −(1 − 2βr +N)U + +N+1 − U + +N−1 +2∆r ++ +�U − +r +− 1 − Λr2 +r +α2U +� +N , +(3.11) +and +˙ϕN = +�(1 − 2βr)U + + U − +2(1 − βr) +� +N . +(3.12) + +8 +In fact, there are only two freely chosen variables among α, βr and U −. From the boundary condition, we have +˙αN = − +� +α +1 − 2βr +� +N +˙βr +N + +� +πrα(U −)2 − (1 − 2βr)2(U +)2 +(1 − 2βr)(1 − βr) +� +N . +(3.13) +This is nothing but the evolution of αN. Our simulation is to provide numerical evidence for the instability of the pure AdS +spacetime. We consider a small perturbation of the scalar field U − on the outer boundary. The reason is that U − is the only +ingoing mode on the outer boundary. +Before choosing the boundary condition for βr +N and U − +N , it is worth mentioning that in our coordinate (Bondi-Sachs gauge), +the pure AdS solution can be expressed as +α(r) = +1 +� +1 − 2Λ +3 r2max + Λ +3 r2 +, +βr(r) = +2Λ(r2 − r2 +max) +6 + 2Λr2 − 4Λr2max +. +(3.14) +Actually, under the above Eq.(3.14) of α and βr, the metric becomes +ds2 = 3(Λr2 − 3) +(Λr2max − 3)2 dv2 + +6 +3 − Λr2max +dvdr + r2(dθ2 + sin2 θdφ2) . +(3.15) +Note that 3/(3 − Λr2 +max) > 0 is a constant. Define +dt′ + +3dr +(3 − Λr2) = +3dv +(3 − Λr2max) , +then the metric becomes +ds2 = − +� +1 − Λr2 +3 +� +dt′2 + +� +1 − Λr2 +3 +�−1 +dr2 + r2(dθ2 + sin2 θdφ2) . +(3.16) +This is the pure AdS spacetime solution in static coordinates. It can be shown that βr < 1/2, as r ∈ [0, rmax]. Therefore, this +pure AdS solution has no apparent horizon as we all know. +Now, we start to consider the boundary condition for βr +N and U − +N . We choose +βr +N(t) = βr +N(0) . +(3.17) +This is a Dirichlet boundary condition for βr +N(t), since βr is a characteristic field at outer boundary with a speed 1. But for U − +N , +it is described by +U − +N (t) = +� +� +� +� +� +� +� +A · N1 +� t − tI +tF − tI +�4� +1 − t − tI +tF − tI +�4 +sin +� +π +� +4 t − tI +tF − tI ++ 1 +�� +, +type I +A · N2 +� t − tI +tF − tI +�4� +1 − t − tI +tF − tI +�4 +, +type II +(3.18) +if t ∈ [tI, tF], and U − +N (t) = 0 otherwise. N1, N2 are the normalization factors and their values are 316.494 and 256. We illustrate +U − +N (t) at type I in Fig.1 and U − +N (t) at type II in Fig.2. Therefore, we construct initial data by setting +α(r) = +1 +� +1 − 2Λ +3 r2max + Λ +3 r2 +, +βr(r) = +2Λ(r2 − r2 +max) +6 + 2Λr2 − 4Λr2max +, +U + = U − = 0 , +ϕ = 1 . +(3.19) +This means that the initial metric configuration is the pure AdS metric. In our numerical simulation, we always choose the pure +AdS solution as the initial data but adjust the form of the ingoing characteristic field U − at outer boundary. +Here, in our simulation, we choose Λ = −1, rmin = 1, tI = 1, tF = 11. In the case of type I, the amplitude of U − +N is chosen +by A = 1×10−3. The number of spatial points we choose is N = 4001. Increasing the position of disturbance (from rmax = 21 +to rmax = 31), we track the position of the apparent horizon. For the final distribution of βr, both results show that βr decreases +as r increases. Moreover, the value of βr for the final moment reaches the maximum at rmin and gradually decreases to 0 (this +is required by the Dirichlet boundary condition we adopted.). These results are shown on the following two figures, see Fig.3 +and Fig.4. In Fig.3, we find that there is no apparent horizon in [rmin, rmax]. In Fig.4, we find that there is an apparent horizon in +[rmin, rmax]. +It should be pointed out that even if there is no apparent horizon in [rmin, rmax], we can not completely rule out the situation +that there is a apparent horizon in [0, rmin]. However, this does not affect our conclusion. Actually, when one sets rmin = 0.5, + +9 +FIG. 1: The configuration of U − +N (t) with tI = 1, tF = 11, A = 1 × 10−3 for the case of type I. +FIG. 2: The configuration of U − +N (t) with tI = 1, tF = 11, A = 1 × 10−3 for the case of type II. +rmax = 21, one can find the apparent horizon is located at rH = 0.7768 < 1 with the same U − +N (t). To put it in a nutshell, the +apparent horizon can emerge by any small perturbation of the scalar field on the boundary far way enough. In other words, one +can always adjust rmin so that the interval [rmin, rmax] contains the apparent horizon rH, and the change of rmin do not affect the +position of the apparent horizon. In fact, we are sure that when the outer boundary perturbation is vanished, there must be no +apparent horizon, because the (analytic) solution obtained at this time is still a pure AdS solution. The apparent horizon rH will +increase as the increasing of amplitude A. These results are shown at Tab.I and Tab.II. However, this is different from the case +without the cosmological constant. Actually, we find no apparent horizon in the case of Λ = 0, when A is not large enough. +A +rH(rmax = 21) rH(rmax = 31) +1 × 10−3 +0.7768(∗) +1.9525 +2 × 10−3 +1.8000 +3.4900 +3 × 10−3 +2.5950 +4.7200 +4 × 10−3 +3.2700 +5.7925 +5 × 10−3 +3.8750 +6.7675 +6 × 10−3 +4.4350 +7.6750 +7 × 10−3 +4.9550 +8.5225 +8 × 10−3 +5.4500 +9.3250 +9 × 10−3 +5.9150 +10.0900 +10 × 10−3 +6.3650 +10.8250 +TABLE I: For rmin = 1, the apparent horizon rH changes under different amplitudes A in this case of type I. We denote that ∗ are calculated +at the case rmin = 0.5, rmax = 21 and rmin do not impact the apparent horizon rH. + +X10-3 +0.8 +0.6 +0.4 +0.2 +0 +-0.2 +-0.4 +-0.6 +-0.8 +-1 +0 +5 +10 +15 +20 +25 +30 +35 +40 +tX10-3 +0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +5 +10 +15 +20 +25 +30 +35 +40 +t10 +FIG. 3: In this case of type I, the value of the shift function βr changes with time at rmin = 1 with outer boundary rmax = 21. There is no +apparent horizon in [rmin, rmax]. +FIG. 4: In this case of type I, the value of the shift function βr changes with time at rmin = 1 with outer boundary rmax = 31. There is a +apparent horizon in [rmin, rmax]. +IV. +CONVERGENCE TEST +Given initial data and boundary data, to calibrate the accuracy of the numerical implementation, we perform simulations at +increasing resolution and compare the results. The initial data in these tests are given by Eq.(3.19) and change the location of +the outer boundary rmax. N = 1001, N = 2001, N = 4001 spatial grid points are carried out in our simulations, respectively. +We then estimate the relative error between different resolutions and define the convergence factor as +Qself = log2 +� ||u∆ − u∆/2|| +||u∆/2 − u∆/4|| +� +, +(4.1) +where u∆ refers to the numerical solution obtained with resolution ∆r = ∆. For a grid function fi, the norm defined in Eq.(4.1) +is +||fi|| ≡ +� 1 +L +N +� +i=1 +(fi)2∆r +�1/2 +, +(4.2) +where L = rmax − rmin. Since the spatial discretization of the radial derivatives is second order and the time integration is +also second order, the convergence factor Qself reduces to 2 as ∆ → 0 in analytical case [24, 25]. The convergence factors for +evolutionary variables are shown in Fig.5. They are approximately 2, indicating second-order convergence. +The constraint Eq.(2.23) is not enforced during the simulation but only used to get the initial data. It is preserved by con- +structing constraint preserving boundary conditions, although our numerical scheme is a free evolution. The norm of constraint +is also defined by Eq.(4.2). However, some subtle changes need to be explained. When we compute the constraint, we have +L = r(N − 1) − r(2) with i = 2, · · · , N − 1 since second order central difference is used to approximate radial derivatives. To + +0.4994 +0.4992 +0.499 +0.4988 +rmin +0.4986 +0.4984 +0.4982 +0.498 +0.4978 +0.4976 +0 +10 +20 +30 +40 +50 +t0.5025 +0.502 +0.5015 +0.501 +0.5005 +0.5 +0.4995 +0.499 +0.4985 +0 +10 +20 +30 +40 +50 +60 +70 +80 +t11 +A +rH(rmax = 21) rH(rmax = 31) +1 × 10−3 +0.8895(∗) +2.1550 +2 × 10−3 +1.9700 +3.7975 +3 × 10−3 +2.8100 +5.1100 +4 × 10−3 +3.5300 +6.2650 +5 × 10−3 +4.1750 +7.3150 +6 × 10−3 +4.7700 +8.2900 +7 × 10−3 +5.3300 +9.2050 +8 × 10−3 +5.8600 +10.0750 +9 × 10−3 +6.3600 +10.9000 +10 × 10−3 +6.8450 +11.6950 +TABLE II: For rmin = 1, the apparent horizon rH changes under different amplitudes A in this case of type II. We denote that ∗ are calculated +at the case rmin = 0.5, rmax = 21 and rmin do not impact the apparent horizon rH. +FIG. 5: The convergence factors Qself of five fields α, βr, ϕ, U − and U + change over time, respectively. The outer boundary is at rmax = 31 +with tI = 1, tF = 11, A = 1 × 10−3 in the case of type I. +tell the truth, the absolute size of the norm of the constraint is meaningless. But, for different resolutions, we will compare with +the change of the norm of the constraint with time by keeping the same initial boundary conditions. These results are shown in +Fig.6 and Fig.7. From these results, we know the constraints are vanished numerically. This is exactly what we expect. +FIG. 6: For rmin = 1, rmax = 31 with tI = 1, tF = 11, A = 1 × 10−3 in the case of type I. L2 norm of the constraint C1 with different +resolutions + +6 +5.5 +5 +U +4.5 +U* +4 +self +3.5 +3 +2.5 +2 +1.5 +0 +10 +20 +30 +40 +50 +60 +70 +80 +t9 +X10-3 +N=1001 +8 +N=2001 +N=4001 +7 +6 +5 +C +4 +3 +2 +1 +0 +0 +10 +20 +30 +40 +50 +60 +70 +t12 +FIG. 7: For rmin = 1, rmax = 31 with tI = 1, tF = 11, A = 1 × 10−3 in the case of type I. L2 norm of the constraint C2 with different +resolutions +V. +CONCLUSIONS AND DISCUSSION +The ADM and Bondi-Sachs frameworks for the Einstein equations have traditionally been looked upon as two separate +approaches for numerical simulations. We have worked out that in the Bondi-Sachs gauge (2.15), the ADM formulation of +Einstein equations with a cosmological constant can be casted into a strongly hyperbolic problem for the lapse and the shift. +The ADM formulation in the Bondi-Sachs gauge has one constraint (2.23) on the initial data for the lapse and shift and one +constraint-preserving boundary condition (2.25). +In our present work, the assumption of spherical symmetry is kept throughout. We use the constraint preserving formulation +to give numerical evidence that the pure AdS spacetime is unstable under slight perturbations. Our simulation is carried out +by changing the boundary conditions which is different from the Ref.[1] in which a slight perturbation is added to the initial +data. In fact, for a pure AdS solution, we find that any small perturbation of the scalar field at the boundary far away enough +can cause the collapse of the pure AdS spacetime. The numerical evidence is provided for the formation of apparent horizons. +The relationship between the perturbation and the position of the apparent horizon has been studied numerically. However, the +situation is different from the case with zero cosmological constant. Actually, there is no apparent horizon in the case of Λ = 0, +when A is not large enough. +From the two tables I and II, we can see that for the same type of perturbation with the same amplitude, the radius of the +apparent horizon will increase as the location of the perturbation staying off. In addition, for the scalar field perturbation placed +at the same position, the larger the amplitude of the perturbation, the larger the apparent horizon formed by collapsing finally. +Two kinds of perturbation with different shapes express similar conclusions. +Acknowledgement +This work was supported in part by the National Natural Science Foundation of China with grants No.12075232, +No.11622543, No.11947301, and No.12047502. This work is also supported by the Fundamental Research Funds for the +Central Universities under Grant No: WK2030000036. +Appendix A: Bianchi identity +In this Appendix, we will give the details for picking out the independent components of Eab. At first, when scalar is satisfied +with ∇a∇aϕ = 0, we have the Bianchi identity which is given by +∇aEab = −8π(∇a∇aϕ)∇bϕ = 0 . +(A1) +The nontrivial components of Eab are Evv, Evˆr = Eˆrv, Evv, Eθθ, Eφφ, respectively. It can be proved that ∇µEµθ = 0 and +∇µEµφ = 0 are trivial under the assumption of spherical symmetry. For b = v, +∇µEµv = gˆrv ∂Eˆrv +∂v ++ gˆrˆr ∂Eˆrv +∂ˆr ++ gvˆr ∂Evv +∂ˆr +− +� +3gvˆrΓˆr +ˆrv + gvˆrΓv +vv + gˆrˆrΓˆr +ˆrˆr + 2gθθΓˆr +θθ +� +Eˆrv +−2gθθΓv +θθEvv − +� +gvˆrΓˆr +vv + gˆrˆrΓˆr +ˆrv +� +Eˆrˆr = 0 . +(A2) + +X106 +5 +N=1001 +4.5 +N=2001 +N=4001 +4 +3.5 +3 +N +C +2.5 +2 +1.5 +1 +0.5 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +t13 +For b = ˆr, +∇µEµˆr = gvˆr ∂Evˆr +∂ˆr ++ gˆrv ∂Eˆrˆr +∂v ++ gˆrˆr ∂Eˆrˆr +∂ˆr +− +� +3gvˆrΓˆr +vˆr + 2gˆrˆrΓˆr +ˆrˆr + 2gθθΓˆr +θθ +� +Eˆrˆr +− +� +gvˆrΓˆr +ˆrˆr + 2gθθΓv +θθ +� +Evˆr − 2gθθΓθ +θˆrEθθ = 0 . +(A3) +From Eq.(A3), when Evˆr = Eˆrˆr = 0, we have Eθθ = 0. Then one gets Eφφ = 0, since Eφφ = sin2 θEθθ. From Eq.(A2), when +Evˆr = Eˆrˆr = 0, we obtain +gvˆr ∂Evv +∂ˆr +− 2gθθΓv +θθEvv = 0 . +(A4) +This equation is equivalent to +∂(Evvgvˆr) +∂ˆr +− ∂gvˆr +∂ˆr Evv − 2gθθΓv +θθEvv = 0 . +(A5) +If Evˆr = 0, we have Ev ˆr = Evµgµˆr = Evvgvˆr. For gvˆr = e−2β, we have ∂gvˆr/∂ˆr = −2gvˆr∂β/∂ˆr. Therefore, Eq.(A5) +becomes +∂Ev ˆr +∂ˆr ++ 2 +�∂β +∂ˆr + 1 +ˆr +� +Ev +ˆr = 0 . +(A6) +Solving this equation (A6), one gets [26] +Ev +ˆr(v, ˆr) = ˆr2 +0 +ˆr2 exp 2 +� +β(v, ˆr0) − β(v, ˆr) +� +Ev +ˆr(v, ˆr0) . +(A7) +Therefore, the independent components of Eab are Eˆrˆr and Evˆr. That means evolution equations are +Eˆrˆr = 0 , +Evˆr = 0 , +∇a∇aϕ = 0 . +(A8) +According to the solution of Eq.(A6), +Ev +ˆr = 0 +(A9) +is recommended to be a constraint equation. + +14 +Appendix B: calculation of basic quantity +In this Appendix, we give results of some basic quantities which we have used in the process of deriving evolution equation. +The non-vanished Christoffel symbols are given as follows, +Γv +vv = +1 +2ˆr2 +� +− V + 2ˆrV β,ˆr + ˆrV,ˆr + 4ˆr2β,v +� +, +Γv +θθ = −ˆre−2β , +Γv +φφ = −ˆre−2β sin2 θ , +Γˆr +vv = +1 +2ˆr3 +� +− V 2 + 2ˆrV 2β,ˆr − ˆr2V,v + ˆrV V,ˆr + 2ˆr2V β,v +� +, +Γˆr +vˆr = Γˆr +ˆrv = +1 +2ˆr2 +� +V − ˆrV,ˆr − 2ˆrV β,ˆr +� +, +Γˆr +ˆrˆr = 2β,ˆr , +Γˆr +θθ = −V e−2β , +Γˆr +φφ = −V e−2β sin2 θ , +Γθ +ˆrθ = Γθ +θˆr = 1 +ˆr , +Γθ +φφ = − cos θ sin θ , +Γφ +ˆrφ = Γφ +φˆr = 1 +ˆr , +Γφ +θφ = Γφ +φθ = cot θ . +(B1) +The components of the Einstein tensor are +Gvv = +1 +ˆr3 +� +2V 2β,ˆr − ˆrV,v + V e2β − V V,ˆr + 2ˆrV β,v +� +, +Gvˆr = Gˆrv = 1 +ˆr2 +� +− 2V β,ˆr + V,ˆr − e2β� +, +Gˆrˆr = 4 +ˆr β,ˆr , +Gθθ = e−2β +2 +� +− 2V β,ˆr + 2ˆrV β,ˆrˆr + 2ˆrV,ˆrβ,ˆr + ˆrV,ˆrˆr + 4ˆr2β,vˆr +� +, +Gφφ = e−2β +2 +� +− 2V β,ˆr + 2ˆrV β,ˆrˆr + 2ˆrV,ˆrβ,ˆr + ˆrV,ˆrˆr + 4ˆr2β,vˆr +� +sin2 θ , +(B2) +and +Gv +ˆr = e−2β +ˆr2 +� +2V β,v − V,v +� +. +(B3) +The components of ∇a∇bϕ are +∇v∇vϕ = ϕ,vv − +1 +2ˆr2 +� +− V + 2ˆrV β,ˆr + ˆrV,ˆr + 4ˆr2β,v +� +ϕ,v +− 1 +2ˆr3 +� +− V 2 + 2ˆrV 2β,ˆr − ˆr2V,v + ˆrV V,ˆr + 2ˆr2V β,v +� +ϕ,ˆr , +∇v∇ˆrϕ = ∇ˆr∇vϕ = ϕ,vˆr − +1 +2ˆr2 +� +V − ˆrV,ˆr − 2ˆrV β,ˆr +� +ϕ,ˆr , +∇ˆr∇ˆrϕ = ϕ,ˆrˆr − 2β,ˆrϕ,ˆr , +∇θ∇θϕ = V e−2βϕ,ˆr + ˆre−2βϕ,v , +∇φ∇φϕ = +� +V e−2βϕ,ˆr + ˆre−2βϕ,v +� +sin2 θ . +(B4) +The expression of ∇a∇aϕ is given by +∇a∇aϕ = e−2β� +2ϕ,vˆr + V +ˆr ϕ,ˆrˆr + +�V,ˆr +ˆr + V +ˆr2 +� +ϕ,ˆr + 2 +ˆr ϕ,v +� +. +(B5) + +15 +The expression of ∇aϕ∇aϕ is given by +∇aϕ∇aϕ = V +ˆr e−2βϕ,ˆrϕ,ˆr + 2e−2βϕ,ˆrϕ,v . +(B6) +[1] P. 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Phys. 105, 337 (1986). + diff --git a/btE5T4oBgHgl3EQfEQ7d/content/tmp_files/load_file.txt b/btE5T4oBgHgl3EQfEQ7d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0977daf10367779edfb68cef1d165ef6763c8ea6 --- /dev/null +++ b/btE5T4oBgHgl3EQfEQ7d/content/tmp_files/load_file.txt @@ -0,0 +1,676 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf,len=675 +page_content='ICTS-USTC/PCFT-23-02 Constraint preserving boundary conditions in Bondi-Sachs gauge: a numerical study of stability of pure AdS spacetime Li-Ming Caoa ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='b∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Liang-Bi Wub†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' and Yu-Sen Zhoub‡ aPeng Huanwu Center for Fundamental Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' China and b Interdisciplinary Center for Theoretical Study and Department of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' China (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 2023) In the Bondi-Sachs gauge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' the Einstein equations with a cosmological constant coupled to a scalar field in spherical symmetry are cast into a first order strongly hyperbolic formulation in which the lapse and shift are the fundamental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For this system of equations, the lapse and shift are ingoing characteristic fields, and the scalar field has three modes: ingoing, outgoing and static, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' A constraint-preserving initial boundary value problem is constructed by using Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Using this scheme, we find that any small perturbation of the scalar field at the boundary far away enough can cause the collapse of the pure AdS spacetime, and we provide the numerical evidence for the formation of apparent horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The numerical evolution is performed with a standard method of lines, second order in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The evolution is performed using the standard second order Runge-Kutta method while the space discrete derivative is second order central difference with fourth order artificial dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' INTRODUCTION Motivated mainly by the AdS/CFT correspondence, a very basic question “Is AdS stable?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' is raised [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In more detail, one may ask “Under the background with a negative cosmological constant, whether the pure AdS solution will collapse to form a black hole under small perturbations of the scalar field?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It is difficult to solve this problem analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, one can attack the problem by numerical relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Actually, it has been shown in [1] that pure (global) AdS spacetime is not stable under the small variation of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Instead of the perturbation on the initial surface, by using the numerical relativity, we give some evidence that any small perturbation of the scalar field located on the boundary far away enough can also give rise the collapse of the pure AdS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Under the perturbation, an apparent horizon will emerge inside the computational domain, and a black hole forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The main task of numerical relativity is to solve Einstein equations numerically under different gauges or coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' There are two common schemes of solving Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In one of the schemes, the spacetime is foliated by spacelike hypersur- faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Each hypersurface represents an instant of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Proper initial data is described on one time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' By using a version of the Einstein equations, one gets the data on a later time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' This scheme is the so-called Cauchy problem for the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The ADM formulation is the original version of the Cauchy problem [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Moreover, in order to get a well-posed strongly hyper- bolic system, different research groups have converted the ADM formulation to the the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulation which is very robust in practice [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For another scheme which is called the characteristic formulation, spacetime is foliated by null hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Appropriate data are prescribed on an initial retarded time slice and possibly on another hypersurface transverse to the retarded slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In numerical relativity, the original version of the characteristic problem is the Bondi-Sachs formulation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Rounded discussion can be found in the review paper by Wincour [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For an actual evolutionary process, there are two alternatives for solving Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' One is called a constrained scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It is a time scheme for integrating the 3+1 Einstein system in which some or all of the four constraints (Hamiltonian and momentum constraints) are used to compute some metric coefficients at each step of the numerical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The other is called a free evolution scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It is a time scheme for integrating 3+1 Einstein system in which the constraint equations are solved only to get the initial data [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' A straightforward manipulation of the Bianchi identities demonstrates that either strategy produces the same solution at the analytical level [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Hence, there is no need to deal with constrained evolution which usually requires solving elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Solving elliptic equations at every time step demands a significant computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For this reason constrained evolu- tions have been, for the most part, avoided beyond the two dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The more direct approach of free evolution can be safely employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, free evolution in numerical implementations display violation of the constraints where only using constraints to get the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Choptuik gives some insights on implementing unconstrained code [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' When the free evolution scheme is considered, one’s purpose is to provide proper boundary conditions so that there exits a unique solution to the evolution equation, while the constraints are preserved in the entire computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In a strongly hyperbolic formulation of gravity, it is sufficient that giving boundary conditions only to the characteristic modes that enter ∗ e-mail address: caolm@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='cn † e-mail address: liangbi@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='cn ‡ e-mail address: zhou ys@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='05413v1 [gr-qc] 13 Jan 2023 2 the computational domain at any given boundary [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Several efforts have illustrated the advantages of providing boundary conditions through the use of constraints in a real numerical simulation [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Now, in our paper, we are dealing with a first-order, quasi-linear strongly hyperbolic formulation of Einstein equations with a negative cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Such a system can be written as ˙u = Mu′ + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' , in the spherically symmetry case, where u is a list of variables, the dot and the prime indicate time and spatial derivatives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The term l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='stands for lower order terms which do not have any derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' M is the so-called principal part which is a diagonalizable matrix that might depend on the variables themselves and the spacetime coordinates, but not depend on the derivatives of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The evolution of the constraints (auxiliary system) is given by ˙uc = Mcu′ c + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' A unique solution to the auxiliary system is fixed by giving initial data to uc and boundary conditions to the ingoing modes of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Note that characteristic fields (eigenvectors of the principal part M) with positive eigenvalues are travelling to the left, on the contrary, characteristic fields (eigenvectors of the principal part M) with negative eigenvalues are travelling to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, for the left boundary, the boundary conditions are needed for the negative characteristic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Accordingly, for the right boundary, the boundary conditions are needed for the positive characteristic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Since in the case of Einstein equations, the auxiliary system is usually supposed to be a homogeneous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The identically zero solution is obtained by providing zero as initial data (This is the process of solving the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=') and zero boundary conditions to the eigenmodes of Mc entering the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' More details can found in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' [8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' As far as we know, there are many papers studying the constraint preserving boundary condition formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Based on its hyperbolic structure, a set of constraint preserving boundary conditions are introduced for the BSSN formulation of the Einstein evolution equations in spherical symmetry [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The case of the first-order version of the Z4 system is considered, and constraint-preserving boundary conditions of the Sommerfeld type are provided in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The Einstein–Christoffel formulation of the Einstein equations in the case of spherical symmetry are discussed in [8, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Constraint preserving boundary condition formulation in the Eddington-Finkelstein coordinates can be found in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Recently, the sufficient conditions required for the PDEs of the system to guarantee the constraint preservation are studied in [16], more meticulously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Under the restriction of spherical symmetry, a constraint preserving formulation of the ADM problem for the Einstein equa- tions transformed from the Bondi-Sachs formulation has been provided in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' After adding a scalar field, the Einstein equations coupled to the scalar field in spherical symmetry are cast into a symmetric-hyperbolic system of equations in which the scalar field, lapse, and shift are fundamental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Without the spherical symmetry, the hyperbolicity of the Bondi-like gauge is studied in the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In our paper, we adhere to the assumption of spherical symmetry, and extend the above Bondi-like formulation to the case with a cosmological constant so as to study the instability of AdS spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We use the constraint preserving formulation to give a numerical evidence that the pure AdS spacetime is unstable under slight perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It should be emphasized that our slight perturbations are located at the outer boundary in our formulation which is different from the one where the perturbation happens in the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The relationship between the perturbation and the position of the apparent horizon is given numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' This work reveals the instability of AdS spacetime from another perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='II, we present the main evolution equations and propagation of the constraints by using the Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Numerical simulations is displayed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We consider two different small perturbations and get the similar conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We perform the numerical convergence test in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='IV and show that our numerical scheme is second order self convergent as we use the second order Runge-Kutta in time integrator and the second order special difference algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='V is the conclusion and the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' MAIN EVOLUTION EQUATIONS AND PROPAGATION OF THE CONSTRAINTS The system we are considering is the same as the one in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The action is given by S = 1 16πG � d4x√−g � R − 2Λ − ∇aϕ∇aϕ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1) where R is the Ricci scalar, g is the determinant of the metric gab, G is the gravitational constant, ϕ is a scalar field, Λ is the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Varying the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1), we get the equations of motion which can be expressed as Rab − 1 2gabR + Λgab = 8πTab , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2) 3 and ∇a∇aϕ = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3) where Tab is the stress-energy tensor, which has a form Tab = ∇aϕ∇bϕ − 1 2gab∇cϕ∇cϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4) We report here that the spherically symmetric ADM formulation is motivated by the Bondi-Sachs formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The line element in the coordinate system (v, ˆr, θ, φ) adapted to the null slicing takes the form ds2 = −e2β V ˆr dv2 + 2e2βdvdˆr + ˆr2(dθ2 + sin2 θdφ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5) where β = β(v, ˆr) and V = V (v, ˆr) are two undetermined functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For convenience, the field equations are denoted in a more compact form Eab ≡ Gab + Λgab − 8πTab = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6) Based on the Bianchi identity, the independent components out of Eab are discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, one gets the equations of evolution Eˆrˆr = 0 , Evˆr = 0 , ∇a∇aϕ = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7) and one constraint Ev ˆr = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8) Substituting the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5) into the above equations, we get β,ˆr − 2πˆr(ϕ,ˆr)2 = 0 , V,ˆr − e2β(1 − Λˆr2) = 0 , 2ϕ,vˆr + V ˆr ϕ,ˆrˆr + �V,ˆr ˆr + V ˆr2 � ϕ,ˆr + 2 ˆr ϕ,v = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9) where the second equation comes from gˆrˆrEˆrˆr + 2gvˆrEvˆr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The constraint variable Ev ˆr has the form Ev ˆr ≡ Gv ˆr + Λδv ˆr − 8πTv ˆr = e−2β ˆr2 � 2V β,v − V,v − 8πˆr2� ϕ,vϕ,v + V ˆr ϕ,vϕ,ˆr �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='10) The essential difference between the Bondi-Sachs problem and the ADM problem is that one of them uses null slices, which is unique in spherical symmetry, whereas the other one uses spacelike slices, which is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' At every point of the spacetime there is one ingoing or outgoing null tangent vector, but there is a one-parameter family of spacelike vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The parameter corresponds to the “slope” of the slice at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Without losing generality, the relevant ADM formulation is obtained by making a coordinate transformation [17, 18] ˆr = r , v = t + f(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='11) Clearly, one gets a unique relationship between the Bondi-Sachs and ADM variables, if one fixes the function f(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Under the coordinate transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='11), the Bondi-Sachs metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5) has the form ds2 = − e2β f ′(2 − f ′V/r)dt2 + e2βf ′(2 − f ′V/r) � dr + (1 − f ′V/r) f ′(2 − f ′V/r)dt �2 + r2(dθ2 + sin2 θdφ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='12) Comparing with the standard ADM metric, ds2 = −α2dt2 + γrr(dr + βrdt)2 + γT r2(dθ2 + sin2 θdφ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='13) 4 one has the identifications α2 = e2β f ′(2 − f ′V/r) , βr = 1 − f ′V/r f ′(2 − f ′V/r) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='14) with γT = 1 , γrr = (f ′)2 α2 (1 − f ′βr)2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='15) where α and βr are referred to as the lapse and shift, f ′ ≡ df(r)/dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='15) are called as the advanced Bondi-Sachs gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In the followings, we will set f(r) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' This slicing corresponds to what is usually referred to as Kerr-Schild coordinates [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In order to obtain an initial value problem in a time slicing, we transform the coordinates from (v, ˆr) into (t, r) by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='11) and the variables from (β, V ) to (α, βr) with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' After some calculation, the initial value problem is found ˙α = α,r + α(1 − 2βr) 2r − α3 2r (1 − Λr2) − 2πrα(P − Q)2 , ˙βr = βr ,r − (1 − βr)(1 − 2βr) r + 1 − βr r α2(1 − Λr2) , ˙P = 2βrP,r + (1 − 2βr)Q,r + 2(1 − βr) r P + �1 − Λr2 r α2 + 1 − 2βr r � (Q − P) , ˙Q = P,r , ˙ϕ = P , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='16) where, to reduce the order of the equation of the scalar field ϕ, we define the derivatives of the field ϕ as new variables P ≡ ˙ϕ , Q ≡ ϕ,r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='17) Since our numerical simulation is performed on the region [rmin, rmax], in order to construct the initial boundary value problem, the boundary conditions have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' To all intents and purposes, the constraint is used to give appropriate boundary conditions, and can be written as Ev ˆr ≡ C1 ≡ 2(1 − 2βr) rα3 α,r + 2 rα2 βr ,r + α2 − 1 + 2βr r2α2 − Λ − 4π α2 � P 2 + (1 − 2βr)Q2� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='18) This is a first-order differential constraint on the lapse and shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' From Appendix A[see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A6)], constraint variable C1 satisfies the following equation ˙C1 − C1,r + · · · = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='19) where “ · · · ” represent undifferentiated terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, the constraint variable C1 propagates at the characteristic speed v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Because our numerical evolution is performed in a finite spatial region [rmin, rmax], the ingoing constraint C1 = 0 must be imposed on the outer boundary rmax [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Note that the definition of Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='17), provides another constraint which is denoted as C2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' But the characteristic speed of this constraint is zero, so the boundary conditions is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' To illustrate the requirement of the boundary conditions for the numerical simulations, diagonalization of principal part is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' First, the system of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='16) has a block-diagonal principal part which can be written as a matrix form M = � ���� 1 0 0 0 0 0 1 0 0 0 0 0 2βr 1 − 2βr 0 0 0 1 0 0 0 0 0 0 0 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='20) The eigenvalues of this matrix are 1, 1 , 1, −(1 − 2βr), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 5 It is not hard to find that the characteristic fields are α, βr, U −, U +, ϕ, where U − = P + (1 − 2βr)Q , U + = P − Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='21) The speed of α, βr, U − is 1, the speed of U + is 2βr − 1 and the speed of ϕ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Due to the fact that the characteristic fields span the whole eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It means that the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='16) is strongly hyperbolic [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Hence, after diagonalization, in terms of these characteristic fields, the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='16) reads ˙α = α,r + α(1 − 2βr) 2r − α3 2r (1 − Λr2) − 2πrα(U +)2 , ˙βr = βr ,r − (1 − βr)(1 − 2βr) r + 1 − βr r α2(1 − Λr2) , ˙U + = −(1 − 2βr)U + ,r + U − r − 1 − Λr2 r α2U + , ˙U − = U − ,r + U − r − 1 − Λr2 r α2U + + (U + − U −)α2(1 − Λr2) − 1 + 2βr r , ˙ϕ = (1 − 2βr)U + + U − 2(1 − βr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='22) Two initial constraints (C1 = 0, C2 = 0) in terms of characteristic fields become 2(1 − 2βr) rα3 α,r + 2 rα2 βr ,r + α2 − 1 + 2βr r2α2 − Λ − 2π α2 �(U −)2 + (1 − 2βr)(U +)2 1 − βr � = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='23) and ϕ,r − −U + + U − 2(1 − βr) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='24) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' On account of the positive characteristic speed of the constraint variable C1, C1 = 0 needs to be enforced at the outer boundary rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, since our numerical simulation is a free evolution, C1 = 0 should be transformed in order to achieve the convenience of the numerical discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Trading spatial derivatives for time derivatives, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='23) in turn becomes ˙βr + 1 − 2βr α ˙α − πr(1 − 2βr)U + + U − 1 − βr � (2βr − 1)U + + U −� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='25) Since the characteristic fields α, βr, U − get into the computing domain at the outer boundary, so the values of these fields are specified freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, they are restricted to C1|rmax = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='26) Actually, there are only two freely chosen variables among α, βr, U −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In order to perform the scalar field perturbation entering the computational domain [rmin, rmax], we enforce the constraint-preserving boundary condition by solving for α, given βr, U +, U − at the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' From the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='25), we have ˙α = − α 1 − 2βr ˙βr + πrα(U −)2 − (1 − 2βr)2(U +)2 (1 − 2βr)(1 − βr) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='27) For the inner boundary, we know that the characteristic variable U + has characteristic speed 2βr − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' When βr < 1/2, or 2βr − 1 < 0, strictly speaking, we should add an inner boundary condition for the characteristic variable U +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' But in our simulation, extrapolation is always used at the inner boundary whenever U + is an ingoing mode or an outgoing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For actual simulations, some reasonable assumption is supposed to be made, whose validity can only be verified after the fact [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Our simulation tracks whether the apparent horizon appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In Bondi-Sachs gauge, the expansion Θ is [23] Θ = √ 2 r√γT � 1 α∂t(√γT r) + � 1 √γrr − βr α � ∂r(√γT r) � = √ 2(1 − 2βr) rα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='28) 6 Therefore, the location of the apparent horizon is given by βr(rH) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Numerical simulations will be shown in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Until now, we have constructed the initial boundary value problem which is suitable for numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' NUMERICAL SIMULATIONS To implement the proposed boundary treatment, a straightforward second-order dissipative method of lines (MOL) is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Spatial derivatives are discretized with second-order centered differences plus fourth-order dissipation as discussed in [24], while for the time integrator we use second order Runge-Kutta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Our uniform grid structure consists of points i = 1, · · · , N, with grid spacing ∆r = L/(N − 1), where L = rmax − rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Spatial derivatives are applied according to standard formulas [8] Mf ′ → MD0f − ϵi ∆t(∆r)4D+D+D−D−f , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1) where (D0f)i = fi+1 − fi−1 2∆r , (D+f)i = fi+1 − fi ∆r , (D−f)i = fi − fi−1 ∆r , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2) and − ϵi ∆t(∆r)4D+D+D−D−f = − ϵi ∆t(fi+2 − 4fi+1 + 6fi − 4fi−1 + fi−2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3) M is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='20), and ϵi is the dissipative factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In order to evaluate derivatives at the boundaries, ghost zones which are artificial points beyond the boundaries where field values are defined via extrapolation (second order one-sided difference is actually the same as second order central difference with a third order extrapolation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Spatial indicators of these ghost points are i = 0 and i = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Field values at these ghost points are defined via third order extrapolations and the same derivative operator is applied at i = 1, · · · , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' After using standard finite differences for the spatial derivatives, we can rewrite our original differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='22) as a coupled system of ordinary differential equations of the form du dt = S(u, t) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4) where u is a vector constructed from the values of the function u in the spatial grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The second order Runge-Kutta algorithm used in our simulation takes the form u∗ = un + ∆tS(un, tn)/2 , un+1 = un + ∆tS(u∗, tn + ∆t/2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5) where un = u(tn) and tn = n∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For our simulation, u is a 5N − 2 dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' They are α1 , · · · , αN , βr 1 , · · · , βr N−1 , U + 1 , · · · , U + N , U − 1 , · · · , U − N−1 , ϕ1 , · · · , ϕN , which are variables of MOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' βr N and U − N are the free variables at the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' They do not iterate in the RK algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Since the time-integration method is explicit, the time-step is limited by the Courant-Friedrichs-Lewy (CFL) condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The CFL condition takes the form as follow max|λa|∆t ≤ ∆r , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6) where λa is the characteristic speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We choose ∆t/∆r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='25, it is stable enough for the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It should be pointed out that since four order derivative cannot be obtained at the boundary points i = 1, i = N, so there is no dissipation at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It means that ϵ1 = ϵN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For the dissipation factor ϵi, it is chosen based on a von Neumann stability analysis [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We choose 7 ϵi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Now, we write the form of ordinary differential equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4) definitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For i = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' N − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' ˙αi = αi+1 − αi−1 2∆r + �α(1 − 2βr) 2r − α3 2r (1 − Λr2) − 2πrα(U +)2� i − ϵi ∆t(αi+2 − 4αi+1 + 6αi − 4αi−1 + αi−2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' ˙βr i = βr i+1 − βr i−1 2∆r + � − (1 − βr)(1 − 2βr) r + 1 − βr r α2(1 − Λr2) � i − ϵi ∆t(βr i+2 − 4βr i+1 + 6βr i − 4βr i−1 + βr i−2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' ˙U + i = −(1 − 2βr i )U + i+1 − U + i−1 2∆r + �U − r − 1 − Λr2 r α2U +� i − ϵi ∆t(U + i+2 − 4U + i+1 + 6U + i − 4U + i−1 + U + i−2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' ˙U − i = U − i+1 − U − i−1 2∆r + �U − r − 1 − Λr2 r α2U + + (U + − U −)α2(1 − Λr2) − 1 + 2βr r � i − ϵi ∆t(U − i+2 − 4U − i+1 + 6U − i − 4U − i−1 + U − i−2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' ˙ϕi = �(1 − 2βr)U + + U − 2(1 − βr) � i − ϵi ∆t(ϕi+2 − 4ϕi+1 + 6ϕi − 4ϕi−1 + ϕi−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7) We have pointed out that for the inner boundary, the values of ghost points are obtained by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, at the inner boundary i = 1, ˙α1 = α2 − α0 2∆r + �α(1 − 2βr) 2r − α3 2r (1 − Λr2) − 2πrα(U +)2� 1 , ˙βr 1 = βr 2 − βr 0 2∆r + � − (1 − βr)(1 − 2βr) r + 1 − βr r α2(1 − Λr2) � 1 , ˙U + 1 = −(1 − 2βr 1)U + 2 − U + 0 2∆r + �U − r − 1 − Λr2 r α2U +� 1 , ˙U − 1 = U − 2 − U − 0 2∆r + �U − r − 1 − Λr2 r α2U + + (U + − U −)α2(1 − Λr2) − 1 + 2βr r � 1 , ˙ϕ1 = �(1 − 2βr)U + + U − 2(1 − βr) � 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8) where α0 = 3α1 − 3α2 + α3 , βr 0 = 3βr 1 − 3βr 2 + βr 3 , U + 0 = 3U + 1 − 3U + 2 + U + 3 , U − 0 = 3U − 1 − 3U − 2 + U − 3 , ϕ0 = 3ϕ1 − 3ϕ2 + ϕ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9) At the outer boundary, since only the characteristic field U + leaves out of the computing domain, we extrapolate U + at the ghost zone point i = N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' That is U + N+1 = U + N−2 − 3U + N−1 + 3U + N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='10) So at i = N, ˙U + N = −(1 − 2βr N)U + N+1 − U + N−1 2∆r + �U − r − 1 − Λr2 r α2U +� N , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='11) and ˙ϕN = �(1 − 2βr)U + + U − 2(1 − βr) � N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='12) 8 In fact, there are only two freely chosen variables among α, βr and U −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' From the boundary condition, we have ˙αN = − � α 1 − 2βr � N ˙βr N + � πrα(U −)2 − (1 − 2βr)2(U +)2 (1 − 2βr)(1 − βr) � N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='13) This is nothing but the evolution of αN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Our simulation is to provide numerical evidence for the instability of the pure AdS spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We consider a small perturbation of the scalar field U − on the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The reason is that U − is the only ingoing mode on the outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Before choosing the boundary condition for βr N and U − N , it is worth mentioning that in our coordinate (Bondi-Sachs gauge), the pure AdS solution can be expressed as α(r) = 1 � 1 − 2Λ 3 r2max + Λ 3 r2 , βr(r) = 2Λ(r2 − r2 max) 6 + 2Λr2 − 4Λr2max .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='14) Actually, under the above Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='14) of α and βr, the metric becomes ds2 = 3(Λr2 − 3) (Λr2max − 3)2 dv2 + 6 3 − Λr2max dvdr + r2(dθ2 + sin2 θdφ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='15) Note that 3/(3 − Λr2 max) > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Define dt′ + 3dr (3 − Λr2) = 3dv (3 − Λr2max) , then the metric becomes ds2 = − � 1 − Λr2 3 � dt′2 + � 1 − Λr2 3 �−1 dr2 + r2(dθ2 + sin2 θdφ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='16) This is the pure AdS spacetime solution in static coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It can be shown that βr < 1/2, as r ∈ [0, rmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, this pure AdS solution has no apparent horizon as we all know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Now, we start to consider the boundary condition for βr N and U − N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We choose βr N(t) = βr N(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='17) This is a Dirichlet boundary condition for βr N(t), since βr is a characteristic field at outer boundary with a speed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' But for U − N , it is described by U − N (t) = � � � � � � � A · N1 � t − tI tF − tI �4� 1 − t − tI tF − tI �4 sin � π � 4 t − tI tF − tI + 1 �� , type I A · N2 � t − tI tF − tI �4� 1 − t − tI tF − tI �4 , type II (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='18) if t ∈ [tI, tF], and U − N (t) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' N1, N2 are the normalization factors and their values are 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='494 and 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We illustrate U − N (t) at type I in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1 and U − N (t) at type II in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, we construct initial data by setting α(r) = 1 � 1 − 2Λ 3 r2max + Λ 3 r2 , βr(r) = 2Λ(r2 − r2 max) 6 + 2Λr2 − 4Λr2max , U + = U − = 0 , ϕ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='19) This means that the initial metric configuration is the pure AdS metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In our numerical simulation, we always choose the pure AdS solution as the initial data but adjust the form of the ingoing characteristic field U − at outer boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Here, in our simulation, we choose Λ = −1, rmin = 1, tI = 1, tF = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In the case of type I, the amplitude of U − N is chosen by A = 1×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The number of spatial points we choose is N = 4001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Increasing the position of disturbance (from rmax = 21 to rmax = 31), we track the position of the apparent horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For the final distribution of βr, both results show that βr decreases as r increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Moreover, the value of βr for the final moment reaches the maximum at rmin and gradually decreases to 0 (this is required by the Dirichlet boundary condition we adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' These results are shown on the following two figures, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3, we find that there is no apparent horizon in [rmin, rmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4, we find that there is an apparent horizon in [rmin, rmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It should be pointed out that even if there is no apparent horizon in [rmin, rmax], we can not completely rule out the situation that there is a apparent horizon in [0, rmin].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, this does not affect our conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Actually, when one sets rmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5, 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 1: The configuration of U − N (t) with tI = 1, tF = 11, A = 1 × 10−3 for the case of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 2: The configuration of U − N (t) with tI = 1, tF = 11, A = 1 × 10−3 for the case of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' rmax = 21, one can find the apparent horizon is located at rH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7768 < 1 with the same U − N (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' To put it in a nutshell, the apparent horizon can emerge by any small perturbation of the scalar field on the boundary far way enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In other words, one can always adjust rmin so that the interval [rmin, rmax] contains the apparent horizon rH, and the change of rmin do not affect the position of the apparent horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In fact, we are sure that when the outer boundary perturbation is vanished, there must be no apparent horizon, because the (analytic) solution obtained at this time is still a pure AdS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The apparent horizon rH will increase as the increasing of amplitude A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' These results are shown at Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='I and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, this is different from the case without the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Actually, we find no apparent horizon in the case of Λ = 0, when A is not large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' A rH(rmax = 21) rH(rmax = 31) 1 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7768(∗) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9525 2 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4900 3 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5950 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7200 4 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2700 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7925 5 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8750 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7675 6 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4350 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6750 7 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9550 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5225 8 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4500 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3250 9 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9150 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='0900 10 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3650 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8250 TABLE I: For rmin = 1, the apparent horizon rH changes under different amplitudes A in this case of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We denote that ∗ are calculated at the case rmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5, rmax = 21 and rmin do not impact the apparent horizon rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' X10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8 1 0 5 10 15 20 25 30 35 40 tX10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1 0 0 5 10 15 20 25 30 35 40 t10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 3: In this case of type I, the value of the shift function βr changes with time at rmin = 1 with outer boundary rmax = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' There is no apparent horizon in [rmin, rmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 4: In this case of type I, the value of the shift function βr changes with time at rmin = 1 with outer boundary rmax = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' There is a apparent horizon in [rmin, rmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' CONVERGENCE TEST Given initial data and boundary data, to calibrate the accuracy of the numerical implementation, we perform simulations at increasing resolution and compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The initial data in these tests are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='19) and change the location of the outer boundary rmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' N = 1001, N = 2001, N = 4001 spatial grid points are carried out in our simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We then estimate the relative error between different resolutions and define the convergence factor as Qself = log2 � ||u∆ − u∆/2|| ||u∆/2 − u∆/4|| � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1) where u∆ refers to the numerical solution obtained with resolution ∆r = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For a grid function fi, the norm defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1) is ||fi|| ≡ � 1 L N � i=1 (fi)2∆r �1/2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2) where L = rmax − rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Since the spatial discretization of the radial derivatives is second order and the time integration is also second order, the convergence factor Qself reduces to 2 as ∆ → 0 in analytical case [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The convergence factors for evolutionary variables are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' They are approximately 2, indicating second-order convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='23) is not enforced during the simulation but only used to get the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It is preserved by con- structing constraint preserving boundary conditions, although our numerical scheme is a free evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The norm of constraint is also defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, some subtle changes need to be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' When we compute the constraint, we have L = r(N − 1) − r(2) with i = 2, · · · , N − 1 since second order central difference is used to approximate radial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' To 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4988 rmin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4976 0 10 20 30 40 50 t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='4985 0 10 20 30 40 50 60 70 80 t11 A rH(rmax = 21) rH(rmax = 31) 1 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8895(∗) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1550 2 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7975 3 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1100 4 × 10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5300 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2650 5 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='1750 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3150 6 × 10−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7700 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2900 7 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3300 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='2050 8 × 10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8600 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='0750 9 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='3600 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='9000 10 × 10−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='8450 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6950 TABLE II: For rmin = 1, the apparent horizon rH changes under different amplitudes A in this case of type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We denote that ∗ are calculated at the case rmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5, rmax = 21 and rmin do not impact the apparent horizon rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 5: The convergence factors Qself of five fields α, βr, ϕ, U − and U + change over time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The outer boundary is at rmax = 31 with tI = 1, tF = 11, A = 1 × 10−3 in the case of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' tell the truth, the absolute size of the norm of the constraint is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' But, for different resolutions, we will compare with the change of the norm of the constraint with time by keeping the same initial boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' These results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' From these results, we know the constraints are vanished numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' This is exactly what we expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 6: For rmin = 1, rmax = 31 with tI = 1, tF = 11, A = 1 × 10−3 in the case of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' L2 norm of the constraint C1 with different resolutions 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 5 U 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 U* 4 self 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 0 10 20 30 40 50 60 70 80 t9 X10-3 N=1001 8 N=2001 N=4001 7 6 5 C 4 3 2 1 0 0 10 20 30 40 50 60 70 t12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 7: For rmin = 1, rmax = 31 with tI = 1, tF = 11, A = 1 × 10−3 in the case of type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' L2 norm of the constraint C2 with different resolutions V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSION The ADM and Bondi-Sachs frameworks for the Einstein equations have traditionally been looked upon as two separate approaches for numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We have worked out that in the Bondi-Sachs gauge (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='15), the ADM formulation of Einstein equations with a cosmological constant can be casted into a strongly hyperbolic problem for the lapse and the shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The ADM formulation in the Bondi-Sachs gauge has one constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='23) on the initial data for the lapse and shift and one constraint-preserving boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In our present work, the assumption of spherical symmetry is kept throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' We use the constraint preserving formulation to give numerical evidence that the pure AdS spacetime is unstable under slight perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Our simulation is carried out by changing the boundary conditions which is different from the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' [1] in which a slight perturbation is added to the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In fact, for a pure AdS solution, we find that any small perturbation of the scalar field at the boundary far away enough can cause the collapse of the pure AdS spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The numerical evidence is provided for the formation of apparent horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The relationship between the perturbation and the position of the apparent horizon has been studied numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' However, the situation is different from the case with zero cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Actually, there is no apparent horizon in the case of Λ = 0, when A is not large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' From the two tables I and II, we can see that for the same type of perturbation with the same amplitude, the radius of the apparent horizon will increase as the location of the perturbation staying off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' In addition, for the scalar field perturbation placed at the same position, the larger the amplitude of the perturbation, the larger the apparent horizon formed by collapsing finally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Two kinds of perturbation with different shapes express similar conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Acknowledgement This work was supported in part by the National Natural Science Foundation of China with grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='12075232, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='11622543, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='11947301, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='12047502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' This work is also supported by the Fundamental Research Funds for the Central Universities under Grant No: WK2030000036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Appendix A: Bianchi identity In this Appendix, we will give the details for picking out the independent components of Eab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' At first, when scalar is satisfied with ∇a∇aϕ = 0, we have the Bianchi identity which is given by ∇aEab = −8π(∇a∇aϕ)∇bϕ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A1) The nontrivial components of Eab are Evv, Evˆr = Eˆrv, Evv, Eθθ, Eφφ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' It can be proved that ∇µEµθ = 0 and ∇µEµφ = 0 are trivial under the assumption of spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For b = v, ∇µEµv = gˆrv ∂Eˆrv ∂v + gˆrˆr ∂Eˆrv ∂ˆr + gvˆr ∂Evv ∂ˆr − � 3gvˆrΓˆr ˆrv + gvˆrΓv vv + gˆrˆrΓˆr ˆrˆr + 2gθθΓˆr θθ � Eˆrv −2gθθΓv θθEvv − � gvˆrΓˆr vv + gˆrˆrΓˆr ˆrv � Eˆrˆr = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A2) X106 5 N=1001 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 N=2001 N=4001 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 3 N C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content='5 0 0 10 20 30 40 50 60 70 80 t13 For b = ˆr, ∇µEµˆr = gvˆr ∂Evˆr ∂ˆr + gˆrv ∂Eˆrˆr ∂v + gˆrˆr ∂Eˆrˆr ∂ˆr − � 3gvˆrΓˆr vˆr + 2gˆrˆrΓˆr ˆrˆr + 2gθθΓˆr θθ � Eˆrˆr − � gvˆrΓˆr ˆrˆr + 2gθθΓv θθ � Evˆr − 2gθθΓθ θˆrEθθ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A3) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A3), when Evˆr = Eˆrˆr = 0, we have Eθθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Then one gets Eφφ = 0, since Eφφ = sin2 θEθθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A2), when Evˆr = Eˆrˆr = 0, we obtain gvˆr ∂Evv ∂ˆr − 2gθθΓv θθEvv = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A4) This equation is equivalent to ∂(Evvgvˆr) ∂ˆr − ∂gvˆr ∂ˆr Evv − 2gθθΓv θθEvv = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A5) If Evˆr = 0, we have Ev ˆr = Evµgµˆr = Evvgvˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' For gvˆr = e−2β, we have ∂gvˆr/∂ˆr = −2gvˆr∂β/∂ˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A5) becomes ∂Ev ˆr ∂ˆr + 2 �∂β ∂ˆr + 1 ˆr � Ev ˆr = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A6) Solving this equation (A6), one gets [26] Ev ˆr(v, ˆr) = ˆr2 0 ˆr2 exp 2 � β(v, ˆr0) − β(v, ˆr) � Ev ˆr(v, ˆr0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A7) Therefore, the independent components of Eab are Eˆrˆr and Evˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' That means evolution equations are Eˆrˆr = 0 , Evˆr = 0 , ∇a∇aϕ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A8) According to the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (A6), Ev ˆr = 0 (A9) is recommended to be a constraint equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 14 Appendix B: calculation of basic quantity In this Appendix, we give results of some basic quantities which we have used in the process of deriving evolution equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' The non-vanished Christoffel symbols are given as follows, Γv vv = 1 2ˆr2 � − V + 2ˆrV β,ˆr + ˆrV,ˆr + 4ˆr2β,v � , Γv θθ = −ˆre−2β , Γv φφ = −ˆre−2β sin2 θ , Γˆr vv = 1 2ˆr3 � − V 2 + 2ˆrV 2β,ˆr − ˆr2V,v + ˆrV V,ˆr + 2ˆr2V β,v � , Γˆr vˆr = Γˆr ˆrv = 1 2ˆr2 � V − ˆrV,ˆr − 2ˆrV β,ˆr � , Γˆr ˆrˆr = 2β,ˆr , Γˆr θθ = −V e−2β , Γˆr φφ = −V e−2β sin2 θ , Γθ ˆrθ = Γθ θˆr = 1 ˆr , Γθ φφ = − cos θ sin θ , Γφ ˆrφ = Γφ φˆr = 1 ˆr , Γφ θφ = Γφ φθ = cot θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (B1) The components of the Einstein tensor are Gvv = 1 ˆr3 � 2V 2β,ˆr − ˆrV,v + V e2β − V V,ˆr + 2ˆrV β,v � , Gvˆr = Gˆrv = 1 ˆr2 � − 2V β,ˆr + V,ˆr − e2β� , Gˆrˆr = 4 ˆr β,ˆr , Gθθ = e−2β 2 � − 2V β,ˆr + 2ˆrV β,ˆrˆr + 2ˆrV,ˆrβ,ˆr + ˆrV,ˆrˆr + 4ˆr2β,vˆr � , Gφφ = e−2β 2 � − 2V β,ˆr + 2ˆrV β,ˆrˆr + 2ˆrV,ˆrβ,ˆr + ˆrV,ˆrˆr + 4ˆr2β,vˆr � sin2 θ , (B2) and Gv ˆr = e−2β ˆr2 � 2V β,v − V,v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (B3) The components of ∇a∇bϕ are ∇v∇vϕ = ϕ,vv − 1 2ˆr2 � − V + 2ˆrV β,ˆr + ˆrV,ˆr + 4ˆr2β,v � ϕ,v − 1 2ˆr3 � − V 2 + 2ˆrV 2β,ˆr − ˆr2V,v + ˆrV V,ˆr + 2ˆr2V β,v � ϕ,ˆr , ∇v∇ˆrϕ = ∇ˆr∇vϕ = ϕ,vˆr − 1 2ˆr2 � V − ˆrV,ˆr − 2ˆrV β,ˆr � ϕ,ˆr , ∇ˆr∇ˆrϕ = ϕ,ˆrˆr − 2β,ˆrϕ,ˆr , ∇θ∇θϕ = V e−2βϕ,ˆr + ˆre−2βϕ,v , ∇φ∇φϕ = � V e−2βϕ,ˆr + ˆre−2βϕ,v � sin2 θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (B4) The expression of ∇a∇aϕ is given by ∇a∇aϕ = e−2β� 2ϕ,vˆr + V ˆr ϕ,ˆrˆr + �V,ˆr ˆr + V ˆr2 � ϕ,ˆr + 2 ˆr ϕ,v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (B5) 15 The expression of ∇aϕ∇aϕ is given by ∇aϕ∇aϕ = V ˆr e−2βϕ,ˆrϕ,ˆr + 2e−2βϕ,ˆrϕ,v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' (B6) [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE5T4oBgHgl3EQfEQ7d/content/2301.05413v1.pdf'} +page_content=' 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0000000000000000000000000000000000000000..3fa55da0b74bb9d78116fc59c9b4214bc2815a38 --- /dev/null +++ b/d9A0T4oBgHgl3EQfHP8Z/content/tmp_files/2301.02057v1.pdf.txt @@ -0,0 +1,174 @@ +arXiv:2301.02057v1 [cs.CL] 5 Jan 2023 +TEXTDESCRIPTIVES: A PYTHON PACKAGE FOR CALCULATING +A LARGE VARIETY OF STATISTICS FROM TEXT +A PREPRINT +Lasse Hansen +Department of Affective Disorders - Psychiatry +Aarhus University Hospital +Aarhus, Denmark +Kenneth Enevoldsen +Center for Humanities Computing +Aarhus University +Aarhus, Denmark +January 6, 2023 +ABSTRACT +TextDescriptives is a Python package for calculating a large variety of statistics from text. It is built +on top of spaCy and can be easily integrated into existing workflows. The package has already been +used for analysing the linguistic stability of clinical texts, creating features for predicting neuropsy- +chiatric conditions, and analysing linguistic goals of primary school students. This paper describes +the package and its features. +Keywords Python • natural language processing • spacy • feature extraction +1 +TextDescriptives: A Python package for calculating a large variety of statistics from text +2 +Summary +Natural language processing (NLP) tasks often require a thorough understanding and description of the corpus. +Document-level metrics can be used to identify low-quality data, assess outliers, or understand differences between +groups. Further, text metrics have long been used in fields such as the digital humanities where e.g. metrics of text +complexity are commonly used to analyse, understand and compare text corpora. However, extracting complex met- +rics can be an error-prone process and is rarely rigorously tested in research implementations. This can lead to subtle +differences between implementations and reduces the reproducibility of scientific results. +TextDescriptives offers a simple and modular approach to extracting both simple and complex metrics from text. +It achieves this by building on the spaCy framework (Honnibal et al. 2020). This means that TextDescriptives +can easily be integrated into existing workflows while leveraging the efficiency and robustness of the spaCy library. +The package has already been used for analysing the linguistic stability of clinical texts (Hansen et al. 2022), creating +features for predicting neuropsychiatric conditions (Hansen 2022), and analysing linguistic goals of primary school +students (Tannert 2023). +3 +Statement of need +Computational text analysis is a broad term that refers to the process of analyzing and understanding text data. This +often involves calculating a set of metrics that describe relevant properties of the data. Dependent on the task at hand, +this can range from simple descriptive statistics related to e.g. word or sentence length to complex measures of text + +TEXTDESCRIPTIVES - JANUARY 6, 2023 +complexity, coherence, or quality. This often requires drawing on multiple libraries and frameworks or writing custom +code. This can be time-consuming and prone to bugs, especially with more complex metrics. +TextDescriptives seeks to unify the extraction of document-level metrics, in a modular fashion. The integration +with spaCy allows the user to seamlessly integrate TextDescriptives in existing pipelines as well as giving the +TextDescriptives package access to model-based metrics such as dependency graphs and part-of-speech tags. The +ease of use and the variety of available metrics allows researchers and practitioners to extend the granularity of their +analyses within a tested and validated framework. +Implementations of the majority of the metrics included in TextDescriptives exist, but none as feature complete. +The textstat library (Ward 2022) implements the same readability metrics, however, each metric has to be extracted +one at a time with no interface for multiple extractions. spacy-readability (Holtzscher 2019) adds readability +metrics to spaCy pipelines, but does not work for new versions of spaCy (>=3.0.0). The textacy (DeWilde 2021) +package has some overlap with TextDescriptives, but with a different focus. TextDescriptives focuses on +document-level metrics, and includes a large number of metrics not included in textacy (dependency distance, co- +herence, and quality), whereas textacy includes components for preprocessing, information extraction, and visual- +ization that are outside the scope of TextDescriptives. What sets TextDescriptives apart is the easy access to +document-level metrics through a simple user-facing API and exhaustive documentation. +4 +Features & Functionality +TextDescriptives +is +a +Python +package +and +provides +the +following +spaCy +pipeline +components: +textdescriptives.descriptive_stats: Calculates the total number of tokens, number of unique tokens, +number of characters, and the proportion of unique tokens, as well as the mean, median, and standard deviation of +token length, sentence length, and the number of syllables per token. textdescriptives.readability: Calculates +the Gunning-Fog index, the SMOG index, Flesch reading ease, Flesch-Kincaid grade, the Automated Readability +Index, the Coleman-Liau index, the Lix score, and the Rix score. textdescriptives.dependency_distance: +Calculates the mean and standard deviation of the dependency distance (the average distance between a word and +its head word), and the mean and the standard deviation of the proportion adjacent dependency relations on the +sentence level. textdescriptives.pos_proportions: Calculates the proportions of all part-of-speech tags in +the documents. textdescriptives.coherence: Calculates the first- and second-order coherence of the document +based on word embedding similarity between sentences. textdescriptives.quality: Calculates the text-quality +metrics proposed in Rae et al. (2022) and Raffel et al. (2020). These measures can be used for filtering out low-quality +text prior to model training or text analysis. These include heuristics such as the number of stop words, ratio of +words containing alphabetic characters, proportion of lines ending with an ellipsis, proportion of lines starting with a +bullet point, ratio of symbols to words, and whether the document contains a specified string (e.g. “lorem ipsum”), as +well as repetitious text metrics such as the proportion of lines that are duplicates, the proportion of paragraphs in a +document that are duplicates, the proportion of n-gram duplicates, and the proportion of characters in a document that +are contained within the top n-grams. +All the components can be added to an existing spaCy pipeline with a single line of code, and jointly extracted to a +dataframe or dictionary with a single call to textdescriptives.extract_{df|dict}(doc). +5 +Example Use Cases +Descriptive statistics can be used to summarize and understand data, such as by exploring patterns and relationships +within the data, getting a better understanding of the data set, or identifying any changes in the distribution of the data. +Readability metrics, which assess the clarity and ease of understanding of written text, have a variety of applications, +including the design of educational materials and the improvement of legal or technical documents (DuBay 2004). +Dependency distance can be used as a measure of language comprehension difficulty or of sentence complexity and +has been used for analysing properties of natural language or for similar purposes as readability metrics (Gibson et +al. 2019; Liu 2008). The proportions of different parts of speech in a document have been found to be predictive +of certain mental disorders and can also be used to assess the quality and complexity of text (Tang et al. 2021). +Semantic coherence, or the logical connection between sentences, has primarily been used in the field of computational +psychiatry to predict the onset of psychosis or schizophrenia (Parola et al. 2022; Bedi et al. 2015), but it also has other +applications in the digital humanities. Measures of text quality are useful cleaning and identifying low-quality data +(Rae et al. 2022; Raffel et al. 2020). +2 + +TEXTDESCRIPTIVES - JANUARY 6, 2023 +6 +Target Audience +The package is mainly targeted at NLP researchers and practitioners. In particular, researchers from fields new to NLP +such as the digital humanities and social sciences as researchers might benefit from the readability metrics as well as +the more complex, but highly useful, metrics such as coherence and dependency distance. +7 +Acknowledgements +The authors thank the contributors of the package including Ludvig Olsen, for his work on the early versions of +TextDescriptives,Martin Bernstorff for his work on the part-of-speech component, and Frida Hæstrup and Roberta +Rocca for important fixes. The authors would also like to Dan Sattrup Nielsen for helpful reviews on early iterations +of the text quality implementations. +References +Bedi, Gillinder, Facundo Carrillo, Guillermo A. Cecchi, Diego Fernández Slezak, Mariano Sigman, Natália +B. Mota, Sidarta Ribeiro, Daniel C. Javitt, Mauro Copelli, and Cheryl M. Corcoran. +2015. +“Automated +Analysis of Free Speech Predicts Psychosis Onset in High-Risk Youths.” +Npj Schizophrenia 1 (1): +1–7. +https://doi.org/10.1038/npjschz.2015.30. +DeWilde, +Burton. +2021. +Textacy: +NLP, +Before +and +After +spaCy +(version +0.12.0). +https://github.com/chartbeat-labs/textacy. +DuBay, William H. 2004. “The Principles of Readability.” Online Submission. +Gibson, Edward, Richard Futrell, Steven P. Piantadosi, Isabelle Dautriche, Kyle Mahowald, Leon Bergen, and Roger +Levy. 2019. “How Efficiency Shapes Human Language.” Trends in Cognitive Sciences 23 (5): 389–407. +Hansen, Lasse. 2022. “Speaking Your Mind: Voice as a Marker for Mental Disorders.” PhD thesis, Aarhus University. +Hansen, Lasse, Kenneth Enevoldsen, Martin Bernstorff, Erik Perfalk, Andreas A. Danielsen, Kristoffer L. Nielbo, and +Søren D. Østergaard. 2022. “Lexical Stability of Psychiatric Clinical Notes from Electronic Health Records over +a Decade.” medRxiv, January, 2022.09.05.22279610. https://doi.org/10.1101/2022.09.05.22279610. +Holtzscher, Michael. 2019. Spacy-Readability: spaCy Pipeline Component for Adding Text Readability Meta Data to +Doc Objects. (version 1.4.1). +Honnibal, Matthew, Ines Montani, Sofie Van Landeghem, and Adriane Boyd. 2020. spaCy: Industrial-Strength +Natural Language Processing in Python. https://doi.org/10.5281/zenodo.1212303. +Liu, Haitao. 2008. “Dependency Distance as a Metric of Language Comprehension Difficulty.” Journal of Cognitive +Science 9 (2): 159–91. +Parola, Alberto, Jessica Mary Lin, Arndis Simonsen, Vibeke Bliksted, Yuan Zhou, Huiling Wang, Lana Inoue, +Katja Koelkebeck, and Riccardo Fusaroli. +2022. “Speech Disturbances in Schizophrenia: Assessing Cross- +Linguistic Generalizability of NLP Automated Measures of Coherence.” +Schizophrenia Research, August. +https://doi.org/10.1016/j.schres.2022.07.002. +Rae, Jack W., Sebastian Borgeaud, Trevor Cai, Katie Millican, Jordan Hoffmann, Francis Song, John Aslanides, et +al. 2022. “Scaling Language Models: Methods, Analysis & Insights from Training Gopher.” arXiv:2112.11446. +arXiv. https://doi.org/10.48550/arXiv.2112.11446. +Raffel, Colin, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, +and Peter J. Liu. 2020. “Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer.” +arXiv:1910.10683 [Cs, Stat], July. http://arxiv.org/abs/1910.10683. +Tang, Sunny X., Reno Kriz, Sunghye Cho, Suh Jung Park, Jenna Harowitz, Raquel E. Gur, Mahendra T. Bhati, +Daniel H. Wolf, João Sedoc, and Mark Y. Liberman. 2021. “Natural Language Processing Methods Are Sen- +sitive to Sub-Clinical Linguistic Differences in Schizophrenia Spectrum Disorders.” Npj Schizophrenia 7 (1): 1–8. +https://doi.org/10.1038/s41537-021-00154-3. +Tannert, Morten. 2023. “Skriftsproglig Udvikling i Grundskolens Danskfag.” PhD thesis, Aarhus University. +Ward, Alex. 2022. Textstat. Textstat. https://github.com/textstat/textstat. +3 + diff --git a/d9A0T4oBgHgl3EQfHP8Z/content/tmp_files/load_file.txt b/d9A0T4oBgHgl3EQfHP8Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b9c40869a29586cd1e416eb222b8e4df7a08290d --- /dev/null +++ b/d9A0T4oBgHgl3EQfHP8Z/content/tmp_files/load_file.txt @@ -0,0 +1,187 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf,len=186 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='02057v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='CL] 5 Jan 2023 TEXTDESCRIPTIVES: A PYTHON PACKAGE FOR CALCULATING A LARGE VARIETY OF STATISTICS FROM TEXT A PREPRINT Lasse Hansen Department of Affective Disorders - Psychiatry Aarhus University Hospital Aarhus, Denmark Kenneth Enevoldsen Center for Humanities Computing Aarhus University Aarhus, Denmark January 6, 2023 ABSTRACT TextDescriptives is a Python package for calculating a large variety of statistics from text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' It is built on top of spaCy and can be easily integrated into existing workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The package has already been used for analysing the linguistic stability of clinical texts, creating features for predicting neuropsy- chiatric conditions, and analysing linguistic goals of primary school students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' This paper describes the package and its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Keywords Python • natural language processing • spacy • feature extraction 1 TextDescriptives: A Python package for calculating a large variety of statistics from text 2 Summary Natural language processing (NLP) tasks often require a thorough understanding and description of the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Document-level metrics can be used to identify low-quality data, assess outliers, or understand differences between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Further, text metrics have long been used in fields such as the digital humanities where e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' metrics of text complexity are commonly used to analyse, understand and compare text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' However, extracting complex met- rics can be an error-prone process and is rarely rigorously tested in research implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' This can lead to subtle differences between implementations and reduces the reproducibility of scientific results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' TextDescriptives offers a simple and modular approach to extracting both simple and complex metrics from text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' It achieves this by building on the spaCy framework (Honnibal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' This means that TextDescriptives can easily be integrated into existing workflows while leveraging the efficiency and robustness of the spaCy library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The package has already been used for analysing the linguistic stability of clinical texts (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2022), creating features for predicting neuropsychiatric conditions (Hansen 2022), and analysing linguistic goals of primary school students (Tannert 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 3 Statement of need Computational text analysis is a broad term that refers to the process of analyzing and understanding text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' This often involves calculating a set of metrics that describe relevant properties of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Dependent on the task at hand, this can range from simple descriptive statistics related to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' word or sentence length to complex measures of text TEXTDESCRIPTIVES - JANUARY 6, 2023 complexity, coherence, or quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' This often requires drawing on multiple libraries and frameworks or writing custom code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' This can be time-consuming and prone to bugs, especially with more complex metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' TextDescriptives seeks to unify the extraction of document-level metrics, in a modular fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The integration with spaCy allows the user to seamlessly integrate TextDescriptives in existing pipelines as well as giving the TextDescriptives package access to model-based metrics such as dependency graphs and part-of-speech tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The ease of use and the variety of available metrics allows researchers and practitioners to extend the granularity of their analyses within a tested and validated framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Implementations of the majority of the metrics included in TextDescriptives exist, but none as feature complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The textstat library (Ward 2022) implements the same readability metrics, however, each metric has to be extracted one at a time with no interface for multiple extractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' spacy-readability (Holtzscher 2019) adds readability metrics to spaCy pipelines, but does not work for new versions of spaCy (>=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The textacy (DeWilde 2021) package has some overlap with TextDescriptives, but with a different focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' TextDescriptives focuses on document-level metrics, and includes a large number of metrics not included in textacy (dependency distance, co- herence, and quality), whereas textacy includes components for preprocessing, information extraction, and visual- ization that are outside the scope of TextDescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' What sets TextDescriptives apart is the easy access to document-level metrics through a simple user-facing API and exhaustive documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 4 Features & Functionality TextDescriptives is a Python package and provides the following spaCy pipeline components: textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='descriptive_stats: Calculates the total number of tokens, number of unique tokens, number of characters, and the proportion of unique tokens, as well as the mean, median, and standard deviation of token length, sentence length, and the number of syllables per token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='readability: Calculates the Gunning-Fog index, the SMOG index, Flesch reading ease, Flesch-Kincaid grade, the Automated Readability Index, the Coleman-Liau index, the Lix score, and the Rix score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='dependency_distance: Calculates the mean and standard deviation of the dependency distance (the average distance between a word and its head word), and the mean and the standard deviation of the proportion adjacent dependency relations on the sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='pos_proportions: Calculates the proportions of all part-of-speech tags in the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='coherence: Calculates the first- and second-order coherence of the document based on word embedding similarity between sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='quality: Calculates the text-quality metrics proposed in Rae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' (2022) and Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' These measures can be used for filtering out low-quality text prior to model training or text analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' These include heuristics such as the number of stop words, ratio of words containing alphabetic characters, proportion of lines ending with an ellipsis, proportion of lines starting with a bullet point, ratio of symbols to words, and whether the document contains a specified string (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' “lorem ipsum”), as well as repetitious text metrics such as the proportion of lines that are duplicates, the proportion of paragraphs in a document that are duplicates, the proportion of n-gram duplicates, and the proportion of characters in a document that are contained within the top n-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' All the components can be added to an existing spaCy pipeline with a single line of code, and jointly extracted to a dataframe or dictionary with a single call to textdescriptives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content='extract_{df|dict}(doc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 5 Example Use Cases Descriptive statistics can be used to summarize and understand data, such as by exploring patterns and relationships within the data, getting a better understanding of the data set, or identifying any changes in the distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Readability metrics, which assess the clarity and ease of understanding of written text, have a variety of applications, including the design of educational materials and the improvement of legal or technical documents (DuBay 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Dependency distance can be used as a measure of language comprehension difficulty or of sentence complexity and has been used for analysing properties of natural language or for similar purposes as readability metrics (Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Liu 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The proportions of different parts of speech in a document have been found to be predictive of certain mental disorders and can also be used to assess the quality and complexity of text (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Semantic coherence, or the logical connection between sentences, has primarily been used in the field of computational psychiatry to predict the onset of psychosis or schizophrenia (Parola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Bedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2015), but it also has other applications in the digital humanities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Measures of text quality are useful cleaning and identifying low-quality data (Rae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 2 TEXTDESCRIPTIVES - JANUARY 6, 2023 6 Target Audience The package is mainly targeted at NLP researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' In particular, researchers from fields new to NLP such as the digital humanities and social sciences as researchers might benefit from the readability metrics as well as the more complex, but highly useful, metrics such as coherence and dependency distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' 7 Acknowledgements The authors thank the contributors of the package including Ludvig Olsen, for his work on the early versions of TextDescriptives,Martin Bernstorff for his work on the part-of-speech component, and Frida Hæstrup and Roberta Rocca for important fixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' The authors would also like to Dan Sattrup Nielsen for helpful reviews on early iterations of the text quality implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' References Bedi, Gillinder, Facundo Carrillo, Guillermo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} +page_content=' Cecchi, Diego Fernández 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9A0T4oBgHgl3EQfHP8Z/content/2301.02057v1.pdf'} diff --git a/ddA0T4oBgHgl3EQfG_-E/vector_store/index.faiss b/ddA0T4oBgHgl3EQfG_-E/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..d00282843b1d1175c9b06edc49d98cc3499a5cc0 --- /dev/null +++ b/ddA0T4oBgHgl3EQfG_-E/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9b97db5eb026253587d783b0be0a83cf389628d6b9f319f6d2692c5980e87e32 +size 3538989 diff --git a/ddE4T4oBgHgl3EQfpg0y/content/tmp_files/2301.05192v1.pdf.txt b/ddE4T4oBgHgl3EQfpg0y/content/tmp_files/2301.05192v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0495bfa7f3f188738e88f7b505570dd48d95544 --- /dev/null +++ b/ddE4T4oBgHgl3EQfpg0y/content/tmp_files/2301.05192v1.pdf.txt @@ -0,0 +1,930 @@ +High-Accuracy Approximation of Evolutionary Pairwise +Games on Complex Networks +Hongyu Wanga, Aming Lia,∗, Long Wanga,∗ +aCenter for Systems and Control, College of Engineering, Peking University, Beijing, +100871, China +Abstract +Previous studies have shown that the topological properties of a complex +network, such as heterogeneity and average degree, affect the evolutionary +game dynamics on it. However, traditional numerical simulations are usu- +ally time-consuming and demand a lot of computational resources. In this +paper, we propose the method of dynamical approximate master equations +(DAMEs) to accurately approximate the evolutionary outcomes on complex +networks. We demonstrate that the accuracy of DAMEs supersedes previ- +ous standard pairwise approximation methods, and DAMEs require far fewer +computational resources than traditional numerical simulations. We use pris- +oner’s dilemma and snowdrift game on regular and scale-free networks to +demonstrate the applicability of DAMEs. Overall, our method facilitates the +investigation of evolutionary dynamics on a broad range of complex networks, +and provides new insights into the puzzle of cooperation. +∗Corresponding authors +Email addresses: wanghongyu1998@pku.edu.cn (Hongyu Wang), +liaming@pku.edu.cn (Aming Li), longwang@pku.edu.cn (Long Wang) +Preprint submitted to Elsevier +January 13, 2023 +arXiv:2301.05192v1 [q-bio.PE] 12 Jan 2023 + +1. +Introduction +Many levels of biological organization, from single-celled organisms to +human society, are based on cooperation [1]. However, in the context of Dar- +winian evolution, natural selection favors defectors over cooperators. Evolu- +tionary game theory is a general mathematical framework for studying the +cooperation between unrelated individuals [2; 3; 4; 5; 6; 7]. As metaphors +for studying the evolution of cooperation, pairwise games such as prisoner’s +dilemma (PD) and snowdrift game (SG) have been widely adopted by re- +searchers from different backgrounds [1; 8; 9; 10; 11]. In infinitely well-mixed +populations, evolution under replicator dynamics leads to a stable fraction +of cooperators for SG but to the complete extinction of cooperators in PD +[12; 13]. +To better understand the emergence of cooperation in more realistic sit- +uations, the importance of studying the behavior of individuals with popu- +lation structure should be highlighted. Graph theory provides a convenient +framework to describe the population structure for studying the evolution +of cooperation [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26], where the +vertices of a graph represent players and the edges define the network of con- +tacts between players. Researchers found that when individuals interacted +only with their neighbors, the fraction of cooperators in both PD and SG +differs from that in well-mixed populations [8; 9; 14; 27; 16; 28; 29]. Regu- +lar graphs ignore the uniqueness of individuals, that is, different individuals +may have different numbers of neighbors they interact with. Scale-free (SF) +networks, whose vertex connectivities follow a power-law distribution, are +often used to represent more realistic heterogeneous networks [30]. Santos +and Pacheco found that the equilibrium frequencies of cooperators are higher +when playing the PD and the SG on SF networks than when playing them on +regular networks [10]. They attributed this phenomenon to the generation +rules of the Barab´asi-Albert model, that is, the vertices in the network are +added sequentially, and the newly added vertice is more likely to connect +to vertices with higher degrees. However, the lack of more in-depth studies, +particularly a theoretical explanation, makes this phenomenon challenging to +understand. Ohtsuki et al. proved that natural selection favors cooperation +if the benefit of altruistic behavior, divided by the costs, exceeds the aver- +age number of neighbors [15]. Recent works explore general formulations of +fixation probabilities for pairwise games under weak selection that apply to +graph-structured populations [31; 32; 33; 34]. In this paper, we propose the +2 + +dynamical approximate master equations (DAMEs) to describe evolutionary +game dynamics on complex networks, starting from a theoretical explanation +of how the heterogeneity of networks affects the evolutionary dynamics and +finally leads to the prevalence of cooperation on SF networks. +Indeed, beyond numerical simulations, we need a theoretical method to +study the evolutionary behavior of various nodes on complex networks. The +most commonly used method for binary-state dynamics on complex networks +is the mean-field (MF) theory [35; 36; 37; 38; 39]. +The pairwise approx- +imation (PA) theory suggested by Dickman improves the accuracy of MF +theory [40; 41; 42; 43; 44] and has been applied to the study of evolution- +ary dynamics on complex networks [45; 46; 47; 48; 49]. The approximate +master equations (AMEs) have achieved high accuracy beyond the PA level +in the binary-state dynamics on complex networks [50; 51; 52; 53], but the +transition probabilities between two states in the AME system are static +(time-invariant). In evolutionary games on complex networks, the transition +probabilities between states of each node depend on the difference between +the node and its neighbors’ payoff, which is time-variant. Here we incor- +porate the transition probabilities in the time-variant form in our DAMEs, +which can accurately approximate evolutionary game dynamics and predict +the equilibrium frequencies of cooperators in a short time. +2. +Evolutionary pairwise games on complex networks +We consider a population captured by the complex network with N nodes. +Each player in the population can be in two states, cooperation or defection. +The degree of a node represents the number of edges between the node and its +neighbors. Degree distribution P(k) = Nk/N represents the probability that +the degree of a randomly selected node in the network is k, where Nk refers +to the number of nodes with degree k. Here we assume that the network +is generated by the configuration model with fixed degree distribution P(k), +which does not have degree correlations [54]. +In both PD and SG, two players have to decide whether to cooperate or +defect in each round. Both players receive R when they cooperate with each +other and P when they defect. A defector exploiting a cooperator receives +T, and the cooperator receives S. Following common practice, researchers +usually adjust the game to rely on a single parameter. For the PD, we have +T > R > P > S. Considering that T + S < 2R, we make 2 > T = b > 1, +R = 1, and P = S = 0, leaving the advantage of defectors b be the single +3 + +parameter. We have tested that if S = −ε < 0 (ε ≪ 1) is set to satisfy S < P, +the result will not change. For the SG, we have T > R > S > P. Considering +that T + S = 2R, we make T = β > 1, R = β − 1/2, S = β − 1, and P = 0, +such that the cost-to-benefit ratio can be written as r = 1/(2β − 1). +Evolution is carried out by implementing the finite population analog of +replicator dynamics through the following transition probabilities: In each +generation, all individuals play a single-round game with all of their neighbors +and accumulate the payoffs. Whenever an individual i desires to update the +strategy, one of its neighbor j will be drawn from its ki neighbors. +The +probability that the individual i copies the strategy of individual j is given +by the Fermi function +Psi→sj = φ(πi, πj) = +1 +1 + eα(πi−πj) +(1) +where α ∈ [0, ∞) denotes the intensity of selection. α → 0 leads to the +random drift and α → ∞ leads to the deterministic imitation dynamics. +3. +Dynamical approximate master equations +Define Ck,m(t) (Dk,m(t)) as the fraction of k-degree nodes that are coop- +erators (defectors) at time t and have m defector neighbors. The DAMEs +consist of M = � +k,m 2 = (1 + kmax − kmin)(2 + kmax + kmin) variables. Then +the fraction of cooperators of k-degree nodes at time t is given by +ρk(t) = +k +� +m=0 +Ck,m(t), +(2) +and the fraction of cooperators in the whole network is +ρ(t) = +� +k +P(k)ρk(t). +(3) +By assuming that the initial cooperators are randomly selected with a +fraction ρ(0), the initial conditions are +Ck,m(0) = ρ(0)Bk,m(1 − ρ(0)), +(4) +Dk,m(0) = (1 − ρ(0))Bk,m(1 − ρ(0)), +(5) +where Bk,m(q) = +� k +m +� +qm(1 − q)k−m is the binomial factor. +4 + +The approximate master equations for the evolution of Ck,m(t) and Dk,m(t) +are +dCk,m(t) +dt += +−PCk,m→Dk,mCk,m(t)m +k ++PDk,m→Ck,mDk,m(t)(k − m) +k +−PCk,m→Ck,m+1(k − m)βCCk,m(t) ++PCk,m−1→Ck,m(k − m + 1)βCCk,m−1(t) +−PCk,m→Ck,m−1mγCCk,m(t) ++PCk,m+1→Ck,m(m + 1)γCCk,m+1(t), +(6) +dDk,m(t) +dt += +−PDk,m→Ck,mDk,m(t)(k − m) +k ++PCk,m→Dk,mCk,m(t)m +k +−PDk,m→Dk,m+1(k − m)βDDk,m(t) ++PDk,m−1→Dk,m(k − m + 1)βDDk,m−1(t) +−PDk,m→Dk,m−1mγDDk,m(t) ++PDk,m+1→Dk,m(m + 1)γDDk,m+1(t). +(7) +The coefficient PCk,m→Dk,m is defined as the transition probability that a +k-degree cooperator, which has m defector neighbors at time t, changes its +strategy to defection by time t + dt, where dt is an infinitesimally small time +interval. Similarly, PDk,m→Ck,m is the transition probability that a k-degree +defector, which has m defector neighbors at time t, changes its strategy to +cooperation by time t + dt. +The coefficient PCk,m→Ck,m+1 is defined as the transition probability that +a cooperator, which has k neighbors and m of them are defectors at time t, +changes its state into Ck,m+1 by time t+dt, which means one of its cooperator +neighbors becomes a defector. The coefficient βC is defined as the probabil- +ity that the randomly selected neighbor is a defector when a cooperator’s +cooperator neighbor updates its strategy. The coefficient PCk,m→Ck,m−1 is de- +fined as the transition probability that a cooperator, which has k neighbors +and m of them are defectors at time t, changes its state into Ck,m−1 by time +t + dt, which means one of its defector neighbors becomes a cooperator. The +5 + +C +C +C +, +−1 +, +−1→ +, +1 +, +→ +, +−1 +, +→ +, ++1 +, ++1→ +, +, +→ +, +−1 +, ++1→ +, +, +−1→ +, +, +→ +, ++1 +, +→ +, +, +→ +, +C +D +C +, +C +D +D +, ++1 +D +C +C +, +−1 +D +D +C +, +D +D +D +, ++1 +Figure 1: Schematic representation of the meaning of each variable in DAMEs. Define +Ck,m(t) (Dk,m(t)) as the fraction of k-degree nodes that are cooperators (defectors) at +time t and have m neighboring defectors. Ck,m(t) and Dk,m(t) change at each time step +due to the node itself or its neighbors’ updating strategies. For example, the coefficient +PCk,m→Dk,m is defined as the transition probability that a k-degree cooperator, which has +m defector neighbors at time t, changes its strategy to defection by time t + dt. A node +may update its strategy only when it plays with nodes that adopt the different strategy, +and the probability that a Ck,m(t) node plays with its defector neighbors is m +k . +coefficient γC is defined as the probability that the randomly selected neigh- +bor is a cooperator when a cooperator’s defector neighbor updates its strat- +egy. PCk,m−1→Ck,m, PCk,m→Ck,m−1, PCk,m+1→Ck,m, PDk,m→Dk,m+1, PDk,m−1→Dk,m, +PDk,m→Dk,m−1 and PDk,m+1→Dk,m, βD, γD are defined in the same way. The +mathematical expressions of these variables are given somewhere below. +In order to compute these transition probabilities, we can directly cal- +culate the payoff of each class of nodes and use the limited information in +the DAMEs system to estimate the payoff of each class of nodes’ first-order +neighbors and second-order neighbors. The approximation in this method is +that we assume that the neighbor configuration of each node is the product +of independent single event probability. That is, the network under study +should have no degree correlations, which will lead to inaccurate results of +our model under some extreme networks (such as social networks and e-mail +6 + +networks). +Step 1: Compute the number of cooperator neighbors and defector neigh- +bors of nodes with different degree. +The number of defector neighbors of a cooperator with degree ki can be +computed by +NC−D(ki) = +�ki +m=0 mCki,m +�ki +m=0 Cki,m +. +(8) +NC−C(ki), ND−C(ki), ND−D(ki) are defined similarly. +Step 2: Compute the degree distribution and payoffs of each class of +nodes’ first-order neighbors. +We can compute the degree distribution of cooperators which have at +least a cooperator neighbor as +PkC−C(ki) = +P(ki) �ki−1 +m=0 Cki,m +� +k P(k) +��k−1 +m=0 Ck,m +�, +(9) +ki ∈ [kmin, kmax]. The reason that the integral has an upper bound of ki − 1 +is that a node with degree ki can have at most ki − 1 defector neighbors +to ensure that it has at least one cooperator neighbor. The probability of +a node connecting with nodes with degree ki depends on the ratio of the +number of its edges to the number of all edges. Thus the degree distribution +of first-order cooperator neighbors of a cooperator can be computed as: +PC−C(ki) = +kiPkC−C(ki) +� +k kPkC−C(k). +(10) +The payoffs of a cooperator’s first-order cooperator neighbors depend on +the number of their cooperator neighbors and defector neighbors. +πC−C(ki) = +�ki−1 +m=0(ki − m)Cki,m +�ki−1 +m=0 Cki,m +· R + +�ki−1 +m=0 mCki,m +�ki−1 +m=0 Cki,m +· S. +(11) +The fraction of defector neighbors of a cooperator’s cooperator neighbor +can be computed by +βC = +� +k P(k)m �k−1 +m=0 Ck,m +� +k P(k)k +��k−1 +m=0 Ck,m +�. +(12) +7 + +βD, γC, γD can be computed similarly. +Step 3: Compute PCk,m→Dk,m and PDk,m→Ck,m, in which case the focal +players are selected to update strategies. +When the focal players is a defector, its payoff is +πD(k, m) = (k − m)T + mP. +(13) +Using the conclusion of Step 2, we can compute +PkC−D(ki) = +P(ki) �ki +m=1 Cki,m +� +k P(k) +��ki +m=1 Ck,m +�, +(14) +PD−C(ki) = +kiPkC−D(ki) +� +k kPkC−D(k), +(15) +πD−C(k) = +�ki +m=1(ki − m)Cki,m +�ki +m=1 Cki,m +· R + +�ki +m=1 mCki,m +�ki +m=1 Cki,m +· S. +(16) +Thus the transition probability PDk,m→Ck,m can be computed as +PDk,m→Ck,m = +kmax +� +ki=kmin +PD−C(ki)φ(πD(k, m), πD−C(ki)). +(17) +Because the probability that a randomly selected neighbor is a coopera- +tor is (k − m)/k, we multiply this coefficient in the second line in Eq. (6). +PCk,m→Dk,m can be computed in the same way. +Step 4: Compute other transition probabilities, in which case the neigh- +bors of the focal players are selected to update strategies. +PCk,m→Ck,m+1 is defined as the probability that one of the cooperator neigh- +bors of a focal player A in class Ck,m becomes a defector by time t + dt. A is +a cooperator, so this first-order cooperator neighbor must be connected with +a second-order defector neighbor. We can compute A’s first-order cooperator +neighbors’ degree distribution PC−C(ki) and payoff πC−C(ki) by the method +in Step 2. We can also compute A’s second-order defector neighbors’ degree +distribution PC−D(kj) and payoff πC−D(kj). Thus PCk,m→Ck,m+1 is given by +PCk,m→Ck,m+1 = +� +ki +� +kj +PC−C(ki)PC−D(kj) +×φ(πC−C(ki), πC−D(kj)). +(18) +8 + +PCk,m−1→Ck,m, PDk,m→Dk,m−1 and PDk,m+1→Dk,m can be computed similarly +because in these cases first-order neighbors can only update their strategy +by imitating the strategies of second-order neighbors. +PCk,m→Ck,m−1 is defined as the probability that one of the defector neigh- +bors of a focal player B in class Ck,m becomes a cooperator by time t + dt. +We must consider that the first-order defector neighbors have probabilities +to imitate B’s strategy. Similarly, we can compute the first-order defector +neighbors’ degree distribution PC−D(ki) and their payoff πC−D(ki) and the +second-order cooperator neighbors’ degree distribution PD−C(kj) and pay- +off πD−C(kj). +The number of cooperator neighbors of first-order defector +neighbors(with degree ki) can be computed by the method in Step 1: +ND−C(ki) = +�ki−1 +m=0(ki − m)Dki,m +�ki−1 +m=0 Dki,m +. +(19) +Thus first-order defector neighbors with degree ki have a probability of +1/ND−C(ki) to imitate B’s strategy, and have a probability of [ND−C(ki) − +1]/ND−C(ki) to imitate a second-order cooperator node’ strategy. Therefore +PCk,m→Ck,m−1 is computed by +PCk,m→Ck,m−1(k, m) = +� +ki +PC−D(ki) +� +1 +ND−C(ki) +×φ(πC−D(ki), πC(k, m)) + ND−C(ki) − 1 +ND−C(ki) +× +� +kj +PD−C(kj)φ(πC−D(ki), πD−C(kj)) +� +� . +(20) +PCk,m+1→Ck,m, PDk,m→Dk,m+1 and PDk,m−1→Dk,m can be computed similarly. +4. +Simulations of evolutionary pairwise games on networks +To test the accuracy of the approximations, we will now compare the +results of the theory explained in the previous section to the numerical sim- +ulations. We consider the simulated dynamics on different types of networks +with N = 10000 nodes. +We consider two kinds of networks with differ- +ent degree distributions: regular ring networks and scale-free networks. For +9 + +scale-free networks, the degree distribution P(k) ∼ k−γ, 2 ≤ γ ≤ 3 obeys +the power law. For convenience, we use the configuration model to generate +scale-free networks with degree fixed distribution P(k), which is given by +the Barab´asi-Albert model. The initial fraction of cooperators is ρ = 50%. +Our simulation results are the average of more than 1000 independent sim- +ulations. PAs and DAMEs used networks with the same degree distribution +P(k) as simulations. +4.1. Equilibrium frequencies of cooperators +We first consider the influence of network structure and average connec- +tivity z on the evolution of cooperation. The role of average connectivity in +cooperation has been examined by several previous works [10; 14]. We calcu- +late the average equilibrium frequencies of cooperators of different models in +the steady state. We will focus on the differences between simulation results +and theoretical results given by DAMEs and the PA method. +Firstly, we consider the evolutionary games on regular ring networks. As +the average connectivity z of the network increases, the equilibrium frequen- +cies of cooperators decrease rapidly for the PD (top left panel of Fig. 2). For +the SG (top right panel of Fig. 2), the gradual increase of z results in the +equilibrium frequencies of cooperators gradually approaching 1 − r, which is +consistent with the result concluded by the replicator equation in well-mixed +populations. In some parameter spaces, the equilibrium frequencies of coop- +erators for the SG on regular ring networks are significantly lower than 1−r. +This phenomenon is well described by DAMEs, but PAs do not give correct +predictions. We show that DAMEs clearly give a better approximation to +evolutionary dynamics than the PA method for both cases on regular ring +networks. It is easy to find that when z is large enough, the properties of +regular ring networks are similar to well-mixed populations. We also find +that the PA method will give a higher equilibrium frequency of cooperators +than the simulation value in most cases. We speculate that this is because +the PA method classifies nodes only by their degree and can not capture the +behavior differences between nodes with different types of neighbors. +Aside from regular ring networks, we now turn our attention to the study +of evolutionary dynamics on scale-free networks. Different from results ob- +tained by Santos and Pacheco on SF networks [14], for both the PD and the +SG, the equilibrium frequencies of cooperators decreased significantly when +the average connectivity changed from 4 to 8 (lower panel of Fig. 2), which +is the same as that observed in regular ring networks. In scale-free networks, +10 + +nodes with cooperative strategies often form clusters of various sizes. These +clusters will protect internal cooperators from being invaded by defectors, +even if the payoffs of these cooperators are not high enough. DAMEs focus +on characterizing the state of each type of node and its first-order neighbors, +resulting in a lower prediction of the equilibrium frequencies of cooperators. +Although there are differences between the results of DAMEs, PAs, and sim- +ulation results, they show the same trend variation. With the increase of +average connectivity z and the intensity of social dilemma, the equilibrium +frequencies of cooperators gradually decrease. +Figure 2: The equilibrium frequencies of cooperators on different types of networks. Re- +sults are shown as functions of advantage of defectors b for the PD (left panels) and +cost-to-benefit ratio r for the SG (right panels). Results for regular ring networks are +shown on top panels and for scale-free networks on lower panels. Markers, dashed lines, +and solid lines indicate the results of simulations, PAs, and DAMEs. Different average +connectivity z is distinguished by different colors. +11 + +4.2. The behavior of various nodes in SF networks +In order to study how scale-free networks promote the emergence of coop- +eration in evolutionary games, we must pay attention to the main difference +between scale-free networks, well-mixed populations, regular networks, and +random networks, that is, the heterogeneity of networks. Several previous +works have investigated the key role of network topology in the evolution of +cooperation [10; 14; 17; 18; 55; 26]. Specifically, we focus on the frequencies +of cooperators with different degrees and how they behave during the process +of evolution, and we study the evolutionary dynamics of the PD with b = 1.4 +on scale-free networks. In contrast, when we replace scale-free networks with +regular-ring networks, the equilibrium frequencies of cooperators become 0. +We divide nodes into four types according to their degrees: Central nodes +with degree ki ⩾ 10 (top 5.5%), Large nodes with degree 10 > ki ⩾ 5 (5.5% - +20%), Middle nodes with degree 5 > ki ⩾ 3 (20% - 50%), and Small: nodes +with degree ki < 3 (50% - 100%). +Figure 3: Evolutionary dynamics of the prisoner’s dilemma with b = 1.4 on scale-free +networks with 10,000 nodes and an average connectivity z = 4. Results are shown as +functions of generations. +The initial fraction of cooperators is ρ = 50%. +Nodes with +different degrees are distinguished by different colors. +At about the tenth generation, we find that the frequency of cooperators +in the networks reaches the lowest level, and then the frequency of coopera- +tors slowly rises. After about 500 generations, the vast majority of nodes in +the networks are cooperators. A remarkable feature of evolutionary dynam- +ics is that the frequency of cooperators of the Central nodes is significantly +higher than those of other nodes, and the frequencies of cooperators of all +other nodes are almost the same at all generations. DAMEs (middle panel of +Fig. 2) capture these characteristics of evolutionary dynamics quantitatively, +12 + +but PAs (right panel of Fig. 2) can not even capture these features qualita- +tively. By analyzing Ck,m(t) and Dk,m(t) in DAME approximations, we can +divide the evolution of cooperation in this situation into three phases: +The evolution begins with the alienation phase. In the beginning, the co- +operators are randomly distributed in the network. Since the average payoff +of the defectors is 1.4 times higher than the cooperators’, the frequency of +cooperators decreases, but the Central nodes are less affected due to their +higher payoffs. Then, Central nodes with higher payoffs gradually propagate +their own strategies. Central cooperators gradually turn their neighbors into +cooperators while Central defectors gradually turn their neighbors into de- +fectors. At this phase, clusters of nodes of the same type appear. Clusters +generally have several Central nodes and Large nodes. +Then the rising phase appears. Central cooperators get higher payoffs +and become more stable, while Central defectors get lower payoffs. When +their payoffs are low enough, there is a high probability of learning the strate- +gies from their cooperator neighbors. Through the above process, the fre- +quency of cooperators of the Central node gradually increases. The Central +nodes have many neighbors, which means that as the evolution progresses, +the probability of other nodes contacting high-payoff cooperators increases. +Nodes that contact the Central nodes gradually become cooperators, bring- +ing the frequency of cooperators in the network increases. +Increased fre- +quency of cooperators makes it easier for defectors to contact higher-payoff +cooperators. +The last is the balance phase. After a long period of evolution, the vast +majority of nodes in the network become cooperators (when b = 1.4). In +most cases, there are still some defectors in the network, and the transition +between cooperators and defectors reaches a balance. That is, the fraction of +cooperators in the network remains stable. Even at this phase, the frequency +of cooperators of the Central nodes is still significantly higher than that of +other nodes. +In regular ring networks, there are no Central nodes that provide leader- +ship, and all cooperators turn into defectors shortly because their payoffs are +much lower than those of the defectors. The analysis of nodes’ behavior in the +simulations confirms the above process. Compared with traditional numeri- +cal simulations that consume a lot of computing resources (proportional to +network size N 2 and usually require a large number of repeated experiments), +the DAMEs can approximate evolutionary dynamics with high accuracy in +13 + +a very short time (proportional to k2 +max). For that kmax ∝ N +1 +γ−1 in scale-free +networks and kmax ∝ ln N in regular networks, it is easy to find that DAMEs +save a lot of computational resources. The DAMEs provide a convenient +theoretical analysis framework, which can accurately demonstrate the criti- +cal role of a few highly connected nodes (hubs) in SF networks. Moreover, +it can easily capture the evolutionary behavior of various nodes in complex +networks. +4.3. The behavior of edges in complex networks +Having explored the equilibrium frequencies of cooperators and the be- +havior of nodes during evolution, we next explore the behavior of edges in +the network as the network dynamics. The initial fraction of cooperators, +CC edges, and CD edges in each case are 50%, 25%, and 50%. As evolu- +tion progresses, these fractions change, and the trajectories of (C, CD) and +(CC, CD) coordinates are shown in Fig. 4. To demonstrate the applicability +of DAMEs, we study the evolutionary dynamics of the prisoner’s dilemma on +scale-free networks with different b and that of the snowdrift game on regular +ring networks with different r. +We first demonstrate the evolution of the CC and CD edges of the pris- +oner’s dilemma on scale-free networks (left two columns of Fig. 4). Both +PAs and DAMEs capture the rapid decline of the fraction of cooperators +during the alienation phase, but both approximation methods overestimate +the rate of decline in the fraction of CD edges. We also find that both ap- +proximations, in general, can qualitatively give the change in the fraction of +the edges, and the DAME approximations have better overall accuracy. +We further show the evolution of the CC and CD edges of the snowdrift +game on regular ring networks (right two columns of Fig. 4). For the case +of r = 0.1, PAs and DAMEs achieve equally high approximation accuracy. +PAs fail to give correct evolutionary dynamics as the cost-to-benefit ratio r +increases, but DAMEs still maintain pretty high accuracy. +5. +Discussion and Conclusion +In this paper, we have proposed the method DAMEs to theoretically pre- +dict the evolutionary game dynamics on complex networks. In particular, we +apply it to study how network heterogeneity influences the evolutionary tra- +jectories. On strongly heterogeneous networks such as scale-free networks, a +small proportion of nodes are more highly connected than the majority, while +14 + +Figure 4: The trajectories of (C, CD) and (CC, CD) coordinates. Markers, dashed lines, +and solid lines indicate simulation results, PA results, and DAME results. All cases have +the same average connectivity z = 4. +We can find that DAMEs have better overall +accuracy than PAs for the PD on SF networks. For the SG on RR networks, both DAMEs +and PAs perform well when r = 0.1. DAMEs accurately describe the evolution dynamics +when r = 0.3 or r = 0.9, but PAs give wrong results. f(C) is the frequency of cooperators +and f(CC) is the frequency of CC edges. +the connectivity of nodes tends to be similar as the decreasing of network +heterogeneity. Our results help to understand why network heterogeneity +acts as a cooperation-promotor: Hubs are more likely to propagate their own +strategies than the less influential nodes. +Hubs adopting the cooperative +strategy become more stable during evolution. Hubs adopting the defecting +strategy reduce their payoff while propagating their own strategies, so that +they are more likely to learn the cooperative strategy from their neighbors. +Hubs will gradually become cooperators and propagate their own strategies, +promoting the evolution of cooperation in the network. Furthermore, our +findings also inspire a few possibilities to enhance the establishment of co- +operation by network surgery or connection modification — reconnect the +edges so that the network has multiple, evenly distributed hubs with moder- +ate influence, or set a suitable cut-off limit for the maximum connectivity of +15 + +the nodes during network generation. The main objective of these solutions +is to increase the proportion of hubs in the network. +By comparing the evolutionary dynamics given by numerical simulations, +DAMEs, and PA methods with different spatial structures and payoff ma- +trices, we demonstrated that the accuracy of DAMEs supersedes standard +PA methods. MF methods, as a rather simple analytical approach, are of- +ten inaccurate on sparse networks due to the lack of dynamic correlations, +which means that the state of a focal player’s neighbors is assumed to be +independent of the state of itself [37]. PA methods consider dynamic corre- +lations at a pairwise level but do not capture dynamical correlations beyond +nearest neighbors [46; 42]. Classical AMEs achieve higher accuracy than MF +methods and PA methods [52], but the transition probabilities between two +states in it are static and cannot be applied to evolutionary games. DAMEs +consider the state of nodes and their first-order neighbors and have transi- +tion probabilities that depend on the difference between nodes’ payoffs and +estimated payoffs of the nodes’ neighbors. +To sum up, DAMEs, as a new tool for studying evolutionary games on +complex networks, can better approximate evolutionary outcomes through +large systems of differential equations. Using DAMEs for computing the equi- +librium frequency of cooperators and the behavior of nodes and edges during +evolution has been shown. 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E 80 (4) (2009) 046106. +doi:10.1103/PhysRevE.80.046106. +21 + diff --git a/ddE4T4oBgHgl3EQfpg0y/content/tmp_files/load_file.txt b/ddE4T4oBgHgl3EQfpg0y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ecdac7fa2b225381945e8d2bf60968d21f385bef --- /dev/null +++ b/ddE4T4oBgHgl3EQfpg0y/content/tmp_files/load_file.txt @@ -0,0 +1,891 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf,len=890 +page_content='High-Accuracy Approximation of Evolutionary Pairwise Games on Complex Networks Hongyu Wanga, Aming Lia,∗, Long Wanga,∗ aCenter for Systems and Control, College of Engineering, Peking University, Beijing, 100871, China Abstract Previous studies have shown that the topological properties of a complex network, such as heterogeneity and average degree, affect the evolutionary game dynamics on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' However, traditional numerical simulations are usu- ally time-consuming and demand a lot of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In this paper, we propose the method of dynamical approximate master equations (DAMEs) to accurately approximate the evolutionary outcomes on complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We demonstrate that the accuracy of DAMEs supersedes previ- ous standard pairwise approximation methods, and DAMEs require far fewer computational resources than traditional numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We use pris- oner’s dilemma and snowdrift game on regular and scale-free networks to demonstrate the applicability of DAMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Overall, our method facilitates the investigation of evolutionary dynamics on a broad range of complex networks, and provides new insights into the puzzle of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' ∗Corresponding authors Email addresses: wanghongyu1998@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='cn (Hongyu Wang), liaming@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='cn (Aming Li), longwang@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='cn (Long Wang) Preprint submitted to Elsevier January 13, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='05192v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='PE] 12 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Introduction Many levels of biological organization, from single-celled organisms to human society, are based on cooperation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' However, in the context of Dar- winian evolution, natural selection favors defectors over cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Evolu- tionary game theory is a general mathematical framework for studying the cooperation between unrelated individuals [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' As metaphors for studying the evolution of cooperation, pairwise games such as prisoner’s dilemma (PD) and snowdrift game (SG) have been widely adopted by re- searchers from different backgrounds [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In infinitely well-mixed populations, evolution under replicator dynamics leads to a stable fraction of cooperators for SG but to the complete extinction of cooperators in PD [12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' To better understand the emergence of cooperation in more realistic sit- uations, the importance of studying the behavior of individuals with popu- lation structure should be highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Graph theory provides a convenient framework to describe the population structure for studying the evolution of cooperation [14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 26], where the vertices of a graph represent players and the edges define the network of con- tacts between players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Researchers found that when individuals interacted only with their neighbors, the fraction of cooperators in both PD and SG differs from that in well-mixed populations [8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Regu- lar graphs ignore the uniqueness of individuals, that is, different individuals may have different numbers of neighbors they interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Scale-free (SF) networks, whose vertex connectivities follow a power-law distribution, are often used to represent more realistic heterogeneous networks [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Santos and Pacheco found that the equilibrium frequencies of cooperators are higher when playing the PD and the SG on SF networks than when playing them on regular networks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' They attributed this phenomenon to the generation rules of the Barab´asi-Albert model, that is, the vertices in the network are added sequentially, and the newly added vertice is more likely to connect to vertices with higher degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' However, the lack of more in-depth studies, particularly a theoretical explanation, makes this phenomenon challenging to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Ohtsuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' proved that natural selection favors cooperation if the benefit of altruistic behavior, divided by the costs, exceeds the aver- age number of neighbors [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Recent works explore general formulations of fixation probabilities for pairwise games under weak selection that apply to graph-structured populations [31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In this paper, we propose the 2 dynamical approximate master equations (DAMEs) to describe evolutionary game dynamics on complex networks, starting from a theoretical explanation of how the heterogeneity of networks affects the evolutionary dynamics and finally leads to the prevalence of cooperation on SF networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Indeed, beyond numerical simulations, we need a theoretical method to study the evolutionary behavior of various nodes on complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The most commonly used method for binary-state dynamics on complex networks is the mean-field (MF) theory [35;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 36;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 38;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The pairwise approx- imation (PA) theory suggested by Dickman improves the accuracy of MF theory [40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 42;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 43;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 44] and has been applied to the study of evolution- ary dynamics on complex networks [45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The approximate master equations (AMEs) have achieved high accuracy beyond the PA level in the binary-state dynamics on complex networks [50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 51;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 52;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 53], but the transition probabilities between two states in the AME system are static (time-invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In evolutionary games on complex networks, the transition probabilities between states of each node depend on the difference between the node and its neighbors’ payoff, which is time-variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Here we incor- porate the transition probabilities in the time-variant form in our DAMEs, which can accurately approximate evolutionary game dynamics and predict the equilibrium frequencies of cooperators in a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Evolutionary pairwise games on complex networks We consider a population captured by the complex network with N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Each player in the population can be in two states, cooperation or defection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The degree of a node represents the number of edges between the node and its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Degree distribution P(k) = Nk/N represents the probability that the degree of a randomly selected node in the network is k, where Nk refers to the number of nodes with degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Here we assume that the network is generated by the configuration model with fixed degree distribution P(k), which does not have degree correlations [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In both PD and SG, two players have to decide whether to cooperate or defect in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Both players receive R when they cooperate with each other and P when they defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' A defector exploiting a cooperator receives T, and the cooperator receives S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Following common practice, researchers usually adjust the game to rely on a single parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For the PD, we have T > R > P > S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Considering that T + S < 2R, we make 2 > T = b > 1, R = 1, and P = S = 0, leaving the advantage of defectors b be the single 3 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We have tested that if S = −ε < 0 (ε ≪ 1) is set to satisfy S < P, the result will not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For the SG, we have T > R > S > P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Considering that T + S = 2R, we make T = β > 1, R = β − 1/2, S = β − 1, and P = 0, such that the cost-to-benefit ratio can be written as r = 1/(2β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Evolution is carried out by implementing the finite population analog of replicator dynamics through the following transition probabilities: In each generation, all individuals play a single-round game with all of their neighbors and accumulate the payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Whenever an individual i desires to update the strategy, one of its neighbor j will be drawn from its ki neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The probability that the individual i copies the strategy of individual j is given by the Fermi function Psi→sj = φ(πi, πj) = 1 1 + eα(πi−πj) (1) where α ∈ [0, ∞) denotes the intensity of selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' α → 0 leads to the random drift and α → ∞ leads to the deterministic imitation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Dynamical approximate master equations Define Ck,m(t) (Dk,m(t)) as the fraction of k-degree nodes that are coop- erators (defectors) at time t and have m defector neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The DAMEs consist of M = � k,m 2 = (1 + kmax − kmin)(2 + kmax + kmin) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Then the fraction of cooperators of k-degree nodes at time t is given by ρk(t) = k � m=0 Ck,m(t), (2) and the fraction of cooperators in the whole network is ρ(t) = � k P(k)ρk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (3) By assuming that the initial cooperators are randomly selected with a fraction ρ(0), the initial conditions are Ck,m(0) = ρ(0)Bk,m(1 − ρ(0)), (4) Dk,m(0) = (1 − ρ(0))Bk,m(1 − ρ(0)), (5) where Bk,m(q) = � k m � qm(1 − q)k−m is the binomial factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4 The approximate master equations for the evolution of Ck,m(t) and Dk,m(t) are dCk,m(t) dt = −PCk,m→Dk,mCk,m(t)m k +PDk,m→Ck,mDk,m(t)(k − m) k −PCk,m→Ck,m+1(k − m)βCCk,m(t) +PCk,m−1→Ck,m(k − m + 1)βCCk,m−1(t) −PCk,m→Ck,m−1mγCCk,m(t) +PCk,m+1→Ck,m(m + 1)γCCk,m+1(t), (6) dDk,m(t) dt = −PDk,m→Ck,mDk,m(t)(k − m) k +PCk,m→Dk,mCk,m(t)m k −PDk,m→Dk,m+1(k − m)βDDk,m(t) +PDk,m−1→Dk,m(k − m + 1)βDDk,m−1(t) −PDk,m→Dk,m−1mγDDk,m(t) +PDk,m+1→Dk,m(m + 1)γDDk,m+1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (7) The coefficient PCk,m→Dk,m is defined as the transition probability that a k-degree cooperator, which has m defector neighbors at time t, changes its strategy to defection by time t + dt, where dt is an infinitesimally small time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Similarly, PDk,m→Ck,m is the transition probability that a k-degree defector, which has m defector neighbors at time t, changes its strategy to cooperation by time t + dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The coefficient PCk,m→Ck,m+1 is defined as the transition probability that a cooperator, which has k neighbors and m of them are defectors at time t, changes its state into Ck,m+1 by time t+dt, which means one of its cooperator neighbors becomes a defector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The coefficient βC is defined as the probabil- ity that the randomly selected neighbor is a defector when a cooperator’s cooperator neighbor updates its strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The coefficient PCk,m→Ck,m−1 is de- fined as the transition probability that a cooperator, which has k neighbors and m of them are defectors at time t, changes its state into Ck,m−1 by time t + dt, which means one of its defector neighbors becomes a cooperator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The 5 C C C , −1 , −1→ , 1 , → , −1 , → , +1 , +1→ , , → , −1 , +1→ , , −1→ , , → , +1 , → , , → , C D C , C D D , +1 D C C , −1 D D C , D D D , +1 Figure 1: Schematic representation of the meaning of each variable in DAMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Define Ck,m(t) (Dk,m(t)) as the fraction of k-degree nodes that are cooperators (defectors) at time t and have m neighboring defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Ck,m(t) and Dk,m(t) change at each time step due to the node itself or its neighbors’ updating strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For example, the coefficient PCk,m→Dk,m is defined as the transition probability that a k-degree cooperator, which has m defector neighbors at time t, changes its strategy to defection by time t + dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' A node may update its strategy only when it plays with nodes that adopt the different strategy, and the probability that a Ck,m(t) node plays with its defector neighbors is m k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' coefficient γC is defined as the probability that the randomly selected neigh- bor is a cooperator when a cooperator’s defector neighbor updates its strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PCk,m−1→Ck,m, PCk,m→Ck,m−1, PCk,m+1→Ck,m, PDk,m→Dk,m+1, PDk,m−1→Dk,m, PDk,m→Dk,m−1 and PDk,m+1→Dk,m, βD, γD are defined in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The mathematical expressions of these variables are given somewhere below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In order to compute these transition probabilities, we can directly cal- culate the payoff of each class of nodes and use the limited information in the DAMEs system to estimate the payoff of each class of nodes’ first-order neighbors and second-order neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The approximation in this method is that we assume that the neighbor configuration of each node is the product of independent single event probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' That is, the network under study should have no degree correlations, which will lead to inaccurate results of our model under some extreme networks (such as social networks and e-mail 6 networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Step 1: Compute the number of cooperator neighbors and defector neigh- bors of nodes with different degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The number of defector neighbors of a cooperator with degree ki can be computed by NC−D(ki) = �ki m=0 mCki,m �ki m=0 Cki,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (8) NC−C(ki), ND−C(ki), ND−D(ki) are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Step 2: Compute the degree distribution and payoffs of each class of nodes’ first-order neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We can compute the degree distribution of cooperators which have at least a cooperator neighbor as PkC−C(ki) = P(ki) �ki−1 m=0 Cki,m � k P(k) ��k−1 m=0 Ck,m �, (9) ki ∈ [kmin, kmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The reason that the integral has an upper bound of ki − 1 is that a node with degree ki can have at most ki − 1 defector neighbors to ensure that it has at least one cooperator neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The probability of a node connecting with nodes with degree ki depends on the ratio of the number of its edges to the number of all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Thus the degree distribution of first-order cooperator neighbors of a cooperator can be computed as: PC−C(ki) = kiPkC−C(ki) � k kPkC−C(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (10) The payoffs of a cooperator’s first-order cooperator neighbors depend on the number of their cooperator neighbors and defector neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' πC−C(ki) = �ki−1 m=0(ki − m)Cki,m �ki−1 m=0 Cki,m R + �ki−1 m=0 mCki,m �ki−1 m=0 Cki,m S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (11) The fraction of defector neighbors of a cooperator’s cooperator neighbor can be computed by βC = � k P(k)m �k−1 m=0 Ck,m � k P(k)k ��k−1 m=0 Ck,m �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (12) 7 βD, γC, γD can be computed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Step 3: Compute PCk,m→Dk,m and PDk,m→Ck,m, in which case the focal players are selected to update strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' When the focal players is a defector, its payoff is πD(k, m) = (k − m)T + mP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (13) Using the conclusion of Step 2, we can compute PkC−D(ki) = P(ki) �ki m=1 Cki,m � k P(k) ��ki m=1 Ck,m �, (14) PD−C(ki) = kiPkC−D(ki) � k kPkC−D(k), (15) πD−C(k) = �ki m=1(ki − m)Cki,m �ki m=1 Cki,m R + �ki m=1 mCki,m �ki m=1 Cki,m S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (16) Thus the transition probability PDk,m→Ck,m can be computed as PDk,m→Ck,m = kmax � ki=kmin PD−C(ki)φ(πD(k, m), πD−C(ki)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (17) Because the probability that a randomly selected neighbor is a coopera- tor is (k − m)/k, we multiply this coefficient in the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PCk,m→Dk,m can be computed in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Step 4: Compute other transition probabilities, in which case the neigh- bors of the focal players are selected to update strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PCk,m→Ck,m+1 is defined as the probability that one of the cooperator neigh- bors of a focal player A in class Ck,m becomes a defector by time t + dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' A is a cooperator, so this first-order cooperator neighbor must be connected with a second-order defector neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We can compute A’s first-order cooperator neighbors’ degree distribution PC−C(ki) and payoff πC−C(ki) by the method in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We can also compute A’s second-order defector neighbors’ degree distribution PC−D(kj) and payoff πC−D(kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Thus PCk,m→Ck,m+1 is given by PCk,m→Ck,m+1 = � ki � kj PC−C(ki)PC−D(kj) ×φ(πC−C(ki), πC−D(kj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (18) 8 PCk,m−1→Ck,m, PDk,m→Dk,m−1 and PDk,m+1→Dk,m can be computed similarly because in these cases first-order neighbors can only update their strategy by imitating the strategies of second-order neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PCk,m→Ck,m−1 is defined as the probability that one of the defector neigh- bors of a focal player B in class Ck,m becomes a cooperator by time t + dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We must consider that the first-order defector neighbors have probabilities to imitate B’s strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Similarly, we can compute the first-order defector neighbors’ degree distribution PC−D(ki) and their payoff πC−D(ki) and the second-order cooperator neighbors’ degree distribution PD−C(kj) and pay- off πD−C(kj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The number of cooperator neighbors of first-order defector neighbors(with degree ki) can be computed by the method in Step 1: ND−C(ki) = �ki−1 m=0(ki − m)Dki,m �ki−1 m=0 Dki,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (19) Thus first-order defector neighbors with degree ki have a probability of 1/ND−C(ki) to imitate B’s strategy, and have a probability of [ND−C(ki) − 1]/ND−C(ki) to imitate a second-order cooperator node’ strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Therefore PCk,m→Ck,m−1 is computed by PCk,m→Ck,m−1(k, m) = � ki PC−D(ki) � 1 ND−C(ki) ×φ(πC−D(ki), πC(k, m)) + ND−C(ki) − 1 ND−C(ki) × � kj PD−C(kj)φ(πC−D(ki), πD−C(kj)) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' (20) PCk,m+1→Ck,m, PDk,m→Dk,m+1 and PDk,m−1→Dk,m can be computed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Simulations of evolutionary pairwise games on networks To test the accuracy of the approximations, we will now compare the results of the theory explained in the previous section to the numerical sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We consider the simulated dynamics on different types of networks with N = 10000 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We consider two kinds of networks with differ- ent degree distributions: regular ring networks and scale-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For 9 scale-free networks, the degree distribution P(k) ∼ k−γ, 2 ≤ γ ≤ 3 obeys the power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For convenience, we use the configuration model to generate scale-free networks with degree fixed distribution P(k), which is given by the Barab´asi-Albert model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The initial fraction of cooperators is ρ = 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Our simulation results are the average of more than 1000 independent sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PAs and DAMEs used networks with the same degree distribution P(k) as simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Equilibrium frequencies of cooperators We first consider the influence of network structure and average connec- tivity z on the evolution of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The role of average connectivity in cooperation has been examined by several previous works [10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We calcu- late the average equilibrium frequencies of cooperators of different models in the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We will focus on the differences between simulation results and theoretical results given by DAMEs and the PA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Firstly, we consider the evolutionary games on regular ring networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' As the average connectivity z of the network increases, the equilibrium frequen- cies of cooperators decrease rapidly for the PD (top left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For the SG (top right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 2), the gradual increase of z results in the equilibrium frequencies of cooperators gradually approaching 1 − r, which is consistent with the result concluded by the replicator equation in well-mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In some parameter spaces, the equilibrium frequencies of coop- erators for the SG on regular ring networks are significantly lower than 1−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' This phenomenon is well described by DAMEs, but PAs do not give correct predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We show that DAMEs clearly give a better approximation to evolutionary dynamics than the PA method for both cases on regular ring networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' It is easy to find that when z is large enough, the properties of regular ring networks are similar to well-mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We also find that the PA method will give a higher equilibrium frequency of cooperators than the simulation value in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We speculate that this is because the PA method classifies nodes only by their degree and can not capture the behavior differences between nodes with different types of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Aside from regular ring networks, we now turn our attention to the study of evolutionary dynamics on scale-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Different from results ob- tained by Santos and Pacheco on SF networks [14], for both the PD and the SG, the equilibrium frequencies of cooperators decreased significantly when the average connectivity changed from 4 to 8 (lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 2), which is the same as that observed in regular ring networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In scale-free networks, 10 nodes with cooperative strategies often form clusters of various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' These clusters will protect internal cooperators from being invaded by defectors, even if the payoffs of these cooperators are not high enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' DAMEs focus on characterizing the state of each type of node and its first-order neighbors, resulting in a lower prediction of the equilibrium frequencies of cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Although there are differences between the results of DAMEs, PAs, and sim- ulation results, they show the same trend variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' With the increase of average connectivity z and the intensity of social dilemma, the equilibrium frequencies of cooperators gradually decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Figure 2: The equilibrium frequencies of cooperators on different types of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Re- sults are shown as functions of advantage of defectors b for the PD (left panels) and cost-to-benefit ratio r for the SG (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Results for regular ring networks are shown on top panels and for scale-free networks on lower panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Markers, dashed lines, and solid lines indicate the results of simulations, PAs, and DAMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Different average connectivity z is distinguished by different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The behavior of various nodes in SF networks In order to study how scale-free networks promote the emergence of coop- eration in evolutionary games, we must pay attention to the main difference between scale-free networks, well-mixed populations, regular networks, and random networks, that is, the heterogeneity of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Several previous works have investigated the key role of network topology in the evolution of cooperation [10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Specifically, we focus on the frequencies of cooperators with different degrees and how they behave during the process of evolution, and we study the evolutionary dynamics of the PD with b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='4 on scale-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In contrast, when we replace scale-free networks with regular-ring networks, the equilibrium frequencies of cooperators become 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We divide nodes into four types according to their degrees: Central nodes with degree ki ⩾ 10 (top 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='5%), Large nodes with degree 10 > ki ⩾ 5 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='5% - 20%), Middle nodes with degree 5 > ki ⩾ 3 (20% - 50%), and Small: nodes with degree ki < 3 (50% - 100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Figure 3: Evolutionary dynamics of the prisoner’s dilemma with b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='4 on scale-free networks with 10,000 nodes and an average connectivity z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Results are shown as functions of generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The initial fraction of cooperators is ρ = 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Nodes with different degrees are distinguished by different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' At about the tenth generation, we find that the frequency of cooperators in the networks reaches the lowest level, and then the frequency of coopera- tors slowly rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' After about 500 generations, the vast majority of nodes in the networks are cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' A remarkable feature of evolutionary dynam- ics is that the frequency of cooperators of the Central nodes is significantly higher than those of other nodes, and the frequencies of cooperators of all other nodes are almost the same at all generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' DAMEs (middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 2) capture these characteristics of evolutionary dynamics quantitatively, 12 but PAs (right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 2) can not even capture these features qualita- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' By analyzing Ck,m(t) and Dk,m(t) in DAME approximations, we can divide the evolution of cooperation in this situation into three phases: The evolution begins with the alienation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In the beginning, the co- operators are randomly distributed in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Since the average payoff of the defectors is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='4 times higher than the cooperators’, the frequency of cooperators decreases, but the Central nodes are less affected due to their higher payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Then, Central nodes with higher payoffs gradually propagate their own strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Central cooperators gradually turn their neighbors into cooperators while Central defectors gradually turn their neighbors into de- fectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' At this phase, clusters of nodes of the same type appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Clusters generally have several Central nodes and Large nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Then the rising phase appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Central cooperators get higher payoffs and become more stable, while Central defectors get lower payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' When their payoffs are low enough, there is a high probability of learning the strate- gies from their cooperator neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Through the above process, the fre- quency of cooperators of the Central node gradually increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The Central nodes have many neighbors, which means that as the evolution progresses, the probability of other nodes contacting high-payoff cooperators increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Nodes that contact the Central nodes gradually become cooperators, bring- ing the frequency of cooperators in the network increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Increased fre- quency of cooperators makes it easier for defectors to contact higher-payoff cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The last is the balance phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' After a long period of evolution, the vast majority of nodes in the network become cooperators (when b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In most cases, there are still some defectors in the network, and the transition between cooperators and defectors reaches a balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' That is, the fraction of cooperators in the network remains stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Even at this phase, the frequency of cooperators of the Central nodes is still significantly higher than that of other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In regular ring networks, there are no Central nodes that provide leader- ship, and all cooperators turn into defectors shortly because their payoffs are much lower than those of the defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The analysis of nodes’ behavior in the simulations confirms the above process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Compared with traditional numeri- cal simulations that consume a lot of computing resources (proportional to network size N 2 and usually require a large number of repeated experiments), the DAMEs can approximate evolutionary dynamics with high accuracy in 13 a very short time (proportional to k2 max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For that kmax ∝ N 1 γ−1 in scale-free networks and kmax ∝ ln N in regular networks, it is easy to find that DAMEs save a lot of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The DAMEs provide a convenient theoretical analysis framework, which can accurately demonstrate the criti- cal role of a few highly connected nodes (hubs) in SF networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Moreover, it can easily capture the evolutionary behavior of various nodes in complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The behavior of edges in complex networks Having explored the equilibrium frequencies of cooperators and the be- havior of nodes during evolution, we next explore the behavior of edges in the network as the network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The initial fraction of cooperators, CC edges, and CD edges in each case are 50%, 25%, and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' As evolu- tion progresses, these fractions change, and the trajectories of (C, CD) and (CC, CD) coordinates are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' To demonstrate the applicability of DAMEs, we study the evolutionary dynamics of the prisoner’s dilemma on scale-free networks with different b and that of the snowdrift game on regular ring networks with different r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We first demonstrate the evolution of the CC and CD edges of the pris- oner’s dilemma on scale-free networks (left two columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Both PAs and DAMEs capture the rapid decline of the fraction of cooperators during the alienation phase, but both approximation methods overestimate the rate of decline in the fraction of CD edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We also find that both ap- proximations, in general, can qualitatively give the change in the fraction of the edges, and the DAME approximations have better overall accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We further show the evolution of the CC and CD edges of the snowdrift game on regular ring networks (right two columns of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For the case of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='1, PAs and DAMEs achieve equally high approximation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PAs fail to give correct evolutionary dynamics as the cost-to-benefit ratio r increases, but DAMEs still maintain pretty high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Discussion and Conclusion In this paper, we have proposed the method DAMEs to theoretically pre- dict the evolutionary game dynamics on complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' In particular, we apply it to study how network heterogeneity influences the evolutionary tra- jectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' On strongly heterogeneous networks such as scale-free networks, a small proportion of nodes are more highly connected than the majority, while 14 Figure 4: The trajectories of (C, CD) and (CC, CD) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Markers, dashed lines, and solid lines indicate simulation results, PA results, and DAME results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' All cases have the same average connectivity z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' We can find that DAMEs have better overall accuracy than PAs for the PD on SF networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' For the SG on RR networks, both DAMEs and PAs perform well when r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' DAMEs accurately describe the evolution dynamics when r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='3 or r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content='9, but PAs give wrong results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' f(C) is the frequency of cooperators and f(CC) is the frequency of CC edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' the connectivity of nodes tends to be similar as the decreasing of network heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Our results help to understand why network heterogeneity acts as a cooperation-promotor: Hubs are more likely to propagate their own strategies than the less influential nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Hubs adopting the cooperative strategy become more stable during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Hubs adopting the defecting strategy reduce their payoff while propagating their own strategies, so that they are more likely to learn the cooperative strategy from their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Hubs will gradually become cooperators and propagate their own strategies, promoting the evolution of cooperation in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Furthermore, our findings also inspire a few possibilities to enhance the establishment of co- operation by network surgery or connection modification — reconnect the edges so that the network has multiple, evenly distributed hubs with moder- ate influence, or set a suitable cut-off limit for the maximum connectivity of 15 the nodes during network generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' The main objective of these solutions is to increase the proportion of hubs in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' By comparing the evolutionary dynamics given by numerical simulations, DAMEs, and PA methods with different spatial structures and payoff ma- trices, we demonstrated that the accuracy of DAMEs supersedes standard PA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' MF methods, as a rather simple analytical approach, are of- ten inaccurate on sparse networks due to the lack of dynamic correlations, which means that the state of a focal player’s neighbors is assumed to be independent of the state of itself [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' PA methods consider dynamic corre- lations at a pairwise level but do not capture dynamical correlations beyond nearest neighbors [46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Classical AMEs achieve higher accuracy than MF methods and PA methods [52], but the transition probabilities between two states in it are static and cannot be applied to evolutionary games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' DAMEs consider the state of nodes and their first-order neighbors and have transi- tion probabilities that depend on the difference between nodes’ payoffs and estimated payoffs of the nodes’ neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' To sum up, DAMEs, as a new tool for studying evolutionary games on complex networks, can better approximate evolutionary outcomes through large systems of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Using DAMEs for computing the equi- librium frequency of cooperators and the behavior of nodes and edges during evolution has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' Compared with traditional numerical methods, DAMEs may handle evolutionary games on large-scale networks with great efficiency and give reasonable evolutionary outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} +page_content=' DAMEs can also pro- vide quick tests and helpful information for newly developed evolutionary game models.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddE4T4oBgHgl3EQfpg0y/content/2301.05192v1.pdf'} diff --git a/ddE5T4oBgHgl3EQfgA-d/content/2301.05631v1.pdf b/ddE5T4oBgHgl3EQfgA-d/content/2301.05631v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fbb0d536f64ca22c5c334d4fa2c1419ec911cf04 --- /dev/null +++ b/ddE5T4oBgHgl3EQfgA-d/content/2301.05631v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7bb07e5abdc7ab8e623642c6ecad3b3d6af73f29033d52c11c61bc9fcd4f38c4 +size 847479 diff --git a/ddE5T4oBgHgl3EQfgA-d/vector_store/index.faiss b/ddE5T4oBgHgl3EQfgA-d/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..95bb037d2589cbf82f8d6fa81827c1aeb148e15f --- /dev/null +++ b/ddE5T4oBgHgl3EQfgA-d/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e70d69cd270d0fc3fe1cbe122293379a681eac108a481e26ea67b6d970507727 +size 2359341 diff --git 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Laboratoire de Physique, F-69342 Lyon, France +(Dated: January 5, 2023) +The study of non-equilibrium dynamics is one of the most important challenges of modern quan- +tum many-body physics, in relationship with fundamental questions in quantum statistical mechan- +ics, as well as with the fields of quantum simulation and computing. In this work we propose a +Gaussian Ansatz for the study of the nonequilibrium dynamics of quantum spin systems. Within +our approach, the quantum spins are mapped onto Holstein-Primakoff bosons, such that a coherent +spin state – chosen as the initial state of the dynamics – represents the bosonic vacuum. The state of +the system is then postulated to remain a bosonic Gaussian state at all times, an assumption which +is exact when the bosonic Hamiltonian is quadratic; and which is justified in the case of a nonlinear +Hamiltonian if the boson density remains moderate. We test the accuracy of such an Ansatz in the +paradigmatic case of the S = 1/2 transverse-field Ising model, in one and two dimensions, initialized +in a state aligned with the applied field. We show that the Gaussian Ansatz, when applied to the +bosonic Hamiltonian with nonlinearities truncated to quartic order, is able to reproduce faithfully +the evolution of the state, including its relaxation to the equilibrium regime, for fields larger than +the critical field for the paramagnetic-ferromagnetic transition in the ground state. In particular +the spatio-temporal pattern of correlations reconstructed via the Gaussian Ansatz reveals the dis- +persion relation of quasiparticle excitations, exhibiting the softening of the excitation gap upon +approaching the critical field. Our results suggest that the Gaussian Ansatz correctly captures the +essential effects of nonlinearities in quantum spin dynamics; and that it can be applied to the study +of fundamental phenomena such as quantum thermalization and its breakdown. +I. +INTRODUCTION +The non-equilibrium unitary dynamics of quantum +many-body systems represents a central topic of mod- +ern quantum physics: it lies at the core of the coher- +ent manipulation of complex quantum states with quan- +tum devices [1–4]; and it represents the mechanism by +which equilibration and the emergence of statistical en- +sembles occurs [5–10]. +In the case of systems with a +time-independent Hamiltonian (which will be the focus +of this work), central questions concern the propagation +of correlations and entanglement, and the scrambling of +quantum information [4, 11]; how such phenomena man- +ifest the nature of elementary excitations [12–16]; and +how they lead to the onset of equilibration [7, 8, 10]. +Answering to the above questions quantitatively for sys- +tems of increasing size is a significant challenge, due to +the exponential increase of the Hilbert-space dimensions +with system size, making an exact numerical treatment +impractical for systems going beyond e.g. a few tens of +qubits. The development of efficient numerical schemes +which approximate the quantum many-body evolution is +therefore the only realistic strategy to approach the prob- +lem theoretically – the only alternative solution comes +from the direct experimental implementation of the dy- +namics of interest via an analog or digital quantum simu- +lator [2]. By “efficient scheme” we mean here a numerical +approach whose computational cost scales polynomially +with system size; and which ideally can be improved sys- +tematically, while maintaining a polynomial cost. +A general framework for the development of efficient +numerical schemes is the one of wavefunction Ans¨atze, +positing that the state vector has an explicit functional +form depending on a number of variational parame- +ters which is much smaller than the Hilbert-space di- +mensions. Relevant examples of variational Ans¨atze for +the unitary evolution are tensor network states [17, 18], +neural-network quantum states [19], Jastrow-like wave- +functions [20–22], etc. Among these families of states, +the tensor-network states can be systematically improved +by increasing the bond dimension, whose logarithm reg- +ulates the maximum amount of subsystem entanglement +entropy that the state can accommodate. Nonetheless +unitary evolutions lead systematically to extensive en- +tanglement entropies, leading to the so-called entangle- +ment barrier, namely the exponential increase of the +bond dimension required to reproduce the state faith- +fully. Other family of states allow for systematical im- +provements without facing an entanglement barrier [23]; +nonetheless, all these approaches are in general very de- +manding from a computational point of view, and their +computational cost further scales significantly with the +dimensions of the local Hilbert space describing individ- +ual degrees of freedom. An alternative to wavefunction +Ans¨atze, valid for spin or bosonic systems, is offered by +the discrete truncated Wigner approximation (DTWA). +This approach postulates an evolved state of the system +expressed in terms of the Wigner-function representation +of the initial state; and of phase-point operators depen- +dent on parameters evolved according to classical equa- +tions of motion [24]. The DTWA approach is very pow- +erful in representing evolutions arbitrarily far from equi- +librium [25], as it is based on a Monte Carlo sampling +of classical trajectories initialized via the Wigner func- +arXiv:2301.01363v1 [cond-mat.quant-gas] 3 Jan 2023 + +2 +tion of the initial state, without making any assumption +on the statistics of quantum fluctuations. +Yet it rep- +resents instantaneous expectation values as incoherent +Monte Carlo averages, and therefore it misses many-body +interference effects which are at the basis of fundamental +phenomena (such as e.g. many-body localization [26]). +In this work we propose and validate an alternative ap- +proximation schemes for quantum evolutions of generic +quantum systems – bosonic or fermionic alike – based on +Gaussian states. +Gaussian many-body states have the +property that the reduced density matrix of each sub- +system has a Gaussian form, such that all observables +obey Wick’s theorem – namely they can be reconstructed +from the knowledge of the average fields and the covari- +ance matrix, whose elements are given by the regular +and anomalous two-point Green’s function. +Hence re- +constructing the evolution of the state amounts to track- +ing the dynamics of the average fields and covariance +matrix, obeying coupled non-linear equations of motion. +This scheme is well known for fermionic (bosonic) sys- +tems as the time-dependent Hartree-Fock (Hartree-Fock- +Bogolyubov) approach; yet in this work we propose its +application to quantum spin systems, specifically based +on the Holstein-Primakoff (HP) spin-boson mapping. A +Gaussian Ansatz for the equilibrium state of the non- +linear bosonic Hamiltonian resulting from the HP map- +ping is at the core of the so-called modified spin-wave +theory [27]. Our approach can therefore be viewed as a +time-dependent version of modified spin-wave theory. On +the other hand, conventional time-dependent spin-wave +theory amounts to discarding all non-linear terms in the +bosonic Hamiltonian. +We apply the Gaussian Ansatz approach to the non- +equilibrium evolution of a paradigmatic model for quan- +tum simulation, namely the S = 1/2 transverse-field +Ising model (TFIM), initialized in a state aligned with +the applied field. This dynamics corresponds to a quan- +tum quench starting from the ground state at infinite +field, and triggered by changing the field abruptly to a +finite value. +We first benchmark our approach in the +case of the one-dimensional TFIM, which can be ex- +actly solved by mapping it onto free fermions. Our re- +sults show that the dynamics of the free-fermion sys- +tem can be mimicked quantitatively by that of a sys- +tem of strongly interacting Holstein-Primakoff bosons +for quenches to fields as low as the critical one for the +ground-state phase transition from paramagnet to fer- +romagnet. +Even more convincing results are obtained +for the case of the two-dimensional TFIM, which is not +exactly solvable; but whose dynamics can be quantita- +tively reconstructed via time-dependent variational cal- +culations [28]. In both cases (1d and 2d) we show that the +Gaussian Ansatz is able to capture fundamental features +of the spatio-temporal structure of the quantum corre- +lations developing after the quench, as observed in their +Fourier transform (via a so-called quench spectroscopy +scheme [14–16]), which allows one to reconstruct the dis- +persion relation of elementary excitations. The disper- +sion relation reconstructed via quench spectroscopy is in +very good agreement with the best available benchmark +results down to the critical field, and it captures in partic- +ular the softening of the excitation gap upon approaching +the critical field, suggesting that the non-equilibrium dy- +namics is sensitive to ground-state quantum criticality. +Our paper is structured as follows: Sec. II introduces +the spin-boson mapping and the Gaussian-Ansatz ap- +proach to the resulting nonlinear bosonic Hamiltoinian; +Sec. III discusses the results for the 1d TFIM, while +Sec. III B 2 presents the results for the 2d TFIM. Con- +clusions and pespectives are offered in Sec. V. +II. +GAUSSIAN-STATE ANSATZ FOR +QUANTUM SPIN SYSTEMS +In this section we illustrate the spin-boson mapping +and the Gaussian-Ansatz (GA) approach which allows +us to cast the evolution of the system in terms of the +evolution of the average fields and covariance matrix for +the bosonic operators. +A. +Spin-to-boson mapping +We shall consider a general linear/bilinear spin Hamil- +tonian, with the form +ˆH = +� +µ,ν=x,y,z +� +i 2S. +After the time t∗ the physics of the bosonic system would +depart from that of the spin system, and the approach is +no longer predictive. This pathology is clearly encoun- +tered by LSW theory for Ω < Ω∗ (see Sec. II D); yet, as +already mentioned above, the inclusion of non-linearities +removes this problem within the GA approach for all the +evolutions studied in this work. + +5 +FIG. 1. Evolution of the transverse magnetization mz of the 1d TFIM after a quantum quench to three field values Ω = 3Ωc, +Ωc and 0.1Ωc. All the data refer to a chain of N = 50 spins with periodic boundary conditions. The leftmost panel compares +the GA results with the LSW and the exact ones; the other panels only report the GA and the exact results. +III. +A SOLVABLE CASE: THE +TRANSVERSE-FIELD ISING CHAIN +We shall first benchmark the GA approach for the ex- +actly solvable case of the S = 1/2 one-dimensional TFIM +with nearest-neighbor interactions, namely Eq. (9) with +Jij = Jδj,i+1 defined on a periodic chain. +The one- +dimensional TFIM can be exactly solved by mapping +spins onto spinless fermions via the non-local Jordan- +Wigner transformation [37, 38]. The density of Jordan- +Wigner fermions corresponds to the deviation of the +spins from the configuration fully polarized with the +field, ˆSz = 1/2 − ˆf † +i ˆfi – where ˆfi, ˆf † +i are fermionic op- +erators. +The transverse spin components are instead +related in a non-local way to the fermionic operators +( ˆS+ +i += ˆfi exp(iπ � +j Ωc and a ferromagnetic phase for Ω < Ωc. +The spinless fermions onto which the spins are mapped +can be Bogolyubov diagonalized to give the dispersion +relation [38] +ϵk = +� +Ω(Ω + J cos k) + (J/2)2 +(15) +which becomes gapless for k = π at Ω = Ωc. This dis- +persion relation clearly differs from that of the linearized +bosons (Eq. (13)). Nonetheless the bosonic and fermionic +populations should be identical (when solving the bosonic +problem exactly, beyond LSW theory). +Therefore the +bosonic approach to the 1d TFIM amounts to reproduc- +ing the physics of non-interacting spinless fermions in +terms of the dynamics of strongly interacting bosons. It +is rather obvious that accounting for the non-linearities +in the bosonic problem – attempted in this work within +the GA approach – is an essential ingredient for this en- +deavor to be successful. +The solvable 1d TFIM clearly offers a very valuable +reference for approximate methods such as the GA ap- +proach. +At the same time, the dynamics of an inte- +grable system such as the 1d TFIM is highly non-generic, +and it poses a significant challenge to any approximation +scheme. Indeed the dynamics is strongly dominated by +a feature – the presence of an extensive number of con- +served quantities – which can only be captured by the +exact treatment of the problem [39] . In the following we +will focus on a few selected aspects of the dynamics, re- +vealing the emergence of a generalized Gibbs ensemble [9] +in the long-time dynamics of the transverse magnetiza- +tion; and fundamental spectral features from the Fourier +analysis of the spatio-temporal correlation pattern. +A. +Transverse magnetization +The transverse magnetization mz = +1 +N +� +i ⟨Sz +i ⟩ = +S − 1 +N +� +i Gii is a measure of the density of the inter- +acting gas of HP bosons (see Eq. (3a)). Monitoring its +dynamics offers a fundamental test of the hypothesis of + +6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Ω/J +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +mz +LSW +GA +Exact +FIG. 2. +Time-averaged transverse magnetization along the +quench dynamics of the 1d TFIM, comparing the GA, LSW +and exact values. The time window for the average is τJ = 40. +All the other parameters as in Fig. 1. +diluteness of HP bosons, which is at the core of the Taylor +expansion of the square roots in the HP transformation; +and which offers the best conditions for the GA approach +to be successful. Here we shall probe to what extent the +GA can cope with the proliferation of bosons and the +corresponding increase of non-linear effects. +Starting from the bosonic vacuum – namely the ground +state at Ω = ∞ – as initial state, the diluteness condi- +tion should be met at best when quenching the field Ω +to large values, Ω ≫ J. The dynamics of the transverse +magnetization is displayed on Fig 1 for three field values +(Ω = 3Ωc, Ωc, 0.1Ωc). In the case of a large field Ω = 3Ωc +we observe that the agreement of the GA with the ex- +act solution is rather remarkable. For that field value we +can also perform a LSW calculation, since LSW theory +becomes unstable only for Ω < Ω∗ = J = 2Ωc. +The +comparison with the GA result and the exact one shows +that nonlinear effects beyond LSW theory are already +sizable, in spite of the fact that the boson density is only +⟨n⟩ ∼ 0.03: LSW theory predicts a boson population +which is significantly larger, with much larger fluctua- +tions and with a dominant frequency which is about half +of that exhibited by the exact solution. When quench- +ing to a smaller field Ω = Ωc, we lose the LSW solution +(Ω < Ω∗) because of the uncontrolled proliferation of +bosons. On the other hand the GA approach still gives +a stable solution, which underestimates the boson popu- +lation generated by the quench, while still capturing the +right dominant frequency of oscillations and the presence +of damping in the oscillations. For an even larger quench +to Ω = 0.1Ωc, the GA approach agrees with the exact +solution only at short times, while it clearly deviates sig- +nificantly from the exact results at longer times, albeit +exhibiting large oscillations similar to those of the exact +solution. It is very important to remark here that the +boson population is found to rise to densities ⟨n⟩ ≈ 0.45, +which can no longer be considered small with respect to +their maximum value nmax = 1 allowed by the spin-to- +boson mapping. This means that not only the assump- +tion of a Gaussian state is to be called into question, but +also the truncation of the bosonic Hamiltonian to the +lowest nonlinear order. Hence it is imaginable that push- +ing the bosonic Hamiltonian to higher order of expansion +would lead to a significantly better agreement – although +the calculation would become rather cumbersome. +A summarizing picture of the dynamics of the magne- +tization is offered in Fig. 2, showing the time-averaged +magnetization mz = 1 +τ +� τ +0 dt mz(t) (with τJ = 40 – as a +function of the quench field Ω. We first analyze the ex- +act result, which clearly exhibits a behavior incompatible +with standard thermalization. Indeed, if the eigenstate +thermalization hypothesis applied [8], the time-averaged +magnetization would be expected to converge to the ther- +mal average value within the Gibbs ensemble, at a tem- +perature T such that ⟨ ˆH⟩T = ⟨ψ(0)| ˆH|ψ(0)⟩. +This is +clearly not the case for Ω < Ωc: even though the ap- +plied field decreases below Ωc, the time-averaged magne- +tization remains locked at the value ≈ 0.25. This non- +thermal behavior is clearly due to the non-integrable na- +ture of the TFIM at finite field; and also to the fact that +the Ω = 0 limit is equally special, given that it corre- +sponds to another integrable limit in which all individual +spin operators ˆSx +i commute with the Hamiltonian. +Fig. 2 compares the exact result with the prediction +of the GA approach as well as with that of LSW the- +ory. The two approaches match with the exact result for +high fields Ω ≫ Ωc; yet the LSW results start deviating +from the exact ones when Ω approaches Ω∗ = 2Ωc from +above, at which LSW theory develops its instability and +stops being predictive. On the other hand the GA ap- +proach remains quantitative for all values of Ω, although +the agreement with the exact result deteriorates upon ap- +proaching Ωc. We reiterate the fact that this deviation +may be due to the failure of the Gaussian approxima- +tion as well as to the truncation of the non-linearities in +the bosonic Hamiltonian. In particular the GA approach +appears to capture the salient feature of the non-thermal +behavior of the system, namely the fact that mz does not +vanish even when Ω → 0. +So far we have dealt with a single-spin property (the +average spin polarization); in the next section we shall ex- +plore instead the dynamics of correlations, and how this +dynamics reveals fundamental features of the excitation +spectrum. +B. +Dynamics of correlations and quench +spectroscopy +The dynamics of two-point correlations is one of the +most insightful aspects of the non-equilibrium evolution +of quantum many-body systems. Indeed, in systems with +elementary excitations having the nature of well-defined +quasi-particles, the development of correlations starting +e.g. from an uncorrelated state proceeds via the propa- +gation of such quasi-particles; and it possesses a causal +light-cone structure [40, 41] revealing the existence of a + +7 +FIG. 3. Evolution of the spin-spin correlation function for the z component of spins in the 1d TFIM: (a-c) false color plot for +the dynamics of Czz(i − j; t) as predicted by (a) LSW theory, (b) GA approach, and (c) the exact solution; (d) time-averaged +instantaneous structure factor Szz +k . All the data refer to a quench to the field value Ω = 3Ωc for a chain of length N = 50. +maximum speed of propagation for the excitations. A +more detailed analysis of the structure of correlations +within the light-cone allows one to inspect the excita- +tion spectrum of the system via the so-called quench +spectroscopy scheme [14–16]. In the following we shall +consider the time evolution of the spin-spin correlation +function +Cµµ(i, j; t) = ⟨ ˆSµ +i ˆSµ +j ⟩ − ⟨ ˆSµ +i ⟩⟨ ˆSµ +j ⟩ +(16) +focusing on the case µ = z and µ = x. +1. +⟨SzSz⟩ correlations and structure factor +We first focus on the spin-spin correlation function for +the z spin components. This quantity is easily accessi- +ble via the exact solution of the problem using fermion- +ization, as it corresponds to the density-density corre- +lation function for the fermions, Czz(i, j; t) = ⟨(1/2 − +ˆf † +i ˆfi)(1/2 − ˆf † +j ˆfj)⟩. +The GA expression for this quan- +tity is instead Czz(i, j; t) = Gij(δij + G∗ +ij) + |Fij|2; the +same expression can be used as well within LSW theory, +but evolving the G and F functions by using linearized +equations of motion. Fig. 3(a-c) shows the comparison +between the two bosonic approaches (LSW theory and +GA approach) with the exact solution for a moderate +quench at a field Ω = 3Ωc . We observe that both LSW +and GA approach correctly predict the light-cone struc- +ture of correlations, with the aperture of the light-cone +reflecting the speed of the fastest quasi-particle excita- +tions. Yet only the GA approach correctly captures the +structure of correlations within the light-cone, exhibiting +damped oscillations of the correlations at each distance +after the correlation front has passed; while LSW theory +predicts undamped oscillations. +A global picture on the correlations developing at long +times can be gathered by looking at the time-averaged +structure factor. +We first introduce the instantaneous +structure factor, namely the Fourier transform of the +spin-spin correlation function: +Sµµ +k (t) = 1 +N +� +i,j +eik·(ri−rj)Cµµ(i, j; t) . +(17) +Then we shall examine its time average for µ = z, +Szz +k += +1 +Nτ +� τ +0 dt Szz +k (t). Fig. 3(d) shows this quantity +as predicted by the GA approach, and compared with +LSW theory and the exact result. The agreement of the +GA prediction with the exact result is rather remarkable, +while LSW theory is off by almost a factor of 2, reflecting +the presence of undamped revivals of correlations that we +already commented above. +2. +⟨SxSx⟩ correlations and quench spectroscopy +The spatio-temporal pattern of correlations establish- +ing in the system during the dynamics is fundamentally +dictated by the spectral features of the excitations in the +system. Indeed it not only reveals the maximum group +velocity via the aperture of the light cone, but it can +also reveal the full dispersion relation when inspecting +the spatial structure within the light cone. This insight +is clearly suggested by LSW theory, when considering the +evolution of the instantaneous structure factor – namely +the spatial Fourier transform of the correlation pattern +at time t, as in Eq. (17). In particular, for µ = x, LSW +predicts that Sxx +k (t) oscillates at a frequency 2ωk (plus +a constant term), where ωk is the dispersion relation of +Eq. (13) [14, 42]; therefore its time-like Fourier transform +reveals the full dispersion relation. +More generally we can introduce the Fourier transform +of the instantaneous structure factor +Sxx +k (ω) = +� +∞ +−∞ +dt +2π e−iωtSxx +k (t) +(18) +=1 +4 +� +m,n +⟨ψ(0)|m⟩⟨n|ψ(0)⟩ δ(ω − ωn,m) +� +⟨m|( ˆS+ +k ˆS+ +−k + h.c.)|n⟩ + 2⟨m| ˆS+ +k ˆS− +k |n⟩ +� +where ωn,m = ωn − ωm; |m⟩, |n⟩ are Hamiltonian eigen- +states; and we have introduced the Fourier transformed +spin operator +ˆS− +k = +1 +√ +N +� +i +eik·ri ˆS− +i +ˆS+ +k = ( ˆS− +−k)† +(19) + +8 +0 +º/2 +º +3º/2 +2º +k +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +! +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +Sxx +k (!) +0 +º/2 +º +3º/2 +2º +k +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +! +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +Sxx +k (!) +0 +º/2 +º +3º/2 +2º +k +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +! +0.0002 +0.0004 +0.0006 +0.0008 +0.0010 +Sxx +k (!) +AB+nicbZ +DLSsNAFIZP6q3W6pLN4NFcFUSKepGKLpxZwV7gTaEyXTSDp1MwsxEKbGP4saFIm59Ene+jdM2C239YeDjP+dwzvxBwpnSjvNtFVZW19Y3ipulre2d3T27vN9Sc +SoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjC6ntbD1QqFot7PU6oF+GBYCEjWBvLt8u924gOMLpEc/CJb1ecqjMTWgY3hwrkavj2V68fkzSiQhOleq6TqK9DEvN +CKeTUi9VNMFkhAe0a1DgiCovm50+QcfG6aMwluYJjWbu74kMR0qNo8B0RlgP1WJtav5X6Y6vPAyJpJU0Hmi8KUIx2jaQ6ozyQlmo8NYCKZuRWRIZaYaJNWyYTg +Ln5GVqnVfesWrurVepXeRxFOIQjOAEXzqEON9CAJhB4hGd4hTfryXqx3q2PeWvBymcO4I+szx/s+JMn⌦ = ⌦c +AB/nicbVDLSsNAFL3xWesrKq7cDBbBVUikPjZC0Y07K9g +HtCFMptN26GQSZiZCRV/xY0LRdz6He78G6dtFtp64MKZc+5l7j1hwpnSrvtLSwuL +a+sFtaK6xubW9v2zm5dxaktEZiHstmiBXlTNCaZprTZiIpjkJOG+Hgeuw3HqhULB +b3ephQP8I9wbqMYG2kwN5v30a0h9El8pzTx+kjIFdch13AjRPvJyUIEc1sL/anZik +ERWacKxUy3MT7WdYakY4HRXbqaIJgPcoy1DBY6o8rPJ+iN0ZJQO6sbSlNBov6eyH +Ck1DAKTWeEdV/NemPxP6+V6u6FnzGRpJoKMv2om3KkYzTOAnWYpETzoSGYSGZ2RaS +PJSbaJFY0IXizJ8+T+onjnTnlu3KpcpXHUYADOIRj8OAcKnADVagBgQye4RXerCfrx +Xq3PqatC1Y+swd/YH3+ADXBlGE=⌦ = 1.5 ⌦c +AB/HicbZDNSsNAFIUn9a/Wv2iXbga +L4KokpagboejGnRVsLbQhTKY37dCZJMx +MhFDaV3HjQhG3Pog738Zpm4W2Hhj4OPde +7p0TJwp7TjfVmFtfWNzq7hd2tnd2z+wD +4/aKk4lhRaNeSw7AVHAWQtzTSHTiKBiI +DYzC6mdUfn0AqFkcPOkvAE2QsZBRo3 +l2+XenYABwVe4Nl2gT3274lSdufAquDlU +UK6mb3/1+jFNBUSacqJU13US7Y2J1Ixy +mJR6qYKE0BEZQNdgRAQobzw/foJPjdPHY +SzNizSeu78nxkQolYnAdAqih2q5NjP/q3 +VTHV56YxYlqYaILhaFKc6xrMkcJ9JoJp +nBgiVzNyK6ZBIQrXJq2RCcJe/vArtWtU9 +r9bv65XGdR5HER2jE3SGXHSBGugWNVELU +ZShZ/SK3qyp9WK9Wx+L1oKVz5TRH1mfP +01Uk+s=⌦ = 2 ⌦c +FIG. 4. +Quench-spectroscopy spectral function Sxx +k (ω) extracted from the quench dynamics of a 1d TFIM with N = 50 for +three field values Ω = 2Ωc, 1.5Ωc and Ωc. The dashed black line in the leftmost panel corresponds to 2ωk as predicted by LSW +theory; while the green dashed line (in all panels) stands for 2ϵk from the exact solution. The time window for the Fourier +transform is τJ = 40. +which create (resp. +destroy) a spin flip delocalized +throughout the system with momentum k. +This spectral function contains therefore frequency +contributions coming from transitions between pairs of +states |n⟩, |m⟩ which are connected by the creation (resp. +destruction) of two delocalized spin flips at wavevectors k +and −k via the operators ˆS− +k ˆS− +−k (resp. ˆS+ +k ˆS+ +−k ); or by +the successive creation and destruction of such a spin flip +(via the operator ˆS+ +k ˆS− +k ). The pairs of states entering in +the spectral function are weighted by their overlap with +the initial state |ψ(0)⟩; if this state has a significant over- +lap with the ground state |n⟩ = |0⟩, the spectral function +will receive contributions from transitions |0⟩ → |m⟩ in- +duced by the creation of two elementary excitations at +opposite wavevectors ±k. +When considering quenches +for Ω ≥ Ωc, the ground state of the TFIM is paramag- +netic and strongly aligned with the applied field, so that +we can imagine that these elementary excitations corre- +spond to the creation of two quasiparticles at opposite +wavevectors. In the specific case of the 1d TFIM, the +transition frequency ωm,n should therefore correspond to +the energy ϵk + ϵ−k = 2ϵk. These considerations suggest +that the double (spatial+temporal) Fourier transform of +the Cxx correlation function should reveal the dispersion +relation of the elementary excitations: this insight is at +the basis of the so-called quench spectroscopy (QS), al- +ready proposed in Refs. [14–16]. +Fig. 4 shows the QS spectral function Sxx +k (ω) as ob- +tained via the GA approach for three values of the trans- +verse field Ω (2Ωc, 1.5Ωc and Ωc). For all three cases +we clearly observe a strong signal, suggesting the ability +of the QS scheme to reconstruct the dispersion relation +of elementary excitations. In the case of the Cxx corre- +lation function and of its Fourier transform, we do not +easily have access to the exact result, given that the ˆSx +i ˆSx +j +operator, when expressed in terms of the free fermions, +contains a string of operators associated with all the sites +between the ith and the jth one. Yet we can compare +the predictions of the GA approach with the dispersion +relation 2ϵk obtained via the Jordan-Wigner transforma- +tion, Eq. (15). The first panel of Fig. 4 shows this com- +parison for Ω = 2Ωc, along with the prediction of the +dispersion relation from LSW theory, Eq. (13). Clearly +there is excellent agreement between the dispersion rela- +tion reconstructed via QS within the GA approach and +the exact prediction; while the LSW dispersion differs +from the fermionic one, predicting a gapless dispersion +mode (as the field in question corresponds to the insta- +bility field Ω∗ for LSW theory). The agreement between +the GA prediction via QS and the exact dispersion rela- +tion persists also at lower fields (down to Ωc) for what +concerns the dispersion relation at moderate to high fre- +quencies; while the GA prediction misses the fact that +the dispersion relation becomes exactly gapless for Ωc. +This may be due to the Gaussian-state approximation, +but also to the truncation of the bosonic nonlinearities +to quartic order, altering the nature of the Hamiltonian +of the system, and the position of its critical point. +C. +Discussion +In this section we have seen that the GA approach +is able to quantitatively capture many aspects of the +quench dynamics in the 1d TFIM for large fields Ω, and +down to the critical one Ωc. This means that a Gaussian- +state Ansatz for strongly interacting HP bosons is able to +mimic the physics of free spinless fermions onto which the +one-dimensional spin model can be exactly mapped. On +the other hand a purely linear theory (the LSW one) only +works at very large fields, while it fails at intermediate +ones due to a dynamical instability. Hence the success of +the GA approach entirely relies on its ability to account +(at least partially) for the nonlinearities of the bosonic +physics. +In particular we observed that the dynamics of cor- +relations allows one to reconstruct the dispersion rela- +tion for elementary excitations via Fourier transforma- + +9 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14 + 0 + 5 + 10 + 15 + 20 +SN/2 +Jt +L=8 +L=10 +L=12 +L=14 +L=16 +AB+nicbZDLSsNAFIZP6q3W6pLN4NFcFUSKepGKLp +xZwV7gTaEyXTSDp1MwsxEKbGP4saFIm59Ene+jdM2C239YeDjP+dwzvxBwpnSjv +NtFVZW19Y3ipulre2d3T27vN9ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjC6n +tbD1QqFot7PU6oF+GBYCEjWBvLt8u924gOMLpEc/CJb1ecqjMTWgY3hwrkavj2 +V68fkzSiQhOleq6TqK9DEvNCKeTUi9VNMFkhAe0a1DgiCovm50+QcfG6aMwlu +YJjWbu74kMR0qNo8B0RlgP1WJtav5X6Y6vPAyJpJU0Hmi8KUIx2jaQ6ozyQlm +o8NYCKZuRWRIZaYaJNWyYTgLn5GVqnVfesWrurVepXeRxFOIQjOAEXzqEON9CA +JhB4hGd4hTfryXqx3q2PeWvBymcO4I+szx/s+JMn⌦ = ⌦c +AB+nicbZD +LSsNAFIZP6q3W6pLN4NFcFUSKepGKLpxZwV7gTaEyXTSDp1MwsxEKbGP4saFIm59Ene+jdM2C239YeDjP+dwzvxBwpnSjvNtFVZW19Y3ipulre2d3T27vN9ScSoJb +ZKYx7ITYEU5E7Spmea0k0iKo4DTdjC6ntbD1QqFot7PU6oF+GBYCEjWBvLt8u924gOMLpEc/CJb1ecqjMTWgY3hwrkavj2V68fkzSiQhOleq6TqK9DEvNCKeTUi9 +VNMFkhAe0a1DgiCovm50+QcfG6aMwluYJjWbu74kMR0qNo8B0RlgP1WJtav5X6Y6vPAyJpJU0Hmi8KUIx2jaQ6ozyQlmo8NYCKZuRWRIZaYaJNWyYTgLn5GVqnV +fesWrurVepXeRxFOIQjOAEXzqEON9CAJhB4hGd4hTfryXqx3q2PeWvBymcO4I+szx/s+JMn⌦ = ⌦c + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 + 0.12 + 0 + 0.2 0.4 0.6 0.8 + 1 + 1.2 1.4 +SN/2 / (N/2) +Jt / L +L=8 +L=10 +L=12 +L=14 +L=16 +AB7nicbVDLSgNBEOz1GeMr +6tHLYBA8xd0Q1GNQBE8SwTwgCWF20p +sMmZ1dZmaFsOQjvHhQxKvf482/cZLsQ +RMLGoqbrq7/FhwbVz321lZXVvf2Mx +t5bd3dvf2CweHDR0limGdRSJSLZ9qF +Fxi3XAjsBUrpKEvsOmPbqZ+8wmV5pF8 +NOMYuyEdSB5wRo2Vmre9P68POkVim +7JnYEsEy8jRchQ6xW+Ov2IJSFKwTV +u25semVBnOBE7ynURjTNmIDrBtqa +Qh6m46O3dCTq3SJ0GkbElDZurviZSGW +o9D3aG1Az1ojcV/PaiQmuimXcWJ +QsvmiIBHERGT6O+lzhcyIsSWUKW5vJ +WxIFWXGJpS3IXiLy+TRrnkXZQqD5V +i9TqLIwfHcAJn4MElVOEOalAHBiN4hl +d4c2LnxXl3PuatK042cwR/4Hz+AJGE +jxQ=EN/2 +AB/HicbVDLSs +NAFL3xWesr2qWbwSK4qkp6rIogiupYB/QhjCZTtqhkwczEyGE+CtuXCji1g9x5984TbvQ1gP3cjnXubO8WLOpLKsb2NldW19Y7O0Vd7e2d3bNw8OzJKBKFtEvFI9DwsKW +chbSumO3FguLA47TrTa6nfveRCsmi8EGlMXUCPAqZzwhWnLNysAXmGQ3bnZ3Vs/zortm1apZBdAysekCnO0XPNrMIxIEtBQEY6l7NtWrJwMC8UIp3l5kEgaYzLBI9rXN +MQBlU5WHJ+jE60MkR8JXaFChfp7I8OBlGng6ckAq7Fc9Kbif14/Uf6lk7EwThQNyewhP+FIRWiaBoyQYniqSaYCKZvRWSMdRpK51XWIdiLX14mnXrNPq817hvV5tU8jhIcw +TGcg0X0IRbaEbCKTwDK/wZjwZL8a78TEbXTHmOxX4A+PzB/QrlFQ=EN/2 +N/2 +AB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi3iqYj+gDWz3bR +LN5uwOxFK6D/w4kERr/4jb/4bt20O2vpg4PHeDPzgkQKg67RWVtfWN4qbpa3tnd298v +5B08SpZrzBYhnrdkANl0LxBgqUvJ1oTqNA8lYwupn6rSeujYjVI4T7kd0oEQoGEUrPeBdr1 +xq+4MZJl4OalAjnqv/NXtxyNuEImqTEdz03Qz6hGwSflLqp4QlIzrgHUsVjbjxs9mlE +3JilT4JY21LIZmpvycyGhkzjgLbGVEcmkVvKv7ndVIMr/xMqCRFrth8UZhKgjGZvk36QnOG +cmwJZVrYWwkbUk0Z2nBKNgRv8eVl0jyrehfV8/vzSu06j6MIR3AMp+DBJdTgFurQAYhPM +rvDkj58V5dz7mrQUnzmEP3A+fwB4po1VtJ +AB63icbVBNSwMxEJ2tX +7V+VT16CRbBU92Voh6LXkQ8VLAf0C4lm2b0CS7JFmhLP0LXjwo4tU/5M1/Y7rdg7Y+GHi8N8PMvCDmTBvX/XYK6tr6xvFzdLW9s7uXn/oKWjRBHaJBGPVCfAmnImadMw2knVhSLgN2ML6 +Z+e0nqjSL5KOZxNQXeChZyAg2mXR3dt8vV9yqmwEtEy8nFcjR6Je/eoOIJIJKQzjWu5sfFTrAwjnE5LvUTGJMxHtKupRILqv0u3WKTqwyQGkbEmDMvX3RIqF1hMR2E6BzUgvejPxP6+b +mPDKT5mME0MlmS8KE45MhGaPowFTlBg+sQTxeytiIywsTYeEo2BG/x5WXSOq96F9XaQ61Sv87jKMIRHMpeHAJdbiFBjSBwAie4RXeHOG8O/Ox7y14OQzh/AHzucPfpqN5A=tJ/L +(a) +(b) +FIG. 5. Evolution of the half-system entanglement entropy EN/2 following a quench to Ω = Ωc in the 2d TFIM, obtained via +the GA approach: (a) entropy evolution for various lattice sizes N = L × L; (b) entropy per spin as a function of the time +rescaled by the linear dimension L of the system. +tion (quench spectroscopy, QS); and that the GA results +for the quench-spectroscopy signal are in very good agree- +ment with the dispersion relation expected for the ele- +mentary fermionic excitations in the 1d TFIM. A word +of caution is in order nonetheless when considering the +QS signal at low fields Ω ≈ Ωc, in spite of the (partial) +agreement between the GA predictions and the exact dis- +persion relation. Indeed in that regime LSW fails com- +pletely, suggesting that the picture of elementary excita- +tions as being bosonic spin flips may no longer be valid. +Indeed the picture of elementary excitations offered by +the Jordan-Wigner mapping is that of fermionic quasi- +particles, namely of Bogolyubov transformations of the +ˆfi, ˆf † +i operators, which in turn do not correspond to sim- +ple spin-flip operators, but to spin operators “dressed” +with operator strings, ˆf † +i = ˆS− +i exp[iπ � +j Ω∗ we +can also compare with LSW theory. For this latter field +we observe that the agreement between the GA results +and the CNN ones is rather remarkable; and that it is +fundamentally due to the inclusion of the quartic non- +linearities in the bosonic theory, since the LSW results, +which ignore all nonlinearities, deviate significantly from +both the GA and the CNN ones. The agreement between +GA and CNN Ansatz is less good for Ω = Ωc, albeit the +most salient features of the evolution (such as the charac- +teristic timescales for the evolution of the magnetization) + +12 +(a) +(b) +(c) +(d) +(e) +(f) +AB/HicbZDNSsNAFIUn9a/Wv2iXbgaL4KokpagboejGnRV +sLbQhTKY37dCZJMxMhFDaV3HjQhG3Pog738Zpm4W2Hhj4OPde7p0TJwp7TjfVmFtf +WNzq7hd2tnd2z+wD4/aKk4lhRaNeSw7AVHAWQtzTSHTiKBiIDYzC6mdUfn0AqFkc +POkvAE2QsZBRo3l2+XenYABwVe4Nl2gT3274lSdufAquDlUK6mb3/1+jFNBUSac +qJU13US7Y2J1IxymJR6qYKE0BEZQNdgRAQobzw/foJPjdPHYSzNizSeu78nxkQolYn +AdAqih2q5NjP/q3VTHV56YxYlqYaILhaFKc6xrMkcJ9JoJpnBgiVzNyK6ZBIQrXJq +2RCcJe/vArtWtU9r9bv65XGdR5HER2jE3SGXHSBGugWNVELUZShZ/SK3qyp9WK9Wx+ +L1oKVz5TRH1mfP01Uk+s=⌦ = 2 ⌦c +AB+nicbZDLSsNAFIZP6q3W6pLN4NFcFUSKepGKLpxZwV +7gTaEyXTSDp1MwsxEKbGP4saFIm59Ene+jdM2C239YeDjP+dwzvxBwpnSjvNtFVZW1 +9Y3ipulre2d3T27vN9ScSoJbZKYx7ITYEU5E7Spmea0k0iKo4DTdjC6ntbD1QqFot +7PU6oF+GBYCEjWBvLt8u924gOMLpEc/CJb1ecqjMTWgY3hwrkavj2V68fkzSiQhOl +eq6TqK9DEvNCKeTUi9VNMFkhAe0a1DgiCovm50+QcfG6aMwluYJjWbu74kMR0qNo8B +0RlgP1WJtav5X6Y6vPAyJpJU0Hmi8KUIx2jaQ6ozyQlmo8NYCKZuRWRIZaYaJNWy +YTgLn5GVqnVfesWrurVepXeRxFOIQjOAEXzqEON9CAJhB4hGd4hTfryXqx3q2PeWv +BymcO4I+szx/s+JMn⌦ = ⌦c +AB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyGRom6Eoht3VrA +PaEOYTG/boTNJmJkIJVT8FTcuFHrd7jzb5y2WjrgQtnzrmXufeECWdKu+63VhaX +ldK6XNja3tnfs3b2GilNJoU5jHstWSBRwFkFdM82hlUgIuTQDIfXE7/5AFKxOLr +XowR8QfoR6zFKtJEC+6BzK6BP8CV2He9x9ghoYJdx50CLxIvJ2WUoxbYX51uTFMBk +acKNX23ET7GZGaUQ7jUidVkBA6JH1oGxoRAcrPpuP8bFRurgXS1ORxlP190RGhFI +jEZpOQfRAzXsT8T+vnerehZ+xKEk1RHT2US/lWMd4kgXuMglU85EhEpmdsV0QCSh2 +iRWMiF48ycvksap4505lbtKuXqVx1FEh+gInSAPnaMqukE1VEcUZegZvaI368l6sd6 +tj1lrwcpn9tEfWJ8/Lf6UXA=⌦ = 0.1 ⌦c +FIG. 8. +Evolution of the spin-spin correlation function for the z component in the 2d TFIM, at two distances |i − j| = 1 (first +column) and |i − j| = 2 second column, for three field values (for each of the three rows) Ω = 2Ωc, Ωc and 0.1Ωc. All the panels +compare the GA results with the CNN ones from Ref. [28], and have been obtained for a N = 10 × 10 lattice. +are correctly captured by the GA. On the other hand for +Ω = 0.1Ωc the agreement between GA and CNN is only in +the overall amplitude of the magnetization fluctuations. +Fig. 7 addresses the question of thermalization of the +many-body dynamics, by showing the comparison be- +tween the time-averaged magnetization mz as obtained +within the GA approach; the same quantity obtained +from the CNN Ansatz (with results averaged over the +limited time window studied in Ref. [28]); and the pre- +dictions of the Gibbs ensemble for a N = 8 × 8 sys- +tem. +The Gibbs-ensemble results have been obtained +via quantum Monte Carlo (based on the stochastic se- +ries expansion [47] approach) at a temperature T such +that ⟨ ˆH⟩T = ⟨ψ(0)| ˆH|ψ(0)⟩. We have used the tempera- +tures corresponding to the expected Gibbs ensemble for +varying Ω as mapped out in Ref. [20]. We clearly ob- +serve that for Ω ≳ 2J the time-averaged magnetization +reproduces rather closely the Gibbs ensemble result. On +the other hand the time-averaged GA results and the +Gibbs-ensemble prediction deviate from each other sig- +nificantly at lower fields, with the dynamical result sys- +tematically overestimating the thermal equilibrium one. +This is clearly not a limitation of the GA approach, as +a similar result holds as well for the limited CNN data +offered in Ref. [28]. This apparent lack of thermalization +at low fields (at least over the limited time window of +interest) can be naturally understood in the limit Ω = 0, +which, as already discussed above, corresponds to an in- +tegrable case in which all Sx +i operators commute with the +Hamiltonian. Therefore it is not entirely surprising that +a non-thermalizing dynamics sets in when Ω → 0 – this is +apparently similar to the case of the 1d TFIM discussed +above, although the 1d TFIM is provably integrable for +any value of Ω. Hence we conclude that the non-thermal +behavior of the averaged magnetization observed at low +fields within the GA – in particular, the fact that the +magnetization does not relax towards a vanishing time- +averaged value when Ω → 0 – may be in fact an actual +feature of the dynamics, and not at all a limitation of the +GA approach. +C. +Correlation dynamics and quench spectroscopy +We complete our study by considering the dynamics of +correlations, and how it can reveal fundamental features +of the excitation spectrum. +Fig. 8 shows the GA and +CNN predictions for the evolution of the Cxx(i, j; t) cor- +relation function for two inter-site distances |i − j| = 1 +and 2, and for the three values of the field already ex- +plored above (Ω = 2Ωc, Ωc and 0.1Ωc). Similar to the +case of the magnetization, the agreement between GA +and CNN results is rather remarkable for the largest field, +while it remains acceptable for Ω = Ωc and it degrades +rather drastically for Ω = 0.1Ωc. This suggest therefore +that the GA predictions remain quantitative at least over +the entire field range Ω ≳ Ωc. +A more global picture of the correlation dynamics +is offered by the quench-spectroscopy analysis of the +Cxx(i, j; t) correlation function. As already discussed in + +13 +(a) +(b) +(c) +(d) +AB+nicbZDLSsNAF +IZP6q3W6pLN4NFcFUSKepGKLpxZwV7gTaEyXTSDp1MwsxEKbGP4saFIm59Ene+jdM2C239YeDjP+dwzvxBwpnSjvNtFVZW19Y3ipulre2d3T27vN9ScSoJbZKYx7ITYEU5E7Spm +ea0k0iKo4DTdjC6ntbD1QqFot7PU6oF+GBYCEjWBvLt8u924gOMLpEc/CJb1ecqjMTWgY3hwrkavj2V68fkzSiQhOleq6TqK9DEvNCKeTUi9VNMFkhAe0a1DgiCovm50+QcfG6 +aMwluYJjWbu74kMR0qNo8B0RlgP1WJtav5X6Y6vPAyJpJU0Hmi8KUIx2jaQ6ozyQlmo8NYCKZuRWRIZaYaJNWyYTgLn5GVqnVfesWrurVepXeRxFOIQjOAEXzqEON9CAJhB4h +Gd4hTfryXqx3q2PeWvBymcO4I+szx/s+JMn⌦ = ⌦c +AB/HicbZDLSsNAFIZP6q3W7RLN4NFcFWSUtSNUHTjzgr2Am0Ik+mkHTq +ZhJmJEp9FTcuFHrg7jzbZy2WjrDwMf/zmHc+YPEs6Udpxvq7C2vrG5Vdwu7ezu7R/Yh0dtF +aeS0BaJeSy7AVaUM0FbmlOu4mkOAo47QTjm1m980ilYrF40FlCvQgPBQsZwdpYvl3u30V0iNE +VqEF+sS3K07VmQutgptDBXI1furP4hJGlGhCcdK9Vwn0d4ES80Ip9NSP1U0wWSMh7RnUOCIK +m8yP36KTo0zQGEszRMazd3fExMcKZVFgemMsB6p5drM/K/WS3V46U2YSFJNBVksClOdIxmSaA +Bk5RonhnARDJzKyIjLDHRJq+SCcFd/vIqtGtV97xav69XGtd5HEU4hM4AxcuoAG30IQWEMjgG +V7hzXqyXqx362PRWrDymTL8kfX5A7vdk40=⌦ = 2⌦c +ACAHicbVDLSgMxFM3UV62vURcu3ASL4GqYKfWxEYpu3FnBPqAdhkyaUO +TzJBkhDLUhb/ixoUibv0Md/6NaTsLbT1w4eSce8m9J0wYVdp1v63C0vLK6lpxvbSxubW9Y+/uN +VWcSkwaOGaxbIdIEUYFaWiqGWknkiAeMtIKh9cTv/VApKxuNejhPgc9QWNKEbaSIF90L3lpI/ +gJfScyil8nD0DHNhl13GngIvEy0kZ5KgH9le3F+OUE6ExQ0p1PDfRfoakpiRcambKpIgPER90 +jFUIE6Un0PGMNjo/RgFEtTQsOp+nsiQ1ypEQ9NJ0d6oOa9ifif10l1dOFnVCSpJgLPopSBnU +MJ2nAHpUEazYyBGFJza4QD5BEWJvMSiYEb/7kRdKsON6ZU72rlmtXeRxFcAiOwAnwDmogRtQB +w2AwRg8g1fwZj1ZL9a79TFrLVj5zD74A+vzBwYtlMc=⌦ = 1.25 ⌦c +AB/3icbVDLSgMxFM3UV6 +2vUcGNm2ARXA0zUh8boejGnRXsA9phyKSZNjTJDElGKGMFf8WNC0Xc+hvu/BvTdhbaeuDCyTn3kntPmDCqtOt+W4WFxaXleJqaW19Y3PL3t5pqDiVmNRxzGLZCpEijApS1Qz0kokQTxkpBkOrs +Z+85IRWNxp4cJ8TnqCRpRjLSRAnuvc8NJD8EL6Dkn8H6CnBgl13HnQDOEy8nZCjFthfnW6MU06Exgwp1fbcRPsZkpiRkalTqpIgvA9UjbUIE4UX42X8ED43ShVEsTQkNJ+rviQxpY8NJ +0c6b6a9cbif1471dG5n1GRpJoIP0oShnUMRyHAbtUEqzZ0BCEJTW7QtxHEmFtIiuZELzZk+dJ49jxTp3KbaVcvczjKIJ9cACOgAfOQBVcgxqoAwewDN4BW/Wk/VivVsf09aClc/sgj+wPn8Aj ++KUiw=⌦ = 1.5 ⌦c +FIG. 9. +Quench-spectroscopy spectral function Sxx +k (ω) extracted from the quench dynamics of a 2d TFIM with N = 24 × 24 +for four field values Ω = 2Ωc, 1.5Ωc, 1.25Ω and Ωc. The x axis refers to a linear trajectory in the 2d Brillouin zone, going from +Γ = (0, 0) to X = (π, π) to M = (π, 0) and then back to Γ. The dashed red line in all panels stands for 2εk from the series +expansion of Ref. [44]. The Fourier transform is performed over the time window τJ = 25. +Sec. III B 2, this analysis is expected to reveal the dis- +persion relation of elementary excitations at least in the +paramagnetic phase for Ω > Ωc, in which elementary ex- +citations should be generated by spin flips along the z +axis, induced by the ˆSx +i operators. Unlike the case of the +1d TFIM, for the 2d system the bosonic picture of the +elementary excitations can be expected to be valid at all +fields, and therefore the GA predictions for the results of +the QS analysis may be remain quantitative down to low +fields. +In order to benchmark the GA results we make use +of the most accurate predictions for the dispersion rela- +tion to our knowledge, namely the ones resulting from +a series expansion in powers of J/Ω around the limit +J → 0 up to order 4 [44]. +The QS spectral function +for the 2d TFIM is shown in Fig. 9 for four field val- +ues (Ω = 2Ωc, 1.5Ωc, 1.25Ωc and Ωc) on approach to the +critical point, and compared with 2εk where εk is the +dispersion relation obtained from the series expansion. +The agreement between the GA results and the series- +expansion predictions is rather striking, bearing two con- +sequences: 1) the fact that the GA approach can faith- +fully reconstruct the nonlinear excitation spectrum of a +strongly interacting system (as a reminder, the linear +excitation spectrum develops imaginary frequencies for +Ω < Ω∗ ≈ 1.3Ωc); and 2) the fact that the correlation dy- +namics far from equilibrium is sensitive to the existence +of a ground-state phase transition, as the QS function +exhibits the softening of the excitation spectrum upon +approaching the quantum critical field. +The fact that +the QS signal from the GA approach does not become +fully gapless at the exact critical field might be due to +the GA approximation to the evolved state; as well as +to the fact that the bosonic Hamiltonian differs from the +exact spin one because of the truncation of nonlinearities +to quartic order. +V. +CONCLUSION AND OUTLOOK +We have introduced and validated the Gaussian- +Ansatz (GA) approach to the non-equilibrium dynam- +ics of quantum spin systems. Our approach is based on +the Holstein-Primakoff (HP) mapping of the spin model +onto a nonlinear bosonic one, and to the truncation of +the nonlinear bosonic Hamiltonian to a finite order. The +Gaussian-state Ansatz allows for the efficient calculation +of the non-equilibrium dynamics: all the information on +the Gaussian state of an N-spin system is contained in +the O(N 2) elements of the covariance matrix, circum- +venting the exponential growth of the Hilbert space di- +mensions. The quantization axis defining the HP trans- + +14 +formation must be chosen so that the initial state of the +evolution is a Gaussian state of the HP bosons – in this +work we have investigated the case of a dynamics ini- +tialized in a coherent spin state aligned with the local +quantization axis, so that the initial bosonic state cor- +responds to the vacuum. The ensuing dynamics is ex- +pected to preserve the Gaussian nature of the state as +long as the density of bosons, proliferating under the non- +equilibrium evolution, remains moderate. More general +initial states can be explored as well – e.g. the ground +states, or even the low-temperature equilibrium states, of +quantum spin Hamiltonians approximated via the linear +or non-linear (i.e. modified) spin-wave theory. +Accounting for nonlinearities turns out to be essen- +tial in order to keep the proliferation of HP bosons +under control during the dynamics; and to reproduce +the emergence of the equilibrium statistical averages +of local observables in the long-time dynamics of the +system. +We investigated the quench dynamics of the +transverse-field Ising model in one and two dimensions +starting from a state aligned with the field, and we ob- +served that the predictions of the GA approach remain +quantitative for quenches to fields down to the critical +field associated with the ground-state paramagnetic-to- +ferromagnetic transition. In particular the GA predic- +tions for the evolution of the spin-spin correlations allow +us to reconstruct the dispersion relation of elementary +excitations via a quench-spectroscopy analysis (Fourier +transform of the spatio-temporal correlation pattern): +the dispersion relation resulting from the GA is in very +good agreement with the most accurate predictions for +the elementary excitation spectrum of the model of in- +terest. +In particular the resulting excitation spectrum +exhibits mode softening upon approaching the quantum +critical field, suggesting that the non-equilibrium dynam- +ics of the system can reveal the existence of ground-state +quantum critical points. +Our results show that the GA approach to the non- +linear bosonic equivalent of quantum spin Hamiltonians +allows one to efficiently investigate the non-equilibrium +dynamics of the system far beyond the linearized treat- +ment, and to account for most salient effects of a) re- +laxation of local observables to an equilibrium (Gibbs or +generalized Gibbs) ensemble; and b) the emergence of +the dispersion relation of elementary excitations in the +correlation dynamics. The main limitations of the GA +approach to quantum spin systems stem from its approx- +imate treatment of quantum fluctuations; and, equally +importantly, from the fact that the nonlinear bosonic +Hamiltonian needs to be truncated to a finite order in its +otherwise infinite expansion in powers of n/(2S) (where +n is the local boson density and S the spin length). While +in this work we chose to consider the minimal nonlinear +Hamiltonian (truncated to quartic order), the expansion +can be pushed to higher order, with the simple effect of +complicating the derivation of the equations of motion +for the elements of the covariance matrix. +The GA approach appears to be a very promising tool +to investigate quantum spin dynamics far beyond the +regimes explored in this work. First of all, the spin length +S enters in the equations of motion for the covariance ma- +trix as a parameter, so that spins of arbitrary length – +namely the quantum dynamics of systems of interacting +qudits – can be studied without any additional computa- +tional cost resulting from the growth of the local Hilbert- +space dimensions. This is particularly relevant when con- +sidering e.g. the dynamics of large-spin magnetic atoms +trapped in optical lattices. The ability of the GA ap- +proach to investigate the equilibration of the system sug- +gests that the same approach can be used to investigate +the failure of the system to relax in the presence of strong +disorder, leading to many-body localization. The latter +phenomenon, forcing the evolved state of the system to +remain parametrically close to the initial state, appears +to be very promising for a GA study, given that the GA +is fully justified when the evolved state of the system +remains close to the initial state if such a state is Gaus- +sian. Further extensions of the GA approach may involve +the study of dynamical phase transitions with Loschmidt +echo singularities [48] (and the Loschmidt echo can be +efficiently evaluated for Gaussian states [49]); the study +of time crystallization under periodic driving [50]; and +the extension of the approach to open-system dynamics, +most importantly within a wavefunction Monte Carlo ap- +proach to the master-equation dynamics, in which each +quantum trajectory can be approximated via a Gaus- +sian state, as already done for models of strongly inter- +acting photons [51]. +Its very moderate computational +cost (amounting at most to the numerical solution of +O(N 2) coupled differential equations in the absence of +any symmetry) makes of the GA approach a very promis- +ing tool to efficiently benchmark experimental quantum +simulators in regimes (of very large system sizes, very +large spins, etc.) in which other approaches – such as +wavefunction-based Ans¨atze – become unpractical. +ACKNOWLEDGMENTS +We would like to thank M. Wouters and W. Verstraelen +for useful discussions. TR is supported by ANR (“EELS” +project), QuantERA (“MAQS” project) and PEPR-Q +(“QubitAF” project). +Appendix A: Equations of motion for the +transverse-field Ising model +Here we report the equations of motion for the covari- +ance matrix when considering the bosonic quartic Hamil- +tonian of Eq. (10) onto which the transverse-field Ising +model is mapped via the Holstein-Primakoff transforma- + +15 +tion. +d +dtGij = i +� +Gij − G∗ +ji +� +, +(A1a) +d +dtFij = i +� +Fij + Fji + jF +ij +� +, +(A1b) +where G, F and jF are N × N matrices. 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Research 2, 022037 (2020). + diff --git a/dtAzT4oBgHgl3EQfZ_z3/content/tmp_files/load_file.txt b/dtAzT4oBgHgl3EQfZ_z3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..adeb3b4583c463b6b1690f01df1db0504718f374 --- /dev/null +++ b/dtAzT4oBgHgl3EQfZ_z3/content/tmp_files/load_file.txt @@ -0,0 +1,1178 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf,len=1177 +page_content='Gaussian-state Ansatz for the non-equilibrium dynamics of quantum spin lattices Rapha¨el Menu1 and Tommaso Roscilde2 1Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Universit¨at des Saarlandes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' D-66123 Saarbr¨ucken,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Germany 2Univ Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' ENS de Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Laboratoire de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' F-69342 Lyon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' France (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' 2023) The study of non-equilibrium dynamics is one of the most important challenges of modern quan- tum many-body physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' in relationship with fundamental questions in quantum statistical mechan- ics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' as well as with the fields of quantum simulation and computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' In this work we propose a Gaussian Ansatz for the study of the nonequilibrium dynamics of quantum spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Within our approach, the quantum spins are mapped onto Holstein-Primakoff bosons, such that a coherent spin state – chosen as the initial state of the dynamics – represents the bosonic vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' The state of the system is then postulated to remain a bosonic Gaussian state at all times, an assumption which is exact when the bosonic Hamiltonian is quadratic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' and which is justified in the case of a nonlinear Hamiltonian if the boson density remains moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' We test the accuracy of such an Ansatz in the paradigmatic case of the S = 1/2 transverse-field Ising model, in one and two dimensions, initialized in a state aligned with the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' We show that the Gaussian Ansatz, when applied to the bosonic Hamiltonian with nonlinearities truncated to quartic order, is able to reproduce faithfully the evolution of the state, including its relaxation to the equilibrium regime, for fields larger than the critical field for the paramagnetic-ferromagnetic transition in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' In particular the spatio-temporal pattern of correlations reconstructed via the Gaussian Ansatz reveals the dis- persion relation of quasiparticle excitations, exhibiting the softening of the excitation gap upon approaching the critical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Our results suggest that the Gaussian Ansatz correctly captures the essential effects of nonlinearities in quantum spin dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' and that it can be applied to the study of fundamental phenomena such as quantum thermalization and its breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' INTRODUCTION The non-equilibrium unitary dynamics of quantum many-body systems represents a central topic of mod- ern quantum physics: it lies at the core of the coher- ent manipulation of complex quantum states with quan- tum devices [1–4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' and it represents the mechanism by which equilibration and the emergence of statistical en- sembles occurs [5–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' In the case of systems with a time-independent Hamiltonian (which will be the focus of this work), central questions concern the propagation of correlations and entanglement, and the scrambling of quantum information [4, 11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' how such phenomena man- ifest the nature of elementary excitations [12–16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' and how they lead to the onset of equilibration [7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Answering to the above questions quantitatively for sys- tems of increasing size is a significant challenge, due to the exponential increase of the Hilbert-space dimensions with system size, making an exact numerical treatment impractical for systems going beyond e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' a few tens of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' The development of efficient numerical schemes which approximate the quantum many-body evolution is therefore the only realistic strategy to approach the prob- lem theoretically – the only alternative solution comes from the direct experimental implementation of the dy- namics of interest via an analog or digital quantum simu- lator [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' By “efficient scheme” we mean here a numerical approach whose computational cost scales polynomially with system size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' and which ideally can be improved sys- tematically, while maintaining a polynomial cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' A general framework for the development of efficient numerical schemes is the one of wavefunction Ans¨atze, positing that the state vector has an explicit functional form depending on a number of variational parame- ters which is much smaller than the Hilbert-space di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Relevant examples of variational Ans¨atze for the unitary evolution are tensor network states [17, 18], neural-network quantum states [19], Jastrow-like wave- functions [20–22], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Among these families of states, the tensor-network states can be systematically improved by increasing the bond dimension, whose logarithm reg- ulates the maximum amount of subsystem entanglement entropy that the state can accommodate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Nonetheless unitary evolutions lead systematically to extensive en- tanglement entropies, leading to the so-called entangle- ment barrier, namely the exponential increase of the bond dimension required to reproduce the state faith- fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Other family of states allow for systematical im- provements without facing an entanglement barrier [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' nonetheless, all these approaches are in general very de- manding from a computational point of view, and their computational cost further scales significantly with the dimensions of the local Hilbert space describing individ- ual degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' An alternative to wavefunction Ans¨atze, valid for spin or bosonic systems, is offered by the discrete truncated Wigner approximation (DTWA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' This approach postulates an evolved state of the system expressed in terms of the Wigner-function representation of the initial state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' and of phase-point operators depen- dent on parameters evolved according to classical equa- tions of motion [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' The DTWA approach is very pow- erful in representing evolutions arbitrarily far from equi- librium [25], as it is based on a Monte Carlo sampling of classical trajectories initialized via the Wigner func- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content='01363v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content='quant-gas] 3 Jan 2023 2 tion of the initial state, without making any assumption on the statistics of quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Yet it rep- resents instantaneous expectation values as incoherent Monte Carlo averages, and therefore it misses many-body interference effects which are at the basis of fundamental phenomena (such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' many-body localization [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' In this work we propose and validate an alternative ap- proximation schemes for quantum evolutions of generic quantum systems – bosonic or fermionic alike – based on Gaussian states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Gaussian many-body states have the property that the reduced density matrix of each sub- system has a Gaussian form, such that all observables obey Wick’s theorem – namely they can be reconstructed from the knowledge of the average fields and the covari- ance matrix, whose elements are given by the regular and anomalous two-point Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Hence re- constructing the evolution of the state amounts to track- ing the dynamics of the average fields and covariance matrix, obeying coupled non-linear equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' This scheme is well known for fermionic (bosonic) sys- tems as the time-dependent Hartree-Fock (Hartree-Fock- Bogolyubov) approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' yet in this work we propose its application to quantum spin systems, specifically based on the Holstein-Primakoff (HP) spin-boson mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' A Gaussian Ansatz for the equilibrium state of the non- linear bosonic Hamiltonian resulting from the HP map- ping is at the core of the so-called modified spin-wave theory [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Our approach can therefore be viewed as a time-dependent version of modified spin-wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' On the other hand, conventional time-dependent spin-wave theory amounts to discarding all non-linear terms in the bosonic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' We apply the Gaussian Ansatz approach to the non- equilibrium evolution of a paradigmatic model for quan- tum simulation, namely the S = 1/2 transverse-field Ising model (TFIM), initialized in a state aligned with the applied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' This dynamics corresponds to a quan- tum quench starting from the ground state at infinite field, and triggered by changing the field abruptly to a finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' We first benchmark our approach in the case of the one-dimensional TFIM, which can be ex- actly solved by mapping it onto free fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Our re- sults show that the dynamics of the free-fermion sys- tem can be mimicked quantitatively by that of a sys- tem of strongly interacting Holstein-Primakoff bosons for quenches to fields as low as the critical one for the ground-state phase transition from paramagnet to fer- romagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Even more convincing results are obtained for the case of the two-dimensional TFIM, which is not exactly solvable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' but whose dynamics can be quantita- tively reconstructed via time-dependent variational cal- culations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' In both cases (1d and 2d) we show that the Gaussian Ansatz is able to capture fundamental features of the spatio-temporal structure of the quantum corre- lations developing after the quench, as observed in their Fourier transform (via a so-called quench spectroscopy scheme [14–16]), which allows one to reconstruct the dis- persion relation of elementary excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' The disper- sion relation reconstructed via quench spectroscopy is in very good agreement with the best available benchmark results down to the critical field, and it captures in partic- ular the softening of the excitation gap upon approaching the critical field, suggesting that the non-equilibrium dy- namics is sensitive to ground-state quantum criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Our paper is structured as follows: Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' II introduces the spin-boson mapping and the Gaussian-Ansatz ap- proach to the resulting nonlinear bosonic Hamiltoinian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' III discusses the results for the 1d TFIM, while Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' III B 2 presents the results for the 2d TFIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Con- clusions and pespectives are offered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' GAUSSIAN-STATE ANSATZ FOR QUANTUM SPIN SYSTEMS In this section we illustrate the spin-boson mapping and the Gaussian-Ansatz (GA) approach which allows us to cast the evolution of the system in terms of the evolution of the average fields and covariance matrix for the bosonic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfZ_z3/content/2301.01363v1.pdf'} +page_content=' Spin-to-boson mapping We shall consider a general linear/bilinear spin Hamil- tonian, with the form ˆH = � µ,ν=x,y,z � i 0 is violated, the di- +electric membrane would thin down uniformly, leading eventually to electric breakdown or other +types of failure (e.g. wrinkling). The result (1.9) provides one counter-example to this common +wisdom – the TK instability may occur first before uniform thickness thinning takes place. In this +paper, we explore another instability mechanism, namely localized axisymmetric necking whereby +thickness thinning is localized near the origin and decays exponentially in the radial direction. Our +preliminary investigations in Wang et al (2022) indicate that the condition for axisymmetric neck- +ing is not given by H = 0 or the limiting point stability criterion although the necking condition in +the case of plane-strain does correspond to the nominal stress reaching a limiting point (Fu et al., +2018a). We observe that in the problem of localized bulging of an inflated hyperelastic tube, the +bifurcation condition corresponds to the inflation pressure reaching a limiting point when the axial +force is fixed or the axial force reaching a maximum when the pressure is fixed (Fu & Il’ichev, +2015). +The rest of this paper is divided into four sections as follows. In the next section we summarise +the governing equations of electroelasticity and derive the incremental governing equations to the +order that is required for the current analysis. Sections 3 and 4 present the linear and weakly +nonlinear analyses, respectively. The paper is concluded in Section 5 with a summary and some +additional comments. +2. Governing equations +2.1. Equations of nonlinear electroelasticity +Consider a dielectric material that is free from volumetric free charges and mechanical body +forces within the material and whose constitutive behavior is governed by the free energy density +function Ω∗(F, D) or Ω(F, E) (=Ω∗(F, D) − D · E), where F is the deformation gradient, D and E +are the nominal electric displacement and electric field vectors, respectively. The nominal electric +field, electric displacement, and the total nominal stress tensor S satisfy the field equations +CurlE = 0, +DivD = 0, +DivS = 0, +(2.1) +where Curl and Div are the curl and divergence operators with respect to X, the position vector +in the undeformed configuration. The constitutive equations are either +S = ∂Ω∗ +∂F − pF −1, +E = ∂Ω∗ +∂D , +(2.2) +or +S = ∂Ω +∂F − pF −1, +D = −∂Ω +∂E, +(2.3) +where we have assumed that the material is incompressible with p denoting the Lagrangian mul- +tiplier enforcing the constraint of incompressibility det F = 1. See Dorfmann & Ogden (2005) or +Zhao & Suo (2007) for further details. +It follows from (2.1)1 that the electric field E can be specified in terms of an electrostatic +potential Φ: +E = −GradΦ. +(2.4) +4 + +We consider the case when the potential Φ is specified on the two surfaces of the membrane through +the coating electrodes. As a result, the jump conditions at the interfaces between the membrane +and surrounding medium need not be considered. +Following common practice, see, e.g., Dorfmann & Ogden (2014b), we consider an energy func- +tion Ω(F, E) that is additively decomposed as a purely mechanical contribution and a part asso- +ciated with the electric field. We further specialize to the case when the electric contribution is +described by an isotropic constitutive formulation with constant permittivity ϵ (the so-called ideal +dielectric). Thus, we have +Ω(F, E) = W(I1, I2) − 1 +2ϵ E · C−1E, +(2.5) +where I1 and I2 are the two principal invariants of FF T . Correspondingly, in terms of the principal +stretches the functions Ω and Ω∗ in (1.1) and (1.2) take the specific forms +Ω(λ1, λ2, E3) = W(λ1, λ2) − 1 +2ϵE2 +3(λ1λ2)2, +(2.6) +Ω∗(λ1, λ2, D3) = W(λ1, λ2) + 1 +2ϵD2 +3(λ1λ2)−2, +(2.7) +where we have used the same symbols Ω and W in (2.5) and (2.6) (although the arguments are +different) to avoid introducing extra notations. In the above equations, the incompressibility con- +dition has been used to eliminate the principal stretch λ3, and W(λ1, λ2) is sometimes referred to +as the reduced strain energy function. +For the above class of free energy functions, the left-hand side of (1.11)1 is always positive and +we have +∂S +∂λ +���� +E3 fixed += ∂S +∂λ +���� +D3 fixed +− 8λ2E2 +3ϵ. +(2.8) +This means that (1.10)2 is always satisfied before (1.10)1 is satisfied. As a result, the conditions +(1.11)1 and (1.10)1 can be neglected, and (1.9), (1.10)2 can be solved explicitly (the condition +(1.11)2 is not independent as remarked earlier). Thus, we have the following two solutions for the +bifurcation values of ϵE2 +3: +ϵE2 +3 +�� +TK = λ−2(W12 − W22), +(2.9) +ϵE2 +3 +��lim = 1 +3λ−2(W12 + W22), +(2.10) +where the subscripts “TK” and “lim” signify associations with the TK and limiting-point instabil- +ities, respectively, and +W12 = ∂2W +∂λ1λ2 +���� +λ1=λ2=λ +, +W22 = ∂2W +∂λ2 +2 +���� +λ1=λ2=λ +. +(2.11) +2.2. Incremental formulation +In this section we derive the equations governing incremental deformations up to and including +quadratic terms. For the linear version, see Dorfmann & Ogden (2010). +We denote the undeformed, uniformly stretched, and bifurcated configurations of the mem- +brane by B0, Be and Bt, and the position vectors of a representative material particle in the three +configurations by X, x and ˜x, respectively. We use F, E, D and S to denote the deformation gra- +dient, the nominal electric field, nominal electric displacement and total nominal stress associated +5 + +with the deformation B0 → Bt. Their counterparts associated with the deformation B0 → Be are +denoted by ¯F, ¯E, ¯D and ¯S. We define the incremental fields η, e, d, and χ through +F = (I + η) ¯F, +E = ¯E + ¯F T e, +(2.12) +D = ¯D + ¯J ¯F −1d, +S = ¯S + ¯J ¯F −1χT . +(2.13) +The determinant ¯J (= det ¯F) is unity but is kept in the above expressions to maintain the generality +of the formulae. With u(x) denoting the incremental displacement from Be to Bt, we have η = +grad u. From the governing equations (2.1) that apply to both the barred and unbarred fields, we +obtain the incremental governing equations +curl e = 0, +div d = 0, +div χT = 0, +(2.14) +where div and curl are evaluated with respect to the position vector x. +We now proceed to derive the incremental forms of the constitutive equations (2.3). We first +expand ∂Ω/∂FiA around F = ¯F, E = ¯E to obtain +� +¯J−1 ¯F ∂Ω +∂F +� +li += ¯J−1 ¯FlA +∂Ω +∂FiA += ¯J−1 ¯FlA +∂Ω +∂FiA +���� ¯F ++ A(1) +lijkηkj + A(1) +li|kek ++ 1 +2A(2) +lijknmηkjηmn + A(2) +lijk|nηkjen + 1 +2A(3) +li|jkejek, +(2.15) +where +A(1) +lijk = ¯J−1 ¯FlA ¯FjB +∂2Ω +∂FiA∂FkB +���� ¯F +, +A(2) +lijknm = ¯J−1 ¯FlA ¯FjB ¯FnC +∂2Ω +∂FiA∂FkB∂FmC +���� ¯F +, +(2.16) +A(1) +li|k = ¯J−1 ¯FlA ¯FkB +∂2Ω +∂FiA∂EB +���� ¯F +, +A(2) +lijk|n = ¯J−1 ¯FlA ¯FjB ¯FnC +∂3Ω +∂FiA∂FkB∂EC +���� ¯F +, +(2.17) +A(3) +li|jk = ¯J−1 ¯FlA ¯FjB ¯FkC +∂3Ω +∂FiA∂EB∂EC +���� ¯F +. +(2.18) +We also have +p ¯FF −1 = (¯p + p∗)(I + η)−1 = ¯p(I − η + η2) + p∗(I − η) + · · · +(2.19) +Thus, it follows from (2.3)1 and (2.12)2 that +(χT )li = A(1) +lijkηkj + A(1) +li|kek + 1 +2A(2) +lijknmηkjηmn + A(2) +lijk|nηkjen + 1 +2A(3) +li|jkejek ++ ¯p(ηli − ηlkηki) − p∗(δli − ηli) + · · · . +(2.20) +For the electric displacement, we can similarly obtain +¯J−1 ¯FlM +∂Ω +∂EM += ¯J−1 ¯FlM +∂Ω +∂EM +���� ¯F ++ ¯J−1 ¯FlM ¯FmA +∂2Ω +∂FiA∂EM +���� ¯F +ηim + ¯J−1 ¯FlM ¯FjA +∂2Ω +∂EA∂EM +���� ¯F +ej ++1 +2 +¯J−1 ¯FlM ¯FmA ¯FnB +∂3Ω +∂FiA∂FkB∂EM +���� ¯F +ηimηkn + ¯J−1 ¯FlM ¯FmA ¯FnC +∂3Ω +∂FiA∂EC∂EM +���� ¯F +ηimen ++ 1 +2 +¯J−1 ¯FlM ¯FiA ¯FnC +∂3Ω +∂EA∂EC∂EM +���� ¯F +eien + · · · . +(2.21) +6 + +It then follows from (2.13)1 and (2.3)2 that +dl = ¯J−1 ¯FlM(− ∂Ω +∂EM ++ +∂Ω +∂EM +���� ¯F +) = −A(1) +mi|lηim − A(1) +jl ej +− 1 +2A(2) +mink|lηimηkn − A(3) +mi|nlηimen − 1 +2A(2) +ilneien + · · · , +(2.22) +where +A(1) +jl = ¯J−1 ¯FjA ¯FlB +∂2Ω +∂EA∂EB +���� ¯F +, +A(2) +iln = ¯J−1 ¯FiA ¯FlM ¯FnC +∂3Ω +∂EA∂EM∂EC +���� ¯F +. +(2.23) +Finally, it follows from the incompressibility conditions det ¯F = 1 and det F = 1 that +Iη + IIη + IIIη = 0, +(2.24) +where the three terms denote the three principal invariants of η, respectively. This is the incremental +incompressibility condition and its linear form is simply tr η = div u = 0. +The governing equation (2.14)1 can be satisfied automatically by writing e = grad ψ where the +scalar function ψ replaces e as one of the new independent variables. The remaining governing +equations (2.14)2,3 are to be solved subjected to the boundary conditions +χ33 = 0, +χ31 = 0, +ψ = 0 +on z = ±h/2. +(2.25) +We take h = 1 in the remaining analysis, which is equivalent to using h as the length unit. +3. Linear analysis +We now consider an axisymmetric perturbation represented by +u = u(r, z)er + v(r, z)ez, +ψ = ψ(r, z), +(3.1) +where r and θ are the cylindrical coordinates for x, er and ez are the unit basis vectors, and u and +v are the associated displacement components. The tensor η (= grad u) now takes the form +η = urer ⊗ er + uzer ⊗ ez + u +r eθ ⊗ eθ + vrez ⊗ er + vzez ⊗ ez, +(3.2) +where ur = ∂u/∂r, uz = ∂u/∂z, etc. +For the current axisymmetric problem, the two components of the equilibrium equation div χT = +0 that are not satisfied automatically are +χ1j,j + 1 +r(χ11 − χ22) = 0, +χ3j,j + 1 +rχ31 = 0, +(3.3) +where (1, 2, 3) corresponds to (r, θ, z). The linearization of the incompressibility condition (2.24), +namely div u = 0, may be written in the form +∂ (ru) +∂r ++ ∂ (rv) +∂z += 0, +(3.4) +which can be satisfied automatically by introducing a ‘stream function’ φ(r, z) such that +u = 1 +rφz, +v = −1 +rφr, +(3.5) +7 + +where as in (3.2) a subscript signifies differentiation (e.g. +φz = ∂φ/∂z). +The non-zero stress +components are given by +χ11 += +A(1) +1122 +u +r + A(1) +1133vz + (A(1) +1111 + ¯p)ur − p∗, +(3.6) +χ22 += +(A(1) +2222 + ¯p)u +r + A(1) +2233vz + A(1) +1122ur − p∗, +(3.7) +χ33 += +A(1) +2233 +u +r + (A(1) +3333 + ¯p)vz + A(1) +1133ur − p∗ − 2E3ϵλ2ψz, +(3.8) +χ13 += +A(1) +3131uz + (A(1) +3113 + ¯p)vr − E3ϵλ2ψr, +(3.9) +χ31 += +A(1) +1313vr + (A(1) +1331 + ¯p)uz − E3ϵλ2ψr, +(3.10) +whereas the linearisation of (2.22) is given by +d1 = −E3ϵλ2(uz + vr) − E3ψr, +d2 = 0, +d3 = −2E3ϵλ2vz − E3ψz. +(3.11) +On substituting these expressions together with (3.5) into (3.3) and then eliminating p∗ by cross- +differentiation, we obtain +α +� +φrrrr − 2 +rφrrr + 3 +r2 φrr − 3 +r3 φr +� ++ 2β +� +φrrzz − 1 +rφrzz +� ++ γφzzzz ++ E3ϵλ2 +� +rψrrr + ψrr − 1 +rψr + rψrzz +� += 0, +(3.12) +where +α = A(1) +2323, +2β = A(1) +2222 + A(1) +3333 − 2A(1) +2233 − 2A(1) +2332, +γ = A(1) +3232. +(3.13) +A second equation for φ and ψ is obtained by substituting (3.11) into (2.14)2: +ψzz + 1 +rψr + ψrr − E3λ2 1 +r3 +� +r2φrrr − rφrr + r2φrzz + φr +� += 0. +(3.14) +Equation (3.12)and (3.14) admit a “normal mode” buckling/wrinkling solution of the form +φ(r, z) = rJ1(kr)S(kz), +ψ(r, z) = J0(kr)K(kz), +(3.15) +where k is a constant playing the role of wavenumber, J0(x) and J1(x) are Bessel’s functions of the +first kind, and the other functions S(kz) and K(kz) are to be determined. +On substituting (3.15) into (3.12) and (3.14) and simplifying by making use of the identity +Jν(x) = 2(ν − 1) +x +Jν−1(x) − Jν−2(x), +the J1(kr) and J0(kr) can be cancelled in the resulting equations and we obtain two ordinary +differential equations: +γS(4)(kz) − 2βS′′(kz) + αS(kz) + k−1E3ϵλ2(K(kz) − K′′(kz)) = 0, +(3.16) +and +� +K′′(kz) − E3kλ2S′′(kz) +� +− +� +K(kz) − E3kλ2S(kz) +� += 0. +(3.17) +The last equation can be integrated straightaway to yield +K(kz) = E3kλ2S(kz) + c5 sinh(kz) + c6 cosh(kz), +(3.18) +8 + +where c5 and c6 are constants. Equation (3.16) then reduces to +γS(4)(kz) − 2β∗S′′(kz) + α∗S(kz) = 0, +(3.19) +where +α∗ = α + E2 +3ϵλ4, +β∗ = β + 1 +2E2 +3ϵλ4. +(3.20) +The general solution of (3.19) may be written in the form +S(kz) = c1 sinh +k +√ζ1 +z + c2 sinh +k +√ζ2 +z + c3 cosh +k +√ζ1 +z + c4 cosh +k +√ζ2 +z, +(3.21) +where c1, c2, c3, c4 are disposable constants, and +ζ1 = 1 +α∗ (β∗ − +� +β∗2 − α∗γ), +ζ2 = 1 +α∗ (β∗ + +� +β∗2 − α∗γ). +(3.22) +The boundary conditions (2.25) take the form +A(1) +3131uz + (A(1) +3113 + ¯p)vr − E3ϵλ2ψr = 0, +on z = ±1/2, +(3.23) +A(1) +2233 +u +r + (A(1) +3333 + ¯p)vz + A(1) +1133ur − p∗ − 2E3ϵλ2ψz = 0, +on z = ±1/2, +(3.24) +ψ = 0, +on z = ±1/2. +(3.25) +The p∗ in (3.24) can be eliminated by first differentiating (3.24) with respect to r and then using +(3.3)1 to eliminate p∗ +r. This gives +1 +r2 (A(1) +2233 − A(1) +2222 − ¯p) +� +r2urr + rur − u +� +− A(1) +3232uzz ++ (A(1) +3333 − A(1) +2332 − A(1) +2233)vrz − E3ϵλ2ψrz = 0, +on z = ±1/2. +(3.26) +On substituting (3.15), (3.18) and (3.21) into the six boundary conditions (3.23), (3.25) and (3.26), +we obtain six algebraic equations. Due to the symmetry of the membrane geometry and external +loads with respect to the mid-plane z = 0, this system of equations admits two types of solutions +corresponding to flexural and extensional modes, respectively. The bifurcation condition for the +extensional modes is what we shall focus on and is given by +d1 tanh +�k +2 +� +tanh +� +k +2√ζ1 +� +− d2 tanh +�k +2 +� +tanh +� +k +2√ζ2 +� ++ d3 tanh +� +k +2√ζ1 +� +tanh +� +k +2√ζ2 +� += 0, +(3.27) +where +d1 += +� +ζ1(1 + ζ1)(ζ2(2β∗ + γ) − γ), +d2 += +� +ζ2(1 + ζ2)(ζ1(2β∗ + γ) − γ), +(3.28) +d3 += +ϵE2 +3λ4� +ζ1ζ2(ζ1 − ζ2). +Expanding (3.27) to order k2, we obtain +γ(β + γ) + k2 +24γ(α − γ) + O(k4) = 0, +(3.29) +9 + +where we have used (3.22) to eliminate ζ1 and ζ2. Note that the coefficient of k2 in the above +asymptotic expression is not unique: we can add an arbitrary multiple of γ(β + γ) to it without +changing the asymptotic order of the second term since the latter expression is of order k2. +As an illustrative example, consider the following two-term Ogden strain-energy function: +W = 2µ1 +m2 +1 +(λm1 +1 ++ λm1 +2 ++ λm1 +3 +− 3) + 2µ2 +m22 (λm2 +1 ++ λm2 +2 ++ λm2 +3 +− 3), +(3.30) +with m1 = 1/2, m2 = 4, µ2 = µ1/80. Fig. 1 displays the bifurcation condition (3.27) and its +two-term approximation (3.29) in the small wavenumber limit. It is seen that the the minimum of +λ is attained at k = 0 in the case of fixed E3 and the minimum of E3 is also attained at k = 0 in +the case of fixed λ. Based on the discussion in Fu (2001), we may postulate that the bifurcation +exact +two-term +1 +2 +3 +4 +k +2.0 +2.1 +2.2 +2.3 +λ +(ϵ/μ1)E3 +2=0.03 +exact +two-term +1 +2 +3 +4 +k +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +(ϵ/μ1)E32 +λ=2.1 +(a) +(b) +Figure 1: Bifurcation condition (3.27) for periodic and symmetric modes, and its two-term approximation (3.29) in +the small wavenumber limit. +condition for localized necking can be obtained by setting the leading order term in (3.29) to zero, +that is β + γ = 0 since γ > 0, or equivalently, +A(1) +2222 + A(1) +3333 + 2A(1) +3232 − 2A(1) +2332 − 2A(1) +2233 = 0. +(3.31) +It can be shown that this condition is equivalent to +∂S1 +∂λ1 +���� +λ1=λ2=λ += 0, +(3.32) +where S1 has the same meaning as in Section 1. Corresponding to the free energy function (2.6), +this equation can be solved explicitly to give +(ϵE2 +3)necking = λ−2W11, +(3.33) +where +W11 = ∂2W +∂λ2 +1 +���� +λ1=λ2=λ +. +(3.34) +The bifurcation condition may be compared with the conditions (2.9) and (2.10) for the TK and +limiting point instabilities. +10 + +LP +necking +TK +2.0 +2.5 +3.0 +3.5 +λ +0.00 +0.01 +0.02 +0.03 +(ϵ/μ1)E32 +LP +necking +TK +1.6 +1.7 +1.8 +1.9 +2.0 +S +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(ϵ/μ1)E32 +(a) +(b) +Figure 2: Bifurcation conditions for the TK, limiting point and necking instabilities corresponding to the strain +energy function (3.30). The alternative representations in (b) are obtained by viewing E3 and S as functions of λ +and varying λ in the interval (1, 3.7). The three lines in (a) intersect at λ = 1.98 and 3.23, and the curve associated +with necking cuts the horizontal axis at λ = 2.44 and 2.92. In (b) the dotted line corresponding to the limiting point +instability is close but always above that for the necking instability. +Corresponding to the strain energy function (3.30), the three bifurcation conditions are shown +in Fig.2(a,b) by viewing E3 as a function of λ or S, respectively. Fig.2 (b) is obtained by eliminating +E3 from S = S(λ, λ, E3) using the bifurcation conditions so that both S and E3 are parametric +functions of λ. +In the absence of an electric field (E3 = 0), there are two bifurcation values for the TK instability +and another two bifurcation values for necking, and limiting points do not exist. +This purely +mechanical case has previously been discussed in Wang et al (2022). In particular, it was shown that +although the first bifurcation value for the TK instability is smaller than the first bifurcation value +for necking, necking can still occur first when the membrane is stretched under edge displacement +control since in this case the TK instability will be suppressed. +When an electric field is applied (E3 ̸= 0), we consider two typical loading scenarios. One is to +first stretch the membrane to a specified value of λ, say λ = 2, in the absence of an electric field, +and then increase the electric field from zero with the edge fixed. This loading scenario corresponds +to displacement control and so TK instability is suppressed. Referring to Fig.2 (a), this means that +the first instability experienced by the membrane is the necking instability although the loading +path crosses the TK instability curve. +The other loading scenario is to first increase the nominal stress S to a specified value, say 1.5, +in the absence of an electric field, and then increase the electric field from zero with S fixed as a +dead load. This is the loading scenario adopted by (Huang et al., 2012). Fig.2 (b) shows that again +the first instability experienced by the membrane is the necking instability. +4. Weakly nonlinear analysis +The linear analysis in the previous section only provides a necessary condition for necking. +Whether a necking solution really bifurcates from the homogeneous solution or not can only be +answered by a near-critical nonlinear analysis. +To fix ideas, we may assume that the strain energy function is given by (3.30) and the case to +be studied is when λ is fixed in the interval (1.98, 2.44) and E3 is gradually increased from zero. +11 + +As pointed out in the previous section, in this parameter regime necking would occur before the +limiting point instability or the TK instability. +We define a non-dimensional load parameter ω through +ω = ϵ +µ1 +E2 +3. +(4.1) +Denoting its bifurcation value by ωcr (which depends on λ), we write +ω = ωcr + εω1, +(4.2) +where ω1 is an O(1) constant and ε is a positive small parameter characterizing the derivation of +ω from ωcr. From the bifurcation condition (3.29) it can be deduced that in this parameter regime +the buckling mode will have k = O(√ε), which means that the dependence of the near-critical +solution on r should be through the stretched variable s defined by +s = √εr. +(4.3) +The relative orders of u, v, p∗ and ψ can be deduced by expanding the linear solutions (3.15) for +small k. The absolute size of v is determined by the fact that the amplitude is expected to be a +linear function of ω − ωcr for the type of bifurcations under consideration. This gives v = O(ε). +Based on this analysis, we look for a near-critical solution of the form +u = √ε +� +u(1)(s, z) + εu(2)(s, z) + ε2u(3)(s, z) + · · · +� +, +v = ε +� +v(1)(s, z) + εv(2)(s, z) + ε2v(3)(s, z) + · · · +� +, +(4.4) +p∗ = ε +� +p(1)(s, z) + εp(2)(s, z) + ε2p(3)(s, z) + · · · +� +, +ψ = ε2 � +ψ(1)(s, z) + εψ(2)(s, z) + ε2ψ(3)(s, z) + · · · +� +, +where all the functions on the right hand sides are to be determined from successive approximations. +To ease descriptions, we scale all the governing equations and boundary conditions so that +the left hand side of each equation becomes of O(1). This is achieved by dividing by ε the elec- +tric equilibrium equation (2.14)2, the mechanical equilibrium equation (3.3)2, the incompressibility +condition (2.24) and the boundary condition (2.25)1, and by √ε the mechanical equilibrium equa- +tion (3.3)1 and boundary condition (2.25)2. On substituting (4.4) into these scaled equations and +then equating the coefficients of like powers of ε, we obtain a hierarchy of boundary value prob- +lems. In the following description, the two equilibrium equations in (3.3) are referred to as the +r- and z-equilibrium equations, respectively. At the n-th order (n = 1, 2 or 3), we integrate the +r-equilibrium equation subject to the boundary condition (2.25)2 to find u(n)(s, z), the incompress- +ibility condition to find v(n)(s, z), and finally the z-equilibrium equation subject to (2.25)1 to find +p(n)(s, z). +At leading order, the above procedure yields +u(1)(s, z) = A(s), +v(1)(s, z) = −z 1 +s(sA(s))′ + B(s), +(4.5) +p(1)(s, z) = −(A(1) +3333 − A(1) +2233 + A(1) +3232 − A(1) +3223)1 +s(sA(s))′, +(4.6) +where A(s) and B(s) are functions to be determined, and here and hereafter in this section all the +moduli are evaluated at ω = ωcr. The electric equilibrium equation (2.14)2 is satisfied automatically. +12 + +At second order, the general solution for u(2)(s, z) contains two new functions C(s) and D(s) +in the form C(s) + zD(s). Subtracting and adding the boundary condition (2.25)2 at z = ±1/2, +respectively, we obtain +A(1) +3333 − 2A(1) +2233 + A(1) +2222 + 2A(1) +3232 − 2A(1) +3223 = 0, +(4.7) +and +D(s) = −B′(s). +(4.8) +The first result (4.7) is equivalent to the bifurcation condition (3.31). The general solutions for +v(2)(s, z) and p2(s, z) contain new functions F(s) and E(s), respectively. On applying the boundary +condition (2.25)1 at z = ±1/2, we obtain sB′′(s) + B′(s) = 0, and an expression for E(s). It then +follows that B(s) = d1 ln s+d2. Since v(1) and hence B(s) should be bounded at s = 0, we must set +d1 = 0. Without loss of generality we may also impose the condition v(1)(0, 0) = 0 to eliminate any +rigid-body displacement. This yields d2 = 0 and hence B(s) = 0. Finally, integrating the electric +equilibrium equation (2.14)2 at this order subject to (2.25)3 at z = ±1/2 yields a unique expression +for ψ1(s, z). +At third order, nonlinear terms come into play and it is at this order that an amplitude equation +for A(s) is derived. We first solve the r-equilibrium equation to find an expression for u(3)(s, z). +It contains two new functions G(s) and H(s) in the form G(s) + zH(s). Subtracting and adding +(2.25)2 evaluated at z = ±1/2, respectively, we obtain the amplitude equation for A(s) and an +expression for H(s). After some simplification, it is found that the amplitude equation takes the +form +c0 +d +ds +1 +s +d +dssP ′(s) + c1ω1P ′(s) + c2 +d +dsP 2(s) + c3A′′(s) +� +A′(s) − 1 +sA(s) +� += 0, +(4.9) +where a prime signifies differentiation, P(s) is defined by +P(s) = 1 +s(sA(s))′, +(4.10) +and the three coefficients are given by +c0 += +1 +12 +� +A(1) +2323 − A(1) +3232 +� +, +c1 += +2A(1)′ +2233 + 2A(1)′ +2332 − A(1)′ +2222 − 2A(1)′ +3232 − A(1)′ +3333, +c2 += +1 +4 +� +−4A(1) +2222 − 2A(1) +2233 + 6A(1) +3333 − A(2) +222222 + 4A(2) +222233 − A(2) +112222 +−6A(2) +223333 + 2A(2) +112233 + 2A(2) +333333 +� +, +c3 += +A(1) +2233 − A(1) +2222 + A(2) +222233 − A(2) +112233 − 1 +2A(2) +222222 + 1 +2A(2) +112222. +In the above expressions, A(1)′ +2233 denotes dA(1) +2233/dω etc., and we have used the bifurcation condition +(4.7) to eliminate A(1) +2332. It can be seen that the amplitude equation (4.9) has the same structure +as its mechanical counterpart derived by Wang et al. (2022). +Corresponding to the specific free energy function (2.5) and (3.30), we have +c0 = −−480λ17/2 + λ12 + 800λ7 + 3 +960λ8 +, +c1 = λ4, +c2 = −56λ17/2 + 240λ7 + 3 +32λ8 +, +c3 = +√ +λ +2 − 3λ4 +80 . +(4.11) +13 + +As a consistency check, we may neglect the nonlinear terms in (4.9) to obtain +c0 +d +ds +1 +s +d +dssP ′(s) + c1ω1P ′(s) = 0. +(4.12) +On substituting a solution of the form P ′(s) = J1(ks/√ε) into (4.12), where k is a constant, we +obtain +c1(ω − ωcr) − c0k2 = 0. +(4.13) +On the other hand, expanding (3.29) around ω = ωcr, we obtain +� d +dω(β + γ) +� +cr +(ω − ωcr) + k2 +24(α − γ) +���� +cr += 0, +(4.14) +where the subscripts “cr” signify evaluation at ω = ωcr. We have verified that (4.13) is indeed +consistent with (4.14). +As another consistency check, we may expand (4.9) out fully and omit all the terms that are +divided by powers of s to obtain its planar counterpart: +c0A(4)(s) + c1ω1A′′(s) + c∗ +2A′(s)A′′(s) = 0, +(4.15) +where +c∗ +2 = 2c2 + c3 = 3A(1) +3333 − 3A(1) +2222 − A(2) +222222 + 3A(2) +222233 − 3A(2) +223333 + A(2) +333333. +(4.16) +It has an exact solution given by +A(s) = 6c0 +c∗ +2 +�−c1ω1 +c0 +tanh +�1 +2 +�−c1ω1 +c0 +s +� +. +(4.17) +This solution has the property A′(s) → 0 as s → ∞ and is the localised necking solution in the 2D +case (Fu et al., 2018a). +It does not seem possible to find a similar analytical solution for the original amplitude equa- +tion (4.9) that is fourth-order with variable coefficients. +We thus resort to finding its numer- +ical solution with the use of the finite difference method. +With the use of the substitution +A(s) → (c0/c2)κ2A(κs), equation (4.9) may be reduced to +d +dt +1 +t +d +dttP ′(t) − P ′(t) + d +dtP 2(t) + c3 +c2 +A′′(t) +� +A′(t) − 1 +t A(t) +� += 0, +(4.18) +where t = κs, κ = +� +−c1ω1/c0 and P(t) is still defined by (4.10). +We replace the semi-infinite interval [0, ∞) by a finite interval [0, L] and discretize the latter +into N equal intervals with node points +ti = i˜h, +˜h = L +N , +i = 0, 1, 2, ..., N. +We apply the central finite difference scheme such that +A′(ti) = Ai+1 − Ai−1 +2˜h +, +A′′(ti) = Ai+1 − 2Ai + Ai−1 +˜h2 +, +(4.19) +A′′′(ti) = Ai+2 − 2Ai+1 + 2Ai−1 − Ai−2 +2˜h3 +, +(4.20) +14 + +A(4)(ti) = Ai+2 − 4Ai+1 + 6Ai − 4Ai−1 + Ai−2 +˜h4 +, +(4.21) +where Ai = A(ti), etc. +Evaluating the amplitude equation (4.9) at the N − 1 interior nodes +t1, t2, ..., tN−1, we obtain N − 1 equations that involve the N + 3 unknowns A−1, A0, ..., and AN+1. +The remaining four equations are obtained as follows. +First, it follows from the symmetry conditions +lim +s→0 u(1)(s, z) = 0 +and lim +s→0 +∂v(1) +∂s (s, 1 +2) = 0 +that limt→0 A(t) = 0 and limt→0 P ′(t) = 0. By trying a series solution for small t, it is found that +the unique solution that satisfies the above conditions has the behavior A(t) ∼ a1t + a2t3 + · · · +for some constants a1 and a2. This gives limt→0 A′′(t) = 0. The two conditions A(0) = A′′(0) = 0 +together with (4.19)2 then yield two additional equations. +Next, we consider the asymptotic behaviour of the solutions as t → ∞. Although the planar +solution (4.17) does not decay, we expect that the solution of (4.9) will experience algebraic decay +due to geometric spreading. Since quadratic terms are expected to decay faster than linear terms, +the decay behavior may be captured by neglecting the nonlinear terms: +d +dt +1 +t +d +dttP ′(t) − P ′(t) = 0, +as t → ∞. +(4.22) +The unique decaying solution of (4.22) is given by +P(t) = P∞(t) ≡ a3K0(t), +A(t) = A∞(t) ≡ a4 +t − a3K1(t), +(4.23) +where a3 and s4 are constants, and K0 and K1 are the modified Bessel function of the second kind +that has the asymptotic behaviour +Kα(x) ∼ +� π +2xe−x +� +1 + 4α2 − 1 +8x ++ · · · +� +, +as x → ∞. +(4.24) +The asymptotic behaviour (4.23) is consistent with our earlier assumption that A(s) decays alge- +braically. We note that the above decay behaviour is based on the assumption that t is a real +variable, or equivalently κ is a real constant. This enables us to deduce that whenever a necking +bifurcation takes place, it is generally subcritical (ω1 < 0 since c1/c0 > 0). +If a function f(x) decays exponentially like e−ax as x → ∞ for some positive constant a, then +it is preferable to impose the “soft” asymptotic condition f′(L) + af(L) = 0 instead of the “hard” +condition f(L) = 0 (since f′(L) + af(L) is much smaller than f(L)). Extending this idea, we +use (4.23) to find the first three derivatives of A∞(s) and by eliminating a1 and a2 express A′′ +∞(s) +and A′′′ +∞(s) in terms of A∞(s) and A′ +∞(s). Replacing A∞(s) by A(s) and evaluating these two +expressions at s = sN = L followed by the use of (4.19)–(4.21), we obtain two more additional +equations. The system of N + 3 quadratic equations can then be solved provided an appropriate +initial guess is given. It is found that one good initial guess is the planar solution (4.17) divided +by 1 + s. For values of λ in the interval (1.979, 2.439), it is found that the coefficient c3/c2 is +positive when λ < 2.096 and negative when λ > 2.096. So we consider two representative cases +corresponding to λ = 2 and 2.2, respectively. +It is found that taking N = 1000 and L = 10 +yields sufficiently accurate results. Fig.3 shows the finite difference solution corresponding to λ = 2 +together with the approximate analytical solution +P(t) = +a +bt2 + 1sech2(ct), +(4.25) +15 + +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +0 +2 +4 +6 +t +0.5 +1.0 +1.5 +2.0 +P(t) +A(t) +Figure 3: FD solution of the amplitude equation (4.9) when λ = 2 and the strain energy function is given by (3.30). +where the constants a, b, c are determined by fitting (4.25) to the finite difference solution. The +maximum relative error over the entire interval is less than 3.4%. The solution corresponding to +λ = 2.2 for which the c3/c2 is of opposite sign is very similar and is thus not displayed here. +5. Discussion and conclusion +Pull-in failure in dielectric elastomer actuators is widely believed to be associated with the +limiting-point behaviour whereby the electric field as a function of the electric displacement or +stretch has a maximum. For the plane-strain or plane-stress case, this connection is well explained +using the analogy with the inflation problem associated with a rubber tube where the limiting-point +behaviour is well-known to be associated with localised bulging that eventually evolves into a “two- +phase” state (Fu et al., 2018a; Huang & Suo, 2012). However, for the case of equibiaxial tension, +this explanation contradicts the fact that at large values of dead load, the limiting-point behaviour +may disappear but pull-in failure can still be observed (Huang et al., 2012). Our current paper +offers an alternative explanation, namely that pull-in failure evolves from axisymmetric necking +through an unstable process. We note that the condition for necking does not necessarily require +limiting-point behaviour. +We have only carried out a linear and weakly nonlinear analysis in the current study, but the +fully nonlinear numerical simulations carried out in our earlier paper (Wang et al., 2022) for the +purely mechanical case should also be indicative of what might be expected in the current elec- +troelastic case. Thus, combining the weakly nonlinear results in the previous section with the fully +nonlinear simulation results in Wang et al. (2022), we may draw the following conclusions. When +the bifurcation condition (3.33) is satisfied, a necking solution will bifurcate from the homogeneous +solution subcritically. If the membrane is gradually pulled further in the radial direction at the +edge, with the electric potential fixed, the necking solution will grow in amplitude, corresponding to +an increased reduction in thickness at the origin, and when a maximum amplitude is approached, +the necking solution will start to propagate in the radial direction in the form of a “two-phase” +deformation. This is very similar to the localised bulging of an inflated rubber tube except that +here the propagation is also accompanied by algebraic decaying of the amplitude due to geometrical +spreading. On the other hand, if the electric potential is increased further from its bifurcation value +16 + +while the membrane edge is fixed, the membrane will snap to a “two-phase” deformation. This is +analogous to the pressure control case in the tube inflation problem. +We wish to highlight the fact that the predictions that can be made are sensitive to the material +model used. To fix ideas, we have used the strain energy function (3.30) as an example. To show +how our results depend on the strain energy function used, we have shown in Fig. 4 the counterpart +of Fig. 2 when the following Gent and Mooney-Rivlin material models are used: +W = −1 +2µJm ln(1 − λ2 +1 + λ2 +2 + λ2 +3 − 3 +Jm +), +(5.1) +W = 1 +2µ +� +λ2 +1 + λ2 +2 + λ2 +3 − 3 + γ(λ−2 +1 ++ λ−2 +2 ++ λ−2 +3 +− 3) +� +. +(5.2) +It is found that the bifurcation curves have a very weak dependence on the value of Jm and the +LP +necking +TK +1.5 +2.0 +2.5 +λ +-0.5 +0.5 +1.0 +1.5 +(ϵ/μ1)E32 +LP +necking +TK +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +λ +-0.2 +0.2 +0.4 +0.6 +0.8 +(ϵ/μ1)E32 +(a) +(b) +Figure 4: Bifurcation conditions for the TK, limiting point and necking instabilities corresponding to (a) the Gent +strain energy function with Jm = 97.2, and (b) the Mooney-Rivlin strain energy function with γ = 0.3. The dashed +line corresponds to zero nominal stress in the radial direction above which the nominal stress is negative. +curves corresponding to Jm = ∞ (the neo-Hookean model) are almost the same as those in Fig. 4(a) +for Jm = 97.2. It is seen that the main effect of increasing the γ in (5.2) is to shift the curves for +the TK and limiting instabilities upwards. As a result, the TK instability is not possible for the +Gent and neo-Hookean material models (since the corresponding E3 is negative) but is possible +for the Mooney-Rivlin material model. This is well-known in the purely mechanical case. The +bifurcation curve for necking is always above the curve corresponding to zero nominal stress in the +radial direction (dashed line). Thus, although necking is theoretically possible, it is unlikely to +be observable when the dielectric membrane has the constitutive behaviour modelled by these two +material models. 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Non-local elastic effects in electroactive polymers. Int. J. Non-Linear Mech., 56, +115–122. +Zurlo, G., Destrade, M., DeTommasi, D., & Puglisi, G. (2017). Catastrophic thinning of dielectric +elastomers. Phys. Rev. Lett., 118, 078001. +21 + diff --git a/fdAzT4oBgHgl3EQfMPug/content/tmp_files/load_file.txt b/fdAzT4oBgHgl3EQfMPug/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4dd0dfb5a1e32224833712a81c46bcba256e7e1 --- /dev/null +++ b/fdAzT4oBgHgl3EQfMPug/content/tmp_files/load_file.txt @@ -0,0 +1,1255 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf,len=1254 +page_content='Axisymmetric necking of a circular electrodes-coated dielectric membrane Yibin Fu∗,a, Xiang Yub aSchool of Computer Science and Mathematics, Keele University, Staffs ST5 5BG, UK bSchool of Computer Science and Technology, Dongguan University of Technology, Dongguan, China Abstract We investigate the stability of a circular electrodes-coated dielectric membrane under the com- bined action of an electric field and all-round in-plane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is known that such a membrane is susceptible to the limiting point instability (also known as pull-in instability) which is widely believed to be a precursor to electric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' However, there is experimental evidence showing that the limiting point instability may not necessarily be responsible for rapid thinning and elec- tric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We explore the possibility that the latter is due to a new instability mechanism, namely localised axisymmetric necking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The bifurcation condition for axisymmetric necking is first derived and used to show that this instability may occur before the Treloar-Kearsley instability or the limiting point instability for a class of free energy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' A weakly nonlinear analysis is then conducted and it is shown that the near-critical behavior is described by a fourth order nonlinear ODE with variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This amplitude equation is solved using the finite dif- ference method and it is demonstrated that a localised solution does indeed bifurcate from the homogeneous solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Based on this analysis and what is already known for the purely mechanical case, we may deduce that the necking evolution follows the same three stages of initiation, growth and propagation as other similar localisation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The insight provided by the current study is expected to be relevant in assessing the integrity of dielectric elastomer actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Key words: Nonlinear electroelasticity, dielectric membranes, localisation, stability, bifurcation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Introduction Dielectric elastomer actuators are believed to hold great potential in a wide range of applications such as human-like robots, stretchable electronics, and energy harvesting (Pelrine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 1998, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Carpi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Carpi & Smela, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Carpi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is known that such actuators are susceptible to a variety of instabilities (Plante & Dubowsky, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhao & Wang, 2014), and before they can be deployed with confidence, a thorough understanding of their stability and buckling properties needs to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Thus, over the past two decades, much effort has been devoted to 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Broderick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Bahreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2022), “two-phase” states (Plante ∗Corresponding author Email address: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='fu@keele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='uk (Yibin Fu) January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='01129v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='soft] 3 Jan 2023 & Dubowsky, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Kollosche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Huang & Suo, 2012), and the interplay between the limiting point instability and Treloar-Kearsley (TK) instability (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We refer to Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2020) for a comprehensive review of the relevant literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The current study is concerned with a different kind of instability, namely necking, that has received relatively less attention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Necking has traditionally been associated with ductile materials and plastic deformations, but in recent years it has been realised that elastic necking can occur in a wide range of soft materials under multiple fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' see, for instance, Na et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2006), Mora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2010), Zhao (2012) and Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The possibility of localised necking in a dielectric elastomer has previously been suggested by Blok & LeGrand (1969) and analysed using an approximate model in a series of papers by Puglisi & Zurlo (2012), Zurlo (2013), De Tommasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2013) and Zurlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The approximate model used in the latter papers is further discussed in Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For the case of uniaxial tension, localised necking was analysed by Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2018a) using analogies with the inflation problem associated with a rubber tube (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It was shown that localised necking would initiate when the limiting point of nominal stress (as a function of stretch with fixed electric potential) or electric potential (as a function of electric displacement with fixed nominal stress) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As in the inflation problem, the localised necking would evolve into a “two-phase” deformation that has been observed experimentally by Plante & Dubowsky (2006), and analysed by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Whereas the connection between the limiting-point instability and localised necking is now well understood in the case of uniaxial tension, this connection no longer exists in the case of equibiaxial tension, as demonstrated recently by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2022) and Yu & Fu (2022) for the purely mechanical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For the case of equibiaxial tension, the limiting-point behaviour may disappear at a large enough dead load, but some kind of snap-through behavior can still be observed that leads to pull-in failure (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' A likely scenario is that even if limiting-point instability does not exist, localised necking can still occur, and it is the axisymmetric necking that leads to a “two-phase” deformation and possible pull-in failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This scenario provides the major motivation for the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This paper may also be viewed as a sequel to our earlier paper, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2022), where the axisymmetric necking was analysed in the purely mechanical context without an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In that paper, the amplitude equation was left unsolved and it was not clear whether the equation did have a well-defined localised solution or not although fully numerical simulations seemed to have answered the question in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In the current paper, we derive the corresponding results for the electroelastic case, and solve the amplitude equation to show that a localised solution does indeed bifurcate from the homogeneous solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' To set the context for our current study, consider a dielectric square membrane that is coated with electrodes and is subject to nominal tresses S1 and S2 in two mutually orthogonal directions within the membrane plane and a nominal electric field E3 in the thickness direction (the 3- direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The associated stretches and nominal electric displacement are denoted by λ1, λ2 and D3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In terms of the free energy function Ω(λ1, λ2, E3), these quantities are related by (Dorfmann & Ogden, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhao & Suo, 2007) S1 = ∂Ω ∂λ1 , S2 = ∂Ω ∂λ2 , D3 = − ∂Ω ∂E3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) Alternatively, defining Ω∗(λ1, λ2, D3) = Ω(λ1, λ2, E3) + E3D3, we have S1 = ∂Ω∗ ∂λ1 , S2 = ∂Ω∗ ∂λ2 , E3 = ∂Ω∗ ∂D3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) 2 The Hessian stability criterion states that the Hessian determinant defined by H = ���������� ∂2Ω∗ ∂λ2 1 ∂2Ω∗ ∂λ1∂λ2 ∂2Ω∗ ∂λ1∂D3 ∂2Ω∗ ∂λ2∂λ1 ∂2Ω∗ ∂λ2 2 ∂2Ω∗ ∂λ2∂D3 ∂2Ω∗ ∂D3∂λ1 ∂2Ω∗ ∂D3∂λ2 ∂2Ω∗ ∂D2 3 ���������� = �������� ∂S1 ∂λ1 ∂S1 ∂λ2 ∂S1 ∂D3 ∂S2 ∂λ1 ∂S2 ∂λ2 ∂S2 ∂D3 ∂E3 ∂λ1 ∂E3 ∂λ2 ∂E3 ∂D3 �������� (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3) should be positive definite for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Since H = 0 is equivalent to J(S1, S2, E3) = 0 where the left-hand side denotes the Jacobian determinant of S1, S2 and E3 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3), marginal violation of the Hessian stability criterion means that the “displacement” (λ1, λ2, D3) cannot uniquely be expressed in terms of the “force” (S1, S2, E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Evaluating the Jacobian determinant at equibiaxial stretching λ1 = λ2 ≡ λ where S1 = S2 ≡ S(λ, D3), E3 ≡ E(λ, D3), ∂S1/∂λ2 = ∂S2/∂λ1, ∂S1/∂λ1 = ∂S2/∂λ2, ∂E3/∂λ1 = ∂E3/∂λ2, etc, we find that J(S1, S2, E3) = �∂S1 ∂λ1 − ∂S1 ∂λ2 � � ∂E ∂D3 ∂S ∂λ − ∂S ∂D3 ∂E ∂λ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4) where all quantities are evaluated at λ1 = λ2 = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The above expression may also be rewritten in two more revealing forms: J(S1, S2, E3) = �∂S1 ∂λ1 − ∂S1 ∂λ2 ���� E3 fixed · ∂S(λ, D3) ∂λ ��� E3 fixed · ∂E3 ∂D3 ��� λ fixed, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) or J(S1, S2, E3) = �∂S1 ∂λ1 − ∂S1 ∂λ2 ���� D3 fixed · ∂S(λ, D3) ∂λ ��� D3 fixed · ∂E3 ∂D3 ��� S fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6) An application of L′Hopital’s rule gives the result �∂S1 ∂λ1 − ∂S1 ∂λ2 ����� D3 fixed = lim λ2→λ1 S2 − S1 λ2 − λ1 ���� D3 fixed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7) It can also be shown that at equibiaxial stretching, �∂S1 ∂λ1 − ∂S1 ∂λ2 ����� D3 fixed = �∂S1 ∂λ1 − ∂S1 ∂λ2 ����� E3 fixed .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8) Thus, H = J(S1, S2, E3) = 0 is satisfied if any one of the following conditions is satisfied: lim λ2→λ1 S2 − S1 λ2 − λ1 ���� E3 fixed = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) ∂S ∂λ ���� D3 fixed = 0, ∂S ∂λ ���� E3 fixed = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10) ∂E3 ∂D3 ���� λ fixed = 0, ∂E3 ∂D3 ���� S fixed = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11) The condition in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) obviously corresponds to the Treloar–Kearsley instability whereby unequal stretches occur at equal nominal stresses (Ogden, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Kearsley, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Ogden, 1987), whereas the other four conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11) correspond to the limiting points of S and E, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Also, it can be shown that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)2 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11)2 imply each other, and so we are left with four independent conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Only a subset of these four conditions can be satisfied depending on the 3 material model adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For instance, when the material is modelled as an ideal dielectric, the left- hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11)1 is always positive and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)1 is satisfied only after (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)2 is already satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As a result, we are only left with two conditions: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11)2 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The former was the focus of study by Zhao & Suo (2007) and Norris (2008), whereas competition between the two conditions was studied by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is commonly believed that when the Hessian stability criterion H > 0 is violated, the di- electric membrane would thin down uniformly, leading eventually to electric breakdown or other types of failure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' wrinkling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) provides one counter-example to this common wisdom – the TK instability may occur first before uniform thickness thinning takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In this paper, we explore another instability mechanism, namely localized axisymmetric necking whereby thickness thinning is localized near the origin and decays exponentially in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Our preliminary investigations in Wang et al (2022) indicate that the condition for axisymmetric neck- ing is not given by H = 0 or the limiting point stability criterion although the necking condition in the case of plane-strain does correspond to the nominal stress reaching a limiting point (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We observe that in the problem of localized bulging of an inflated hyperelastic tube, the bifurcation condition corresponds to the inflation pressure reaching a limiting point when the axial force is fixed or the axial force reaching a maximum when the pressure is fixed (Fu & Il’ichev, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The rest of this paper is divided into four sections as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In the next section we summarise the governing equations of electroelasticity and derive the incremental governing equations to the order that is required for the current analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Sections 3 and 4 present the linear and weakly nonlinear analyses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The paper is concluded in Section 5 with a summary and some additional comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Governing equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Equations of nonlinear electroelasticity Consider a dielectric material that is free from volumetric free charges and mechanical body forces within the material and whose constitutive behavior is governed by the free energy density function Ω∗(F, D) or Ω(F, E) (=Ω∗(F, D) − D · E), where F is the deformation gradient, D and E are the nominal electric displacement and electric field vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The nominal electric field, electric displacement, and the total nominal stress tensor S satisfy the field equations CurlE = 0, DivD = 0, DivS = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) where Curl and Div are the curl and divergence operators with respect to X, the position vector in the undeformed configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The constitutive equations are either S = ∂Ω∗ ∂F − pF −1, E = ∂Ω∗ ∂D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) or S = ∂Ω ∂F − pF −1, D = −∂Ω ∂E, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3) where we have assumed that the material is incompressible with p denoting the Lagrangian mul- tiplier enforcing the constraint of incompressibility det F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' See Dorfmann & Ogden (2005) or Zhao & Suo (2007) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1)1 that the electric field E can be specified in terms of an electrostatic potential Φ: E = −GradΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4) 4 We consider the case when the potential Φ is specified on the two surfaces of the membrane through the coating electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As a result, the jump conditions at the interfaces between the membrane and surrounding medium need not be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Following common practice, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Dorfmann & Ogden (2014b), we consider an energy func- tion Ω(F, E) that is additively decomposed as a purely mechanical contribution and a part asso- ciated with the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We further specialize to the case when the electric contribution is described by an isotropic constitutive formulation with constant permittivity ϵ (the so-called ideal dielectric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Thus, we have Ω(F, E) = W(I1, I2) − 1 2ϵ E · C−1E, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) where I1 and I2 are the two principal invariants of FF T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Correspondingly, in terms of the principal stretches the functions Ω and Ω∗ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) take the specific forms Ω(λ1, λ2, E3) = W(λ1, λ2) − 1 2ϵE2 3(λ1λ2)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6) Ω∗(λ1, λ2, D3) = W(λ1, λ2) + 1 2ϵD2 3(λ1λ2)−2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7) where we have used the same symbols Ω and W in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6) (although the arguments are different) to avoid introducing extra notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In the above equations, the incompressibility con- dition has been used to eliminate the principal stretch λ3, and W(λ1, λ2) is sometimes referred to as the reduced strain energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For the above class of free energy functions, the left-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11)1 is always positive and we have ∂S ∂λ ���� E3 fixed = ∂S ∂λ ���� D3 fixed − 8λ2E2 3ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8) This means that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)2 is always satisfied before (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As a result, the conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11)1 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)1 can be neglected, and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10)2 can be solved explicitly (the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11)2 is not independent as remarked earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Thus, we have the following two solutions for the bifurcation values of ϵE2 3: ϵE2 3 �� TK = λ−2(W12 − W22), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) ϵE2 3 ��lim = 1 3λ−2(W12 + W22), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10) where the subscripts “TK” and “lim” signify associations with the TK and limiting-point instabil- ities, respectively, and W12 = ∂2W ∂λ1λ2 ���� λ1=λ2=λ , W22 = ∂2W ∂λ2 2 ���� λ1=λ2=λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Incremental formulation In this section we derive the equations governing incremental deformations up to and including quadratic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For the linear version, see Dorfmann & Ogden (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We denote the undeformed, uniformly stretched, and bifurcated configurations of the mem- brane by B0, Be and Bt, and the position vectors of a representative material particle in the three configurations by X, x and ˜x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We use F, E, D and S to denote the deformation gra- dient, the nominal electric field, nominal electric displacement and total nominal stress associated 5 with the deformation B0 → Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Their counterparts associated with the deformation B0 → Be are denoted by ¯F, ¯E, ¯D and ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We define the incremental fields η, e, d, and χ through F = (I + η) ¯F, E = ¯E + ¯F T e, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12) D = ¯D + ¯J ¯F −1d, S = ¯S + ¯J ¯F −1χT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='13) The determinant ¯J (= det ¯F) is unity but is kept in the above expressions to maintain the generality of the formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' With u(x) denoting the incremental displacement from Be to Bt, we have η = grad u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' From the governing equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) that apply to both the barred and unbarred fields, we obtain the incremental governing equations curl e = 0, div d = 0, div χT = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14) where div and curl are evaluated with respect to the position vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We now proceed to derive the incremental forms of the constitutive equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We first expand ∂Ω/∂FiA around F = ¯F, E = ¯E to obtain � ¯J−1 ¯F ∂Ω ∂F � li = ¯J−1 ¯FlA ∂Ω ∂FiA = ¯J−1 ¯FlA ∂Ω ∂FiA ���� ¯F + A(1) lijkηkj + A(1) li|kek + 1 2A(2) lijknmηkjηmn + A(2) lijk|nηkjen + 1 2A(3) li|jkejek, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='15) where A(1) lijk = ¯J−1 ¯FlA ¯FjB ∂2Ω ∂FiA∂FkB ���� ¯F , A(2) lijknm = ¯J−1 ¯FlA ¯FjB ¯FnC ∂2Ω ∂FiA∂FkB∂FmC ���� ¯F , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='16) A(1) li|k = ¯J−1 ¯FlA ¯FkB ∂2Ω ∂FiA∂EB ���� ¯F , A(2) lijk|n = ¯J−1 ¯FlA ¯FjB ¯FnC ∂3Ω ∂FiA∂FkB∂EC ���� ¯F , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='17) A(3) li|jk = ¯J−1 ¯FlA ¯FjB ¯FkC ∂3Ω ∂FiA∂EB∂EC ���� ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='18) We also have p ¯FF −1 = (¯p + p∗)(I + η)−1 = ¯p(I − η + η2) + p∗(I − η) + · · · (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='19) Thus, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3)1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12)2 that (χT )li = A(1) lijkηkj + A(1) li|kek + 1 2A(2) lijknmηkjηmn + A(2) lijk|nηkjen + 1 2A(3) li|jkejek + ¯p(ηli − ηlkηki) − p∗(δli − ηli) + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='20) For the electric displacement, we can similarly obtain ¯J−1 ¯FlM ∂Ω ∂EM = ¯J−1 ¯FlM ∂Ω ∂EM ���� ¯F + ¯J−1 ¯FlM ¯FmA ∂2Ω ∂FiA∂EM ���� ¯F ηim + ¯J−1 ¯FlM ¯FjA ∂2Ω ∂EA∂EM ���� ¯F ej +1 2 ¯J−1 ¯FlM ¯FmA ¯FnB ∂3Ω ∂FiA∂FkB∂EM ���� ¯F ηimηkn + ¯J−1 ¯FlM ¯FmA ¯FnC ∂3Ω ∂FiA∂EC∂EM ���� ¯F ηimen + 1 2 ¯J−1 ¯FlM ¯FiA ¯FnC ∂3Ω ∂EA∂EC∂EM ���� ¯F eien + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='21) 6 It then follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='13)1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3)2 that dl = ¯J−1 ¯FlM(− ∂Ω ∂EM + ∂Ω ∂EM ���� ¯F ) = −A(1) mi|lηim − A(1) jl ej − 1 2A(2) mink|lηimηkn − A(3) mi|nlηimen − 1 2A(2) ilneien + · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='22) where A(1) jl = ¯J−1 ¯FjA ¯FlB ∂2Ω ∂EA∂EB ���� ¯F , A(2) iln = ¯J−1 ¯FiA ¯FlM ¯FnC ∂3Ω ∂EA∂EM∂EC ���� ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23) Finally, it follows from the incompressibility conditions det ¯F = 1 and det F = 1 that Iη + IIη + IIIη = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24) where the three terms denote the three principal invariants of η, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This is the incremental incompressibility condition and its linear form is simply tr η = div u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The governing equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14)1 can be satisfied automatically by writing e = grad ψ where the scalar function ψ replaces e as one of the new independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The remaining governing equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14)2,3 are to be solved subjected to the boundary conditions χ33 = 0, χ31 = 0, ψ = 0 on z = ±h/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25) We take h = 1 in the remaining analysis, which is equivalent to using h as the length unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Linear analysis We now consider an axisymmetric perturbation represented by u = u(r, z)er + v(r, z)ez, ψ = ψ(r, z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) where r and θ are the cylindrical coordinates for x, er and ez are the unit basis vectors, and u and v are the associated displacement components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The tensor η (= grad u) now takes the form η = urer ⊗ er + uzer ⊗ ez + u r eθ ⊗ eθ + vrez ⊗ er + vzez ⊗ ez, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) where ur = ∂u/∂r, uz = ∂u/∂z, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For the current axisymmetric problem, the two components of the equilibrium equation div χT = 0 that are not satisfied automatically are χ1j,j + 1 r(χ11 − χ22) = 0, χ3j,j + 1 rχ31 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3) where (1, 2, 3) corresponds to (r, θ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The linearization of the incompressibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24), namely div u = 0, may be written in the form ∂ (ru) ∂r + ∂ (rv) ∂z = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4) which can be satisfied automatically by introducing a ‘stream function’ φ(r, z) such that u = 1 rφz, v = −1 rφr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) 7 where as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) a subscript signifies differentiation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' φz = ∂φ/∂z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The non-zero stress components are given by χ11 = A(1) 1122 u r + A(1) 1133vz + (A(1) 1111 + ¯p)ur − p∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6) χ22 = (A(1) 2222 + ¯p)u r + A(1) 2233vz + A(1) 1122ur − p∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7) χ33 = A(1) 2233 u r + (A(1) 3333 + ¯p)vz + A(1) 1133ur − p∗ − 2E3ϵλ2ψz, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8) χ13 = A(1) 3131uz + (A(1) 3113 + ¯p)vr − E3ϵλ2ψr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) χ31 = A(1) 1313vr + (A(1) 1331 + ¯p)uz − E3ϵλ2ψr, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10) whereas the linearisation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='22) is given by d1 = −E3ϵλ2(uz + vr) − E3ψr, d2 = 0, d3 = −2E3ϵλ2vz − E3ψz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11) On substituting these expressions together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3) and then eliminating p∗ by cross- differentiation, we obtain α � φrrrr − 2 rφrrr + 3 r2 φrr − 3 r3 φr � + 2β � φrrzz − 1 rφrzz � + γφzzzz + E3ϵλ2 � rψrrr + ψrr − 1 rψr + rψrzz � = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12) where α = A(1) 2323, 2β = A(1) 2222 + A(1) 3333 − 2A(1) 2233 − 2A(1) 2332, γ = A(1) 3232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='13) A second equation for φ and ψ is obtained by substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14)2: ψzz + 1 rψr + ψrr − E3λ2 1 r3 � r2φrrr − rφrr + r2φrzz + φr � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12)and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14) admit a “normal mode” buckling/wrinkling solution of the form φ(r, z) = rJ1(kr)S(kz), ψ(r, z) = J0(kr)K(kz), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='15) where k is a constant playing the role of wavenumber, J0(x) and J1(x) are Bessel’s functions of the first kind, and the other functions S(kz) and K(kz) are to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' On substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='15) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14) and simplifying by making use of the identity Jν(x) = 2(ν − 1) x Jν−1(x) − Jν−2(x), the J1(kr) and J0(kr) can be cancelled in the resulting equations and we obtain two ordinary differential equations: γS(4)(kz) − 2βS′′(kz) + αS(kz) + k−1E3ϵλ2(K(kz) − K′′(kz)) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='16) and � K′′(kz) − E3kλ2S′′(kz) � − � K(kz) − E3kλ2S(kz) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='17) The last equation can be integrated straightaway to yield K(kz) = E3kλ2S(kz) + c5 sinh(kz) + c6 cosh(kz), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='18) 8 where c5 and c6 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='16) then reduces to γS(4)(kz) − 2β∗S′′(kz) + α∗S(kz) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='19) where α∗ = α + E2 3ϵλ4, β∗ = β + 1 2E2 3ϵλ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='20) The general solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='19) may be written in the form S(kz) = c1 sinh k √ζ1 z + c2 sinh k √ζ2 z + c3 cosh k √ζ1 z + c4 cosh k √ζ2 z, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='21) where c1, c2, c3, c4 are disposable constants, and ζ1 = 1 α∗ (β∗ − � β∗2 − α∗γ), ζ2 = 1 α∗ (β∗ + � β∗2 − α∗γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='22) The boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25) take the form A(1) 3131uz + (A(1) 3113 + ¯p)vr − E3ϵλ2ψr = 0, on z = ±1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23) A(1) 2233 u r + (A(1) 3333 + ¯p)vz + A(1) 1133ur − p∗ − 2E3ϵλ2ψz = 0, on z = ±1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24) ψ = 0, on z = ±1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25) The p∗ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24) can be eliminated by first differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24) with respect to r and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3)1 to eliminate p∗ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This gives 1 r2 (A(1) 2233 − A(1) 2222 − ¯p) � r2urr + rur − u � − A(1) 3232uzz + (A(1) 3333 − A(1) 2332 − A(1) 2233)vrz − E3ϵλ2ψrz = 0, on z = ±1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='26) On substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='15), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='21) into the six boundary conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='26), we obtain six algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Due to the symmetry of the membrane geometry and external loads with respect to the mid-plane z = 0, this system of equations admits two types of solutions corresponding to flexural and extensional modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The bifurcation condition for the extensional modes is what we shall focus on and is given by d1 tanh �k 2 � tanh � k 2√ζ1 � − d2 tanh �k 2 � tanh � k 2√ζ2 � + d3 tanh � k 2√ζ1 � tanh � k 2√ζ2 � = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='27) where d1 = � ζ1(1 + ζ1)(ζ2(2β∗ + γ) − γ), d2 = � ζ2(1 + ζ2)(ζ1(2β∗ + γ) − γ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='28) d3 = ϵE2 3λ4� ζ1ζ2(ζ1 − ζ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Expanding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='27) to order k2, we obtain γ(β + γ) + k2 24γ(α − γ) + O(k4) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='29) 9 where we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='22) to eliminate ζ1 and ζ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Note that the coefficient of k2 in the above asymptotic expression is not unique: we can add an arbitrary multiple of γ(β + γ) to it without changing the asymptotic order of the second term since the latter expression is of order k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As an illustrative example, consider the following two-term Ogden strain-energy function: W = 2µ1 m2 1 (λm1 1 + λm1 2 + λm1 3 − 3) + 2µ2 m22 (λm2 1 + λm2 2 + λm2 3 − 3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30) with m1 = 1/2, m2 = 4, µ2 = µ1/80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 1 displays the bifurcation condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='27) and its two-term approximation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='29) in the small wavenumber limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is seen that the the minimum of λ is attained at k = 0 in the case of fixed E3 and the minimum of E3 is also attained at k = 0 in the case of fixed λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Based on the discussion in Fu (2001), we may postulate that the bifurcation exact two-term 1 2 3 4 k 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3 λ (ϵ/μ1)E3 2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='03 exact two-term 1 2 3 4 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='06 (ϵ/μ1)E32 λ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1 (a) (b) Figure 1: Bifurcation condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='27) for periodic and symmetric modes, and its two-term approximation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='29) in the small wavenumber limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' condition for localized necking can be obtained by setting the leading order term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='29) to zero, that is β + γ = 0 since γ > 0, or equivalently, A(1) 2222 + A(1) 3333 + 2A(1) 3232 − 2A(1) 2332 − 2A(1) 2233 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='31) It can be shown that this condition is equivalent to ∂S1 ∂λ1 ���� λ1=λ2=λ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='32) where S1 has the same meaning as in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Corresponding to the free energy function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6), this equation can be solved explicitly to give (ϵE2 3)necking = λ−2W11, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='33) where W11 = ∂2W ∂λ2 1 ���� λ1=λ2=λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='34) The bifurcation condition may be compared with the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10) for the TK and limiting point instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 10 LP necking TK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='03 (ϵ/μ1)E32 LP necking TK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='05 (ϵ/μ1)E32 (a) (b) Figure 2: Bifurcation conditions for the TK, limiting point and necking instabilities corresponding to the strain energy function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The alternative representations in (b) are obtained by viewing E3 and S as functions of λ and varying λ in the interval (1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The three lines in (a) intersect at λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='98 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23, and the curve associated with necking cuts the horizontal axis at λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='44 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In (b) the dotted line corresponding to the limiting point instability is close but always above that for the necking instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Corresponding to the strain energy function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30), the three bifurcation conditions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2(a,b) by viewing E3 as a function of λ or S, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 (b) is obtained by eliminating E3 from S = S(λ, λ, E3) using the bifurcation conditions so that both S and E3 are parametric functions of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In the absence of an electric field (E3 = 0), there are two bifurcation values for the TK instability and another two bifurcation values for necking, and limiting points do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This purely mechanical case has previously been discussed in Wang et al (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In particular, it was shown that although the first bifurcation value for the TK instability is smaller than the first bifurcation value for necking, necking can still occur first when the membrane is stretched under edge displacement control since in this case the TK instability will be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' When an electric field is applied (E3 ̸= 0), we consider two typical loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' One is to first stretch the membrane to a specified value of λ, say λ = 2, in the absence of an electric field, and then increase the electric field from zero with the edge fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This loading scenario corresponds to displacement control and so TK instability is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Referring to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 (a), this means that the first instability experienced by the membrane is the necking instability although the loading path crosses the TK instability curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The other loading scenario is to first increase the nominal stress S to a specified value, say 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5, in the absence of an electric field, and then increase the electric field from zero with S fixed as a dead load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This is the loading scenario adopted by (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 (b) shows that again the first instability experienced by the membrane is the necking instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Weakly nonlinear analysis The linear analysis in the previous section only provides a necessary condition for necking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Whether a necking solution really bifurcates from the homogeneous solution or not can only be answered by a near-critical nonlinear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' To fix ideas, we may assume that the strain energy function is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30) and the case to be studied is when λ is fixed in the interval (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='98, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='44) and E3 is gradually increased from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 11 As pointed out in the previous section, in this parameter regime necking would occur before the limiting point instability or the TK instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We define a non-dimensional load parameter ω through ω = ϵ µ1 E2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) Denoting its bifurcation value by ωcr (which depends on λ), we write ω = ωcr + εω1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) where ω1 is an O(1) constant and ε is a positive small parameter characterizing the derivation of ω from ωcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' From the bifurcation condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='29) it can be deduced that in this parameter regime the buckling mode will have k = O(√ε), which means that the dependence of the near-critical solution on r should be through the stretched variable s defined by s = √εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3) The relative orders of u, v, p∗ and ψ can be deduced by expanding the linear solutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='15) for small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The absolute size of v is determined by the fact that the amplitude is expected to be a linear function of ω − ωcr for the type of bifurcations under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This gives v = O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Based on this analysis, we look for a near-critical solution of the form u = √ε � u(1)(s, z) + εu(2)(s, z) + ε2u(3)(s, z) + · · · � , v = ε � v(1)(s, z) + εv(2)(s, z) + ε2v(3)(s, z) + · · · � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4) p∗ = ε � p(1)(s, z) + εp(2)(s, z) + ε2p(3)(s, z) + · · · � , ψ = ε2 � ψ(1)(s, z) + εψ(2)(s, z) + ε2ψ(3)(s, z) + · · · � , where all the functions on the right hand sides are to be determined from successive approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' To ease descriptions, we scale all the governing equations and boundary conditions so that the left hand side of each equation becomes of O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This is achieved by dividing by ε the elec- tric equilibrium equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14)2, the mechanical equilibrium equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3)2, the incompressibility condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24) and the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)1, and by √ε the mechanical equilibrium equa- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3)1 and boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' On substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4) into these scaled equations and then equating the coefficients of like powers of ε, we obtain a hierarchy of boundary value prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In the following description, the two equilibrium equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3) are referred to as the r- and z-equilibrium equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' At the n-th order (n = 1, 2 or 3), we integrate the r-equilibrium equation subject to the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)2 to find u(n)(s, z), the incompress- ibility condition to find v(n)(s, z), and finally the z-equilibrium equation subject to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)1 to find p(n)(s, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' At leading order, the above procedure yields u(1)(s, z) = A(s), v(1)(s, z) = −z 1 s(sA(s))′ + B(s), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) p(1)(s, z) = −(A(1) 3333 − A(1) 2233 + A(1) 3232 − A(1) 3223)1 s(sA(s))′, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6) where A(s) and B(s) are functions to be determined, and here and hereafter in this section all the moduli are evaluated at ω = ωcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The electric equilibrium equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14)2 is satisfied automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 12 At second order, the general solution for u(2)(s, z) contains two new functions C(s) and D(s) in the form C(s) + zD(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Subtracting and adding the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)2 at z = ±1/2, respectively, we obtain A(1) 3333 − 2A(1) 2233 + A(1) 2222 + 2A(1) 3232 − 2A(1) 3223 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7) and D(s) = −B′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8) The first result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7) is equivalent to the bifurcation condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The general solutions for v(2)(s, z) and p2(s, z) contain new functions F(s) and E(s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' On applying the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)1 at z = ±1/2, we obtain sB′′(s) + B′(s) = 0, and an expression for E(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It then follows that B(s) = d1 ln s+d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Since v(1) and hence B(s) should be bounded at s = 0, we must set d1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Without loss of generality we may also impose the condition v(1)(0, 0) = 0 to eliminate any rigid-body displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This yields d2 = 0 and hence B(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Finally, integrating the electric equilibrium equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14)2 at this order subject to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)3 at z = ±1/2 yields a unique expression for ψ1(s, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' At third order, nonlinear terms come into play and it is at this order that an amplitude equation for A(s) is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We first solve the r-equilibrium equation to find an expression for u(3)(s, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It contains two new functions G(s) and H(s) in the form G(s) + zH(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Subtracting and adding (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25)2 evaluated at z = ±1/2, respectively, we obtain the amplitude equation for A(s) and an expression for H(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' After some simplification, it is found that the amplitude equation takes the form c0 d ds 1 s d dssP ′(s) + c1ω1P ′(s) + c2 d dsP 2(s) + c3A′′(s) � A′(s) − 1 sA(s) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) where a prime signifies differentiation, P(s) is defined by P(s) = 1 s(sA(s))′, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10) and the three coefficients are given by c0 = 1 12 � A(1) 2323 − A(1) 3232 � , c1 = 2A(1)′ 2233 + 2A(1)′ 2332 − A(1)′ 2222 − 2A(1)′ 3232 − A(1)′ 3333, c2 = 1 4 � −4A(1) 2222 − 2A(1) 2233 + 6A(1) 3333 − A(2) 222222 + 4A(2) 222233 − A(2) 112222 −6A(2) 223333 + 2A(2) 112233 + 2A(2) 333333 � , c3 = A(1) 2233 − A(1) 2222 + A(2) 222233 − A(2) 112233 − 1 2A(2) 222222 + 1 2A(2) 112222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' In the above expressions, A(1)′ 2233 denotes dA(1) 2233/dω etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', and we have used the bifurcation condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='7) to eliminate A(1) 2332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It can be seen that the amplitude equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) has the same structure as its mechanical counterpart derived by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Corresponding to the specific free energy function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30), we have c0 = −−480λ17/2 + λ12 + 800λ7 + 3 960λ8 , c1 = λ4, c2 = −56λ17/2 + 240λ7 + 3 32λ8 , c3 = √ λ 2 − 3λ4 80 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='11) 13 As a consistency check, we may neglect the nonlinear terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) to obtain c0 d ds 1 s d dssP ′(s) + c1ω1P ′(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12) On substituting a solution of the form P ′(s) = J1(ks/√ε) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='12), where k is a constant, we obtain c1(ω − ωcr) − c0k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='13) On the other hand, expanding (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='29) around ω = ωcr, we obtain � d dω(β + γ) � cr (ω − ωcr) + k2 24(α − γ) ���� cr = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14) where the subscripts “cr” signify evaluation at ω = ωcr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We have verified that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='13) is indeed consistent with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As another consistency check, we may expand (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) out fully and omit all the terms that are divided by powers of s to obtain its planar counterpart: c0A(4)(s) + c1ω1A′′(s) + c∗ 2A′(s)A′′(s) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='15) where c∗ 2 = 2c2 + c3 = 3A(1) 3333 − 3A(1) 2222 − A(2) 222222 + 3A(2) 222233 − 3A(2) 223333 + A(2) 333333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='16) It has an exact solution given by A(s) = 6c0 c∗ 2 �−c1ω1 c0 tanh �1 2 �−c1ω1 c0 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='17) This solution has the property A′(s) → 0 as s → ∞ and is the localised necking solution in the 2D case (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It does not seem possible to find a similar analytical solution for the original amplitude equa- tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) that is fourth-order with variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We thus resort to finding its numer- ical solution with the use of the finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' With the use of the substitution A(s) → (c0/c2)κ2A(κs), equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) may be reduced to d dt 1 t d dttP ′(t) − P ′(t) + d dtP 2(t) + c3 c2 A′′(t) � A′(t) − 1 t A(t) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='18) where t = κs, κ = � −c1ω1/c0 and P(t) is still defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We replace the semi-infinite interval [0, ∞) by a finite interval [0, L] and discretize the latter into N equal intervals with node points ti = i˜h, ˜h = L N , i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We apply the central finite difference scheme such that A′(ti) = Ai+1 − Ai−1 2˜h , A′′(ti) = Ai+1 − 2Ai + Ai−1 ˜h2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='19) A′′′(ti) = Ai+2 − 2Ai+1 + 2Ai−1 − Ai−2 2˜h3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='20) 14 A(4)(ti) = Ai+2 − 4Ai+1 + 6Ai − 4Ai−1 + Ai−2 ˜h4 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='21) where Ai = A(ti), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Evaluating the amplitude equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) at the N − 1 interior nodes t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', tN−1, we obtain N − 1 equations that involve the N + 3 unknowns A−1, A0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', and AN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The remaining four equations are obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' First, it follows from the symmetry conditions lim s→0 u(1)(s, z) = 0 and lim s→0 ∂v(1) ∂s (s, 1 2) = 0 that limt→0 A(t) = 0 and limt→0 P ′(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' By trying a series solution for small t, it is found that the unique solution that satisfies the above conditions has the behavior A(t) ∼ a1t + a2t3 + · · · for some constants a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This gives limt→0 A′′(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The two conditions A(0) = A′′(0) = 0 together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='19)2 then yield two additional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Next, we consider the asymptotic behaviour of the solutions as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Although the planar solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='17) does not decay, we expect that the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) will experience algebraic decay due to geometric spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Since quadratic terms are expected to decay faster than linear terms, the decay behavior may be captured by neglecting the nonlinear terms: d dt 1 t d dttP ′(t) − P ′(t) = 0, as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='22) The unique decaying solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='22) is given by P(t) = P∞(t) ≡ a3K0(t), A(t) = A∞(t) ≡ a4 t − a3K1(t), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23) where a3 and s4 are constants, and K0 and K1 are the modified Bessel function of the second kind that has the asymptotic behaviour Kα(x) ∼ � π 2xe−x � 1 + 4α2 − 1 8x + · · · � , as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='24) The asymptotic behaviour (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23) is consistent with our earlier assumption that A(s) decays alge- braically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We note that the above decay behaviour is based on the assumption that t is a real variable, or equivalently κ is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This enables us to deduce that whenever a necking bifurcation takes place, it is generally subcritical (ω1 < 0 since c1/c0 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' If a function f(x) decays exponentially like e−ax as x → ∞ for some positive constant a, then it is preferable to impose the “soft” asymptotic condition f′(L) + af(L) = 0 instead of the “hard” condition f(L) = 0 (since f′(L) + af(L) is much smaller than f(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Extending this idea, we use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='23) to find the first three derivatives of A∞(s) and by eliminating a1 and a2 express A′′ ∞(s) and A′′′ ∞(s) in terms of A∞(s) and A′ ∞(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Replacing A∞(s) by A(s) and evaluating these two expressions at s = sN = L followed by the use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='19)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='21), we obtain two more additional equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The system of N + 3 quadratic equations can then be solved provided an appropriate initial guess is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is found that one good initial guess is the planar solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='17) divided by 1 + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For values of λ in the interval (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='979, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='439), it is found that the coefficient c3/c2 is positive when λ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='096 and negative when λ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' So we consider two representative cases corresponding to λ = 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is found that taking N = 1000 and L = 10 yields sufficiently accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3 shows the finite difference solution corresponding to λ = 2 together with the approximate analytical solution P(t) = a bt2 + 1sech2(ct), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25) 15 □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ □ 0 2 4 6 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 P(t) A(t) Figure 3: FD solution of the amplitude equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='9) when λ = 2 and the strain energy function is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' where the constants a, b, c are determined by fitting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='25) to the finite difference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The maximum relative error over the entire interval is less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The solution corresponding to λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 for which the c3/c2 is of opposite sign is very similar and is thus not displayed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Discussion and conclusion Pull-in failure in dielectric elastomer actuators is widely believed to be associated with the limiting-point behaviour whereby the electric field as a function of the electric displacement or stretch has a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' For the plane-strain or plane-stress case, this connection is well explained using the analogy with the inflation problem associated with a rubber tube where the limiting-point behaviour is well-known to be associated with localised bulging that eventually evolves into a “two- phase” state (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Huang & Suo, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' However, for the case of equibiaxial tension, this explanation contradicts the fact that at large values of dead load, the limiting-point behaviour may disappear but pull-in failure can still be observed (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Our current paper offers an alternative explanation, namely that pull-in failure evolves from axisymmetric necking through an unstable process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We note that the condition for necking does not necessarily require limiting-point behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We have only carried out a linear and weakly nonlinear analysis in the current study, but the fully nonlinear numerical simulations carried out in our earlier paper (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 2022) for the purely mechanical case should also be indicative of what might be expected in the current elec- troelastic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Thus, combining the weakly nonlinear results in the previous section with the fully nonlinear simulation results in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2022), we may draw the following conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' When the bifurcation condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='33) is satisfied, a necking solution will bifurcate from the homogeneous solution subcritically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' If the membrane is gradually pulled further in the radial direction at the edge, with the electric potential fixed, the necking solution will grow in amplitude, corresponding to an increased reduction in thickness at the origin, and when a maximum amplitude is approached, the necking solution will start to propagate in the radial direction in the form of a “two-phase” deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This is very similar to the localised bulging of an inflated rubber tube except that here the propagation is also accompanied by algebraic decaying of the amplitude due to geometrical spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' On the other hand, if the electric potential is increased further from its bifurcation value 16 while the membrane edge is fixed, the membrane will snap to a “two-phase” deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This is analogous to the pressure control case in the tube inflation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' We wish to highlight the fact that the predictions that can be made are sensitive to the material model used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' To fix ideas, we have used the strain energy function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='30) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' To show how our results depend on the strain energy function used, we have shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 4 the counterpart of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 2 when the following Gent and Mooney-Rivlin material models are used: W = −1 2µJm ln(1 − λ2 1 + λ2 2 + λ2 3 − 3 Jm ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='1) W = 1 2µ � λ2 1 + λ2 2 + λ2 3 − 3 + γ(λ−2 1 + λ−2 2 + λ−2 3 − 3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) It is found that the bifurcation curves have a very weak dependence on the value of Jm and the LP necking TK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='5 (ϵ/μ1)E32 LP necking TK 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='8 (ϵ/μ1)E32 (a) (b) Figure 4: Bifurcation conditions for the TK, limiting point and necking instabilities corresponding to (a) the Gent strain energy function with Jm = 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2, and (b) the Mooney-Rivlin strain energy function with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The dashed line corresponds to zero nominal stress in the radial direction above which the nominal stress is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' curves corresponding to Jm = ∞ (the neo-Hookean model) are almost the same as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 4(a) for Jm = 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is seen that the main effect of increasing the γ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content='2) is to shift the curves for the TK and limiting instabilities upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' As a result, the TK instability is not possible for the Gent and neo-Hookean material models (since the corresponding E3 is negative) but is possible for the Mooney-Rivlin material model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' This is well-known in the purely mechanical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' The bifurcation curve for necking is always above the curve corresponding to zero nominal stress in the radial direction (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Thus, although necking is theoretically possible, it is unlikely to be observable when the dielectric membrane has the constitutive behaviour modelled by these two material models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It then remains an open question whether there exist dielectric materials whose constitutive behaviour allows the type of axisymmetric necking that is described in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' It is hoped that this question will be answered in our future experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant No 12072224).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 17 References References Bahreman, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Maths Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Solids, 20, 1198–1214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' B.' metadata={'source': 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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Solids, 147, 104250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Pearce, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', & Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} 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electrodes-coated dielectric plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Non-linear Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 106, 60–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Gei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Colonnelli, S.' metadata={'source': 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of a dielectric elastomer under dead load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 100, 041911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Huang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', & Suo, Z.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Catastrophic thinning of dielectric elastomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Electrostat, 70, 312–316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Rudykh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', & deBotton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2011).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Broderick, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', & Destrade, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Wrinkles in soft dielectric plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Solids, 119, 298–318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Non-Linear Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 113, 62–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Chen, W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' A, 476, 20200267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', & Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Axisymmetric necking versus Treloar–Kearsley instability in a hyperelastic sheet under equibiaxial stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Solids, 27, 1610–1631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Guo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', 73, 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdAzT4oBgHgl3EQfMPug/content/2301.01129v1.pdf'} +page_content=', & Xie, Y.' metadata={'source': 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b/hNE1T4oBgHgl3EQfzQVT/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:560a9c625f7c3938f1a80b63ccb6597e7618cb2c4bb53f615520e77cea0179db +size 149068 diff --git a/htAzT4oBgHgl3EQfa_y7/content/tmp_files/2301.01379v1.pdf.txt b/htAzT4oBgHgl3EQfa_y7/content/tmp_files/2301.01379v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c7dfc7bf08b5e68daa9d07710496998a5981a2f --- /dev/null +++ b/htAzT4oBgHgl3EQfa_y7/content/tmp_files/2301.01379v1.pdf.txt @@ -0,0 +1,978 @@ +arXiv:2301.01379v1 [cs.AI] 3 Jan 2023 +A Succinct Summary of Reinforcement Learning +Sanjeevan Ahilan∗ +Abstract +This document is a concise summary of many key results in single- +agent reinforcement learning (RL). The intended audience are those who +already have some familiarity with RL and are looking to review, reference +and/or remind themselves of important ideas in the field. +Contents +1 +Acknowledgements +2 +2 +Fundamentals +2 +2.1 +The RL paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2.2 +Agent and environment +. . . . . . . . . . . . . . . . . . . . . . . +2 +2.3 +Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.4 +Markov processes and Markov reward processes . . . . . . . . . . +3 +2.5 +Markov decision processes . . . . . . . . . . . . . . . . . . . . . . +3 +2.6 +Policies, values and models +. . . . . . . . . . . . . . . . . . . . . +3 +2.7 +Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . +5 +3 +Model-free approaches +6 +3.1 +Prediction +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3.2 +Control with action-value functions . . . . . . . . . . . . . . . . . +8 +3.3 +Value function approximation . . . . . . . . . . . . . . . . . . . . +9 +3.4 +Policy gradient methods . . . . . . . . . . . . . . . . . . . . . . . +10 +3.5 +Baselines +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3.6 +Compatible function approximation . . . . . . . . . . . . . . . . . +11 +3.7 +Deterministic policy gradients . . . . . . . . . . . . . . . . . . . . +12 +4 +Model-based Approaches +12 +4.1 +Model Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +4.2 +Combining model-free and model-based approaches . . . . . . . . +13 +5 +Latent variables and partial observability +14 +5.1 +Latent variable models . . . . . . . . . . . . . . . . . . . . . . . . +14 +5.2 +Partially observable Markov decision processes +. . . . . . . . . . +14 +6 +Deep reinforcement learning +15 +6.1 +Experience replay . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +6.2 +Target networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +∗sanjeevanahilan@gmail.com. Much of this work was done at the Gatsby Unit, UCL. +1 + +1 +Acknowledgements +I would like to thank Peter Dayan, David Silver, Chris Watkins and ChatGPT +for helpful feedback. Much of this work was drawn from David Silver’s UCL +course1 and Sutton and Barto’s textbook (Sutton and Barto, 2018) and formed +the introductory chapter of my PhD thesis (Ahilan, 2021). +2 +Fundamentals +2.1 +The RL paradigm +The field of reinforcement learning (RL) (Sutton and Barto, 2018) concerns it- +self with the computational principles underlying goal-directed learning through +interaction. Although primarily seen as a field of machine learning, it has a rich +history spanning multiple fields. In psychology it can be used to model classi- +cal (Pavlovian) and operant (instrumental) conditioning. In neuroscience it has +been used to model the dopamine system of the brain (Schultz et al., 1997). In +economics, it relates to fields such as bounded rationality, and in engineering +it has extensive overlap with the field of optimal control (Bellman, 1957). In +mathematics, investigation has continued under the guise of operations research. +The plethora of perspectives ensures that RL continues to be an exciting and +extraordinarily interdisciplinary field. +2.2 +Agent and environment +RL problems typically draw a separation between the agent and the environ- +ment. The agent receives observation ot and scalar reward rt from the environ- +ment and emits action at, where t indicates the time step. The environment +receives action at from the agent and then emits a reward rt+1 and an ob- +servation ot+1. The cycle then begins again with the agent emitting its next +action. +How the environment responds to the agent’s action is determined by the +environment state st, which is updated at every time step. The conditional +distribution for the next environment state depends only on the present state +and action and therefore satisfies the Markov property: +P(st+1|st, at) = P(st+1|s1, . . . , st, a1, . . . , at) +(1) +The environment state is in general private from the agent, which only re- +ceives observations and rewards. The conditional distribution for the next ob- +servation given the current observation is not in general Markov, and so it may +be beneficial for an agent to construct its own notion of state sα +t , which it uses +to determine its next action. This can be defined as sα +t = f(ht), where ht is the +history of the agent’s sequence of observations, actions and rewards: +ht = a1, o1, r1, . . . , at, ot, rt +(2) +1https://www.davidsilver.uk/teaching/ +2 + +2.3 +Observability +A special case exists when the observation received by the agent ot is identical +to the environment state st (such that there is no need to distinguish between +the two). This is the assumption underlying the formalism of Markov decision +processes covered in the next section. An environment is partially observable +if the agent cannot observe the full environment state, meaning that the con- +ditional distribution for its next observation given its current observation does +not satisfy the Markov property. This assumption underlies the formalism of a +partially observable Markov decision process which we describe in Section 5.2. +2.4 +Markov processes and Markov reward processes +A Markov process (or Markov chain) is a sequence of random states with the +Markov property. It is defined in terms of the tuple ⟨S, P⟩ where S is a finite +set of states and P : S × S → [0, 1] is the state transition probability kernel. +A Markov Reward Process (MRP) ⟨S, P, r, γ⟩ extends the Markov process +by including a reward function r : S × S → R for each state transition and a +discount factor γ. The immediate expected reward in a given state is defined +as: r(s) = � +s′ P(s, s′)r(s, s′). +The discount factor γ ∈ [0, 1] is used to determine the present value of future +rewards. Conventionally, a reward received k steps into the future is of worth +γk times what it would be worth if received immediately. As we will shortly +see, the cumulative sum of discounted rewards is a quantity RL agents often +seek to maximise, and so γ < 1 ensures that this sum is bounded (assuming r +is bounded). +2.5 +Markov decision processes +Single-agent RL can be formalised in terms of Markov decision processes (MDPs). +The idea of an MDP is to capture the key components available to the learning +agent; the agent’s sensation of the state of its environment, the actions it takes +which can affect the state, and the rewards associated with states and actions. +An MDP extends the formalism of an MRP to include a finite set of actions on +which both P and r depend. Discrete-time, infinite-horizon MDPs are described +in terms of the 5-tuple ⟨S, A, P, r, γ⟩ where S is the set of states, A is the +set of actions, P : S × A × S → [0, 1] is the state transition probability kernel, +r : S×A×S → R is the immediate reward function and γ ∈ [0, 1) is the discount +factor. The expected immediate reward for a given state and action is defined +as r(s, a) = � +s′ P(s, a, s′)r(s, a, s′), which we use for convenience subsequently. +2.6 +Policies, values and models +Common components of a reinforcement learning agent are a policy, value func- +tion and a model. The policy π : S ×A → [0, 1] is the agent’s behaviour function +which denotes the probability of taking action a in state s. Agents may also act +according to a deterministic policy µ : S → A. We will assume that policies are +stochastic unless otherwise noted. +3 + +Given an MDP and a policy π, the observed state sequence is a Markov +process ⟨S, Pπ⟩. +Pπ(s, s′) = +� +a∈A +π(s, a)P(s, a, s′) +(3) +Similarly, the state and reward sequence is a MRP ⟨S, Pπ, rπ, γ⟩ in which: +rπ(s) = +� +a∈A +π(s, a)r(s, a) +(4) +Starting from any particular state s at time step t = 0, the value function +vπ(s) is a prediction of the expected discounted future reward given that the +agent starts in state s and follows policy π: +vπ(s) = Eπ +� ∞ +� +t=0 +γtrt+1|s0 = s +� +(5) +where rt+1 = r(st, at, st+1) +which is the solution of an associated Bellman expectation equation: +vπ(s) = +� +a∈A +π(s, a) +� +r(s, a) + γ +� +s′∈S +P(s, a, s′)vπ(s′) +� +(6) +In matrix form the Bellman expectation equation can be expressed in terms +of the induced MRP: +vπ = rπ + γPπvπ = (I − γPπ)−1rπ +(7) +where vπ ∈ R|S| and rπ ∈ R|S| are the vector of values and expected imme- +diate rewards respectively for each state under policy π. We can also define a +Bellman expectation backup operator: +T π(v) = rπ + γPπv +(8) +which has a fixed point of vπ. +An action-value for a policy π can also be defined, which is the expected dis- +counted future reward for executing action a and subsequently following policy +π. +qπ(s, a) = r(s, a) + γ +� +s′∈S +P(s, a, s′)vπ(s′) += r(s, a) + γ +� +s′∈S +P(s, a, s′) +� +a′∈A +π(s′, a′)qπ(s′, a′) +(9) +The process of estimating vπ or qπ is known as policy evaluation. Policies +can be evaluated without directly knowing or estimating a model, using instead +the directly sampled experience of the environment, an approach which is known +as ‘model-free’. However a ‘model-based’ approach is also possible in which a +model is used to predict what the environment will do next. A key component +of a model is an estimate of P(s, a, s′), the probability of the next state given +the current state and action. Another is an estimate of r(s, a), the expected +immediate reward. +4 + +Policy evaluation enables a value function to be learned for a given policy. +However, we often wish to learn the best possible policy. The value function for +this is known as the optimal value function and corresponds to the maximum +value function over all policies: +v∗(s) = max +π +vπ(s) +(10) +The definition of the optimal action-value function (which evaluates the im- +mediate action a in state s) is similarly: +q∗(s, a) = max +π +qπ(s, a) +(11) +A partial ordering over policies can be defined according to: +π ≥ π′ if vπ(s) ≥ vπ′(s), ∀s +(12) +For any MDP there exists an optimal policy π∗ that is better than or equal +to all other policies. All optimal policies achieve the optimal value function and +optimal action-value function and there is always a deterministic optimal policy +for any MDP. The latter is achieved by selecting: +a = arg max +a∈A +q∗(s, a) +(13) +If there are many possible actions which satisfy this, any of these may be +chosen to constitute an optimal policy (of which there may be many). +The +optimal value and state-value functions satisfy Bellman optimality equations: +v∗(s) = max +a∈A q∗(s, a) +v∗(s) = max +a∈A +� +r(s, a) + γ +� +s′∈S +P(s, a, s′)v∗(s′) +� +q∗(s, a) = r(s, a) + γ +� +s′∈S +P(s, a, s′)max +a′ +q∗(s′, a′) +(14) +The Bellman optimality equation is non-linear with no closed form solution +(in general). Solving it therefore requires iterative solution methods. +2.7 +Dynamic programming +Dynamic programming (DP) (Bertsekas et al., 1995) refers to a collection of +algorithms that can be used to compute optimal policies given a perfect model +of the environment as an MDP. In general, DP solves complex problems by +breaking them down into subproblems and then combining the solutions. It is +particularly useful for overlapping subproblems, the solutions to which reoccur +many times when solving the overall problem, making it more computationally +efficient to cache and reuse them. +When applied to MDPs, the recursive decomposition of DP corresponds to +the Bellman equation and the cached solution to the value function. DP assumes +that the MDP is fully known and therefore does not address the full RL problem +but instead addresses the problem of planning. By planning, the prediction +problem can be addressed by finding the value function vπ of a given policy +5 + +π. This can be evaluated by iterative application of the Bellman Expectation +Backup (Equation 8). +This leads to convergence to a unique fixed point vπ, which can be shown +using the contraction mapping theorem (also known as the Banach fixed-point +theorem) (Banach, 1922). When a Bellman expectation backup operator T π +is applied to two value functions u and v over states, we find that it is a γ- +contraction: +||T π(u) − T π(v)||∞ = ||(rπ + γPπu) − (rπ + γPπv)||∞ += ||γPπ(u − v)||∞ +≤ ||γPπ1||u − v||∞||∞ +≤ γ||u − v||∞ +(15) +where 1 is a vector of ones and the infinity norm of a vector a is denoted +||a||∞ and is defined as the maximum value of its components. This contraction +ensures that both u and v converge to the unique fixed point of T π which is vπ. +For control, DP can be used to find the optimal value function v∗ and in turn +the optimal policy π∗. One possibility is policy iteration in which the current +policy π is first evaluated as described and then subsequently improved to π′ +such that: +π′(s) = arg max +a∈A +qπ(s, a) +(16) +This improves the value from any state s over one step: +qπ(s, π′(s)) = max +a∈A qπ(s, a) ≥ +� +a∈A +π(s, a)qπ(s, a) = vπ(s) +(17) +It can be shown that this improves the value function such that that vπ′(s) ≥ +vπ(s) (Silver, 2015). This process is then repeated, with improvements ending +when the Bellman optimality equation (14) has been satisfied and convergence +to π∗ achieved. A generalisation of policy iteration is also possible in which, +instead of waiting for policy evaluation to converge, only n steps of evaluation +are taken before policy improvement occurs and the process is repeated. If n = 1 +this is known as value iteration, as the policy is no longer explicit (being a direct +consequence of the value function). Like policy iteration, value iteration is also +guaranteed to converge to the optimal value function and policy. This can be +demonstrated using the contraction mapping theorem. +3 +Model-free approaches +3.1 +Prediction +As has been outlined, dynamic programming can be used to solve known MDPs +enabling optimal value functions and policies to be found. However, in many +cases the MDP is not directly known - instead an agent taking actions in the +MDP must learn directly from its experiences, as it transitions from state to +state and receives rewards accordingly. One approach, known as ‘model-free’, +seeks to solve MDPs without learning transitions or rewards. For prediction, a +6 + +key quantity to estimate in this setting is the expected discounted future reward. +A sampled estimate of this, starting from state st, is known as the return: +Rt = rt+1 + γrt+2 + γ2rt+3 + ... = +∞ +� +k=0 +γkrt+k+1 +(18) +which depends on the actions sampled from the policy, and states from +transitions. +Monte-Carlo (MC) methods seek to estimate this directly using complete +episodes of experience. Introducing a learning rate αt, the agent’s value function +can therefore be updated according to2: +v(st) ← v(st) + αt +� +Rt − v(st) +� +(19) +The value function updated in this way will converge to a solution with min- +imum mean-square error (best fit to the observed returns), assuming a suitable +sequential decrease in the learning rate. +Temporal-difference (TD) learning methods learn from incomplete episodes +by bootstrapping. For example, if learning occurs after a single step, this is +known as TD(0), which has the following update: +v(st) ← v(st) + αt +� +rt+1 + γv(st+1) − v(st) +� +(20) +where rt+1 + γv(st+1) is known as the target. This approximates the full-width +Bellman expectation backup (Equation 8) in which every successor state and +action is considered, with experiences instead being sampled. TD(0) will con- +verge to the solution of the maximum likelihood Markov model which best fits +the data (again assuming a suitable sequential decrease in the learning rate). +This solution may be different from the minimum mean-square error solution of +MC methods, which do not assume the Markov property. +Unlike MC methods, TD methods introduce bias into the estimated return +as the currently estimated value function may be different from the true value +function. However, they generally have reduced variance relative to MC meth- +ods, as in MC the estimated return depends on a potentially long sequence of +random actions, transitions and rewards. +The distinction between MC and TD methods can be blurred by considering +multi-step TD methods (rather than only TD(0)), in which rewards are sampled +for a number of steps before the value function is used to compute an estimate +of future rewards. The n-step return is defined as: +R(n) +t += rt+1 + γrt+2 + ... + γn−1rt+n + γnv(st+n) +(21) +As n → ∞ it tends towards the unbiased MC return. An algorithm may +seek to find a good bias-variance tradeoff by estimating a weighted combination +of n-step returns; one popular method to do this is known as TD(λ): +Rλ +t = (1 − λ) +∞ +� +n=1 +λn−1R(n) +t +(22) +where λ ∈ [0, 1]. +2assuming a table-based representation rather than use of a function approximator +7 + +3.2 +Control with action-value functions +Model free control concerns itself with optimising rather than evaluating the +RL objective. Policies may be evaluated according to various objectives. In +the case of continuing environments, the objective can be the average value or +the average reward per time-step. We focus instead on episodic environments, +assuming an initial distribution over starting states p0(s) : S → [0, 1]. The +objective is thus: +J(π) = Eπ +� ∞ +� +t=0 +γtrt+1|p0(s) +� +(23) +Note that if the domain of the starting state distribution is only over a single +starting state, the objective is simply the value function (Equation 5) in that +starting state. This objective can equivalently be expressed as: +J(π) = Es∼ρπ,a∼π[r(s, a)] +(24) +where: +ρπ(s) := +� +s′ +∞ +� +t=0 +γtp(st = s|s′, π)p0(s′) +(25) +is the improper discounted state distribution induced by policy π starting from +an initial state distribution p0(s′). In Section 3.4 we describe policy gradient +methods which seek to optimise this objective directly. +However, we first consider model-free approaches which rely on an action- +value function q(s, a) to achieve control (a value function v(s) alone is insufficient +for model-free control). +The optimal action-value function q∗(s, a) must be +learned, with MC and TD methods both viable. +Once it has been learned, +an optimal policy may be achieved by selecting the best action in each state +(Equation 13). +However, unlike dynamic programming, full-width backups are not used and +so if actions are selected greedily (meaning those with highest action-values are +always chosen) then certain states and actions may never be correctly evalu- +ated. Model-free RL methods must therefore allow for enough exploration dur- +ing learning before ultimately exploiting this learning to achieve near-optimal +cumulative reward. +One simple approach, known as ǫ-greedy is to take a random action with +probability ǫ but otherwise act greedily according to the current estimate of +the action-value function. The value of ǫ can be decreased with the number of +episodes. This can satisfy a condition known as greedy in the limit of infinite +exploration in which all state-action pairs are explored infinitely many times +and the policy converges to the greedy policy. +One popular algorithm for model-free control is known as Q-learning (Watkins and Dayan, +1992), which seeks to learn the optimal action-value function whilst using a pol- +icy which also takes exploratory actions (such as epsilon greedy). This learning +is termed off-policy as the policy used to sample experience is different from the +policy being learned (the optimal policy). The resulting update is: +q(st, at) ← q(st, at) + α +� +rt+1 + γmax +a′∈A q(st+1, a′) − q(st, at) +� +(26) +8 + +An alternative to off-policy Q-learning is on-policy SARSA (Rummery and Niranjan, +1994). This uses the sampled sampled state st, action at, reward rt+1, next state +st+1, and next action at+1 for updates3: +q(st, at) ← q(st, at) + α(rt+1 + γq(st+1, at+1) − q(st, at)) +(27) +3.3 +Value function approximation +So far we have assumed a tabular representation of states and actions such +that each state is separately updated. However, in practice we would like value +functions and policies to generalise to new states and actions, and so it is ben- +eficial to use function approximators such as deep neural networks. A common +approach is to approximate the value function or action-value function: +vw(s) = ˆv(s; w) ≈ vπ(s) +qw(s, a) = ˆq(s, a; w) ≈ qπ(s, a) +(28) +where w are the parameters we wish to learn. +If we start by assuming we +know the true value function vπ, we can define a mean square error between the +approximate value function and the true function: +L(w) = Eπ[(vπ(s) − vw(s))2] +(29) +Given a distribution of states s ∼ p(s)4, we can minimise this iteratively +using stochastic gradient descent: +w ← w + α(vπ(st) − vw(st))∇wvw(st) +(30) +In reality we can only use a better estimate of vπ provided by the sampled +reward(s). For example, if we use the TD(0) target the update is: +w ← w + α(rt+1 + γvw(st+1) − vw(st))∇wvw(st) +(31) +Updates like this are known as ‘semi-gradient’ as the gradient of the value +function used to define the target is ignored. +If we use a linear function approximator vw(s) = x(s)T w (where features +x(s) and w are vectors), then we find: +w ← w + α(rt+1 + γvw(st+1) − vw(st))x(st) +(32) +indicating that the linear weights are updated in proportion to the activity +of their corresponding features. Non-linear function approximators can also be +used, but typically have weaker convergence guarantees than linear function +approximators. Nevertheless, due to their flexibility such approximators have +enabled impressive performance in a number of challenging domains, such as +Atari games (Mnih et al., 2015) and Go (Silver et al., 2016). +3and also gives SARSA its name +4we later discuss a method for sampling states +9 + +3.4 +Policy gradient methods +Parameterised stochastic policies πθ may be improved using the policy gradient +theorem (Sutton et al., 2000). This can be derived for any of the common RL +objectives. To demonstrate a derivation of this result we use a starting state +objective J(θ) = vπθ(s0) with a single starting state s0: +∇θJ(θ) = ∇θvπ(s0) += ∇θ +� +a +π(s0, a)qπ(s0, a) += +� +a +∇θπ(s0, a)qπ(s0, a) + π(s0, a)∇θqπ(s0, a) += +� +a +∇θπ(s0, a)qπ(s0, a) + π(s0, a)∇θ +� +r(s0, a) + +� +s′ +γP(s0, a, s′)vπ(s′) +� += +� +a +∇θπ(s0, a)qπ(s0, a) + π(s0, a) +� +s′ +γP(s0, a, s′)∇θvπ(s′) +(33) +We note that we could continue to unroll ∇θvπ(s′) on the R.H.S in the same +way as we have already done. Considering now transitions from starting state +s0 to arbitrary state s we therefore find: +∇θvπ(s0) = +� +s +∞ +� +t=0 +γtp(st = s|s0, π) +� +a +∇θπ(s, a)qπ(s, a) +(34) +where �∞ +t=0 γtp(st = s|s0, π) is the discounted state distribution ρπ(s) from +a fixed starting state s0 (Equation 25). This derivation holds even when there +is a distribution over starting states, and gives us the policy gradient theorem: +∇θJ(θ) = +� +s +ρπ(s) +� +a +∇θπ(s, a)qπ(s, a) +(35) +Using the likelihood ratio trick: +∇θπ(s, a) = π(s, a)∇θπ(s, a) +π(s, a) += π(s, a)∇θ log π(s, a) +(36) +this can be equivalently expressed as: +∇θJ(θ) = +� +s +ρπ(s) +� +a +π(s, a)qπ(s, a)∇θ log π(s, a) += Eπ[qπ(s, a)∇θ log π(s, a)] +(37) +The policy gradient theorem result enables model-free learning as gradi- +ents need only be determined for the policy rather than for properties of the +environment. There are a variety of approaches for determining qπ. If qπ is ap- +proximated using the sample return (Equation 18), this leads to the algorithm +known as REINFORCE (Williams, 1992): +θ ← θ + αRt∇θ log π(st, at) +(38) +10 + +As there is no bootstrapping here, this is also known as MC policy gradient. +An alternative approach is to separately approximate qπ with a ‘critic’ qw giving +rise to what are commonly known as ‘actor-critic’ methods. These introduce two +sets of parameter updates; the critic parameters w are updated to approximate +qπ, and the policy (actor) parameters θ are updated according to the policy +gradient as indicated by the critic. The critic itself can be updated according +to the TD error. An example of this approach is SARSA actor-critic: +w ← w + α1(rt+1 + γqw(st+1, at+1) − qw(st, at))∇wqw(st, at) +θ ← θ + α2qw(st, at)∇θ log π(st, at) +(39) +where different learning rates α1 and α2 may be used for the actor and the +critic. +3.5 +Baselines +Whether we use REINFORCE or an actor-critic based approach to policy gradi- +ents, it is possible to reduce the variance further by the introduction of baselines. +If this baseline depends only on the state s, then we find it introduces no bias: +� +s +ρπ(s) +� +a +∇θπ(s, a)b(s) = +� +s +ρπ(s)b(s)∇θ +� +a +π(s, a) += +� +s +ρπ(s)b(s)∇θ1 += 0 +(40) +A natural choice for the state-dependent baseline is the value function: +∇θJ(θ) = Eπ[(qπ(s, a) − vπ(s))∇θ log π(s, a)] += Eπ[Aπ(s, a)∇θ log π(s, a)] +(41) +where Aπ is known as the advantage, which may in some algorithms be approx- +imated directly (rather than approximating both qπ and vπ). +3.6 +Compatible function approximation +In the general case, our choice to approximate qπ with qw introduces bias such +that there are no guarantees of convergence to a local optimum. However, in the +special case of a compatible function approximator we can introduce no bias and +take steps in the direction of the true policy gradient. This becomes possible +when the critic’s function approximator reaches a minimum in the mean-squared +error: +0 = Eπ[∇w(qπ(s, a) − qw(s, a))2] += Eπ[(qπ(s, a) − qw(s, a))∇wqw(s, a)] +(42) +If we choose qw(s, a) such that ∇wqw(s, a) = ∇θ log π(s, a) we find: +Eπ[qπ(s, a)∇θ log π(s, a)] = Eπ[qw(s, a)∇θ log π(s, a)] +(43) +11 + +where the L.H.S is equal to the true policy gradient and so our function ap- +proximation has introduced no bias. For example, if the policy is a Boltzmann +policy with a linear combination of features, of the form: +π(s, a) = +eθT φ(s,a) +� +a′ eθT φ(s,a′) +(44) +then a compatible value function must be linear in the same features as the +policy except normalised to zero mean for each state using a subtractive baseline +(Sutton et al., 2000). +qw(s, a) = wT [φ(s, a) − +� +a′ +φ(s, a′)π(s, a′)] +(45) +3.7 +Deterministic policy gradients +Rather than have a policy specify a probability for certain actions in certain +states we can instead have it simply be a function mapping states to actions +µθ : S → A and, in the case of continuous actions, seek to find the gradient +of the objective with respect to the policy parameters. An example of an al- +gorithm which uses such an approach is Deterministic Policy Gradients (DPG) +(Silver et al., 2014). +The DPG algorithm builds on the deterministic policy +gradient theorem: +∇θJ(θ) = Es∼ρµ[∇θµθ(s)∇aqµ(s, a)|a=µθ(s)]. +(46) +where the parameters of the policy are adjusted in an off-policy fashion using +an exploratory behavioural policy (which is a noisy version of the deterministic +policy). In practice qµ is approximated by the critic qw, which is differentiable +in the action and updated using Q-learning: +δt = rt+1 + γqw(st+1, µθ(st+1)) − qw(st, at) +w ← w + α1δt∇wqw(st, at) +The parameters of the policy are then updated according to: +θ ← θ + α2∇θµθ(st)∇aqw(st, at)|a=µθ(st) +(47) +4 +Model-based Approaches +In model-free RL agents learn to take actions directly from experiences, without +ever modelling transitions in the environment or reward functions, whereas in +model-based RL the agent attempts to learn these. The key benefit is that if +the agent can perfectly predict the environment ‘in its head’, then it no longer +needs to interact directly with the environment in order to learn an optimal +policy. +4.1 +Model Learning +Recall that MDPs are defined in terms of the 5-tuple ⟨S, A, P, r, γ⟩. Although +models can be predictions about anything, a natural starting point is to ap- +proximate the state transition function Pη ≈ P and immediate reward function +12 + +rη ≈ r. We can then use dynamic programming to learn the optimal policy for +an approximate MDP ⟨S, A, Pη, rη, γ⟩, the performance of which may be worse +than for the true MDP. +Given a fixed set of experiences, a model can be learned using supervised +methods. For predicting immediate expected scalar rewards, this is a regression +problem whereas for predicting the distribution over next states this a density +estimation problem. Given the simplicity of this framing, a range of function +approximators may be employed, including neural networks and Gaussian pro- +cesses. +4.2 +Combining model-free and model-based approaches +Once a model is learned it can be used for planning. However, in many situ- +ations it is computationally infeasible to do the full-width backups of dynamic +programming as the state space is too large. Instead, experiences can be sam- +pled from the model and used as data by a model-free algorithm. +A well known architecture which combines model-based and model-free RL is +the Dyna architecture (Sutton, 1991). Dyna treats samples of simulated and real +experience similarly, using both to learn a value function. Simulated experience +is generated by the model which is itself learned from real experience. In Dyna, +model-free based updates depend on the state the agent is currently in, whereas +for the model-based component starting states can be sampled randomly and +then rolled forwards using the model to update the value function using e.g. +TD learning. +One potential disadvantage of Dyna is that it does not preferentially treat +the state the agent is currently in. +In many cases, such as deciding on the +next move in chess, it is useful to start all rollouts from the current state (the +board position) when choosing the next move. This is known as forward search, +where a search tree is built with the current state as the root. Forward based +search often uses sample based rollouts rather than full-width ones so as to be +computationally tractable and this is known as simulation-based search. +An effective algorithm for simulation-based search is Monte-Carlo Tree search +(Coulom, 2007). It uses the MC return to estimate the action-value function for +all nodes in the search tree using the current policy. It then improves the pol- +icy, for example by being ǫ-greedy with respect to the new action-value function +(or more commonly handling exploration-exploitation using Upper Confidence +Trees, see Kocsis and Szepesv´ari (2006) for a more detailed discussion). MC +Tree Search is equivalent to MC control applied to simulated experience and +therefore is guaranteed to converge on the optimal search tree. Instead of using +MC control for search it is also possible to use TD-based control, which will +increase bias but reduce variance. +Model-based RL is a highly active area of research. Recent advances include +MuZero (Schrittwieser et al., 2020), which extends model-based predictions to +value functions and policies, and Dreamer which plans using latent variable +models (Hafner et al., 2019). +13 + +5 +Latent variables and partial observability +5.1 +Latent variable models +Hidden or ‘latent’ variables correspond to variables which are not directly ob- +served but nevertheless influence observed variables and thus may be inferred +from observation. In reinforcement learning, it can be beneficial for agents to +infer latent variables as these often provide a simpler and more parsimonious +description of the world, enabling better predictions of future states and thus +more effective control. +Latent variable models are common in the field of unsupervised learning. +Given data p(x) we may describe a probability distribution over x according to: +p(x; θx|z, θz) = +� +dz p(x|z; θx|z)p(z; θz) +(48) +where θx|z parameterises the conditional distribution x|z and θz parame- +terises the distribution over z. +Key aims in unsupervised learning include capturing high-dimensional cor- +relations with fewer parameters (as in probabilistic principal components analy- +sis), generating samples from a data distribution, describing an underlying gen- +erative process z which describes causes of x, and flexibly modelling complex +distributions even when the underlying components are simple (e.g. belonging +to an exponential family). +5.2 +Partially observable Markov decision processes +A partially observable Markov decision process (POMDP) (Kaelbling et al., +1998) is a generalisation of an MDP in which the agent cannot directly ob- +serve the true state of the system, the dynamics of which is determined by an +MDP. Formally, a POMDP is a 7-tuple ⟨S, A, P, r, O, Ω, γ⟩ where S is the set +of states, A is the set of actions, P : S × A × S → [0, 1] is the state transition +probability kernel, r : S × A × S → R is the reward function, O is the set of +observations, Ω : S × A × O → [0, 1] is the observation probability kernel and +γ ∈ [0, 1) is the discount factor. As with MDPs, agents in POMDPs seek to +learn a policy π(sα +t ) which maximises some notion of cumulative reward, com- +monly Eπ[�∞ +t=0 γtrt+1]. This policy depends on the agent’s representation of +state sα +t = f(ht), which is a function of its history. +One approach to solving POMDPs is by maintaining a belief state over the +latent environment state - transitions for which satisfy the Markov property. +Maintaining a belief over states only requires knowledge of the previous belief +state, the action taken and the current observation. Beliefs may then be updated +according to: +b′(s′) = ηΩ(o′|s′, a) +� +s∈S +P(s′|s, a)b(s) +(49) +where η = 1/ � +s′ Ω(o′|s′, a) � +s∈S P(s′|s, a)b(s) is a normalising constant. +A Markovian belief state allows a POMDP to be formulated as an MDP +where every belief is a state. However, in practice, maintaining belief states in +POMDPs will be computationally intractable for any reasonably sized problem. +In order to address this, approximate solutions may be used. +Alternatively, +14 + +agents learning using function approximators which condition on the past can +construct their own state representations, which may in turn enable relevant +aspects of the state to be approximately Markov. +6 +Deep reinforcement learning +The policies and value functions used in reinforcement learning can be learned +using artificial neural network function approximators. When such networks +have many layers they are conventionally denoted as ‘deep’, and are typically +trained on large amounts of data using stochastic gradient descent (LeCun et al., +2015). The application of deep networks in model-free reinforcement learning +garnered extensive attention when they were successfully used to learn a variety +of Atari games from scratch (Mnih et al., 2013). For the particular problem +of learning from pixels a convolutional neural network architecture was used +(LeCun et al., 1998), which are highly effective at extracting useful features +from images. They have been extensively used on supervised image classification +tasks due to their ability to scale to large and complex datasets (LeCun et al., +2015). +A deep analysis of deep reinforcement learning (DRL) is beyond the scope +of this summary. However we review two key techniques used to overcome the +technical challenge of stabilising training. +6.1 +Experience replay +As an agent interacts with its environment it receives experiences that can be +used for learning. However, rather than using those experiences immediately, it +is possible to store such experience in a ‘replay buffer’ and sample them at a later +point in time for learning. The benefits of such an approach were introduced by +Mnih et al. (2013) for their ‘deep Q-learning’ algorithm. At each timestep, this +method stores experiences et = (st, at, rt+1, st+1) in a replay buffer over many +episodes. After sufficient experience has been collected, Q-learning updates are +then applied to randomly sampled experiences from the buffer. This breaks the +correlation between samples, reducing the variance of updates and the potential +to overfit to recent experience. Further improvements to the method can be +made by prioritised (as opposed to random) sampling of experiences according to +their importance, determined using the temporal-difference error (Schaul et al., +2015). +6.2 +Target networks +When using temporal difference learning with deep function approximators a +common challenge is stability of learning. A source of instability arises when +the same function approximator is used to evaluate both the value of the current +state and the value of the target state for the temporal difference update. After +such updates, the approximated value of both current and target state change +(unlike tabular methods), which can lead to a runaway target. To address this, +deep RL algorithms often make use of a separate target network that remains +stable even whilst the standard network is updated. As it is not desirable for +the target network to diverge too far from the standard network’s improved +15 + +predictions, at fixed intervals the parameters of the standard network can be +copied to the target network. Alternatively, this transition is made more slowly +using Polyak averaging: +φtarg ← ρφtarg + (1 − ρ)φ +(50) +where φ are the parameters of the standard network and ρ is a hyperparameter +typically close to 1. +16 + +References +Sanjeevan Ahilan. Structures for Sophisticated Behaviour: Feudal Hierarchies +and World Models. PhD thesis, UCL (University College London), 2021. +Stefan Banach. Sur les op´erations dans les ensembles abstraits et leur application +aux ´equations int´egrales. Fund. math, 3(1):133–181, 1922. +Richard Bellman. A markovian decision process. Journal of mathematics and +mechanics, pages 679–684, 1957. +Dimitri P Bertsekas, Dimitri P Bertsekas, Dimitri P Bertsekas, and Dimitri P +Bertsekas. Dynamic programming and optimal control, volume 1. 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Machine learning, 8(3-4):229–256, 1992. +18 + diff --git a/htAzT4oBgHgl3EQfa_y7/content/tmp_files/load_file.txt b/htAzT4oBgHgl3EQfa_y7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34bb04bd21113d8479a4e3553489cd94ffb294e8 --- /dev/null +++ b/htAzT4oBgHgl3EQfa_y7/content/tmp_files/load_file.txt @@ -0,0 +1,851 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf,len=850 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='01379v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='AI] 3 Jan 2023 A Succinct Summary of Reinforcement Learning Sanjeevan Ahilan∗ Abstract This document is a concise summary of many key results in single- agent reinforcement learning (RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The intended audience are those who already have some familiarity with RL and are looking to review, reference and/or remind themselves of important ideas in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Contents 1 Acknowledgements 2 2 Fundamentals 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 The RL paradigm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 12 4 Model-based Approaches 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Model Learning .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 13 5 Latent variables and partial observability 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Latent variable models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Partially observable Markov decision processes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 14 6 Deep reinforcement learning 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Experience replay .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Target networks .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 15 ∗sanjeevanahilan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Much of this work was done at the Gatsby Unit, UCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 1 1 Acknowledgements I would like to thank Peter Dayan, David Silver, Chris Watkins and ChatGPT for helpful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Much of this work was drawn from David Silver’s UCL course1 and Sutton and Barto’s textbook (Sutton and Barto, 2018) and formed the introductory chapter of my PhD thesis (Ahilan, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2 Fundamentals 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 The RL paradigm The field of reinforcement learning (RL) (Sutton and Barto, 2018) concerns it- self with the computational principles underlying goal-directed learning through interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Although primarily seen as a field of machine learning, it has a rich history spanning multiple fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In psychology it can be used to model classi- cal (Pavlovian) and operant (instrumental) conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In neuroscience it has been used to model the dopamine system of the brain (Schultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In economics, it relates to fields such as bounded rationality, and in engineering it has extensive overlap with the field of optimal control (Bellman, 1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In mathematics, investigation has continued under the guise of operations research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The plethora of perspectives ensures that RL continues to be an exciting and extraordinarily interdisciplinary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Agent and environment RL problems typically draw a separation between the agent and the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The agent receives observation ot and scalar reward rt from the environ- ment and emits action at, where t indicates the time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The environment receives action at from the agent and then emits a reward rt+1 and an ob- servation ot+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The cycle then begins again with the agent emitting its next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' How the environment responds to the agent’s action is determined by the environment state st, which is updated at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The conditional distribution for the next environment state depends only on the present state and action and therefore satisfies the Markov property: P(st+1|st, at) = P(st+1|s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' , st, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' , at) (1) The environment state is in general private from the agent, which only re- ceives observations and rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The conditional distribution for the next ob- servation given the current observation is not in general Markov, and so it may be beneficial for an agent to construct its own notion of state sα t , which it uses to determine its next action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This can be defined as sα t = f(ht), where ht is the history of the agent’s sequence of observations, actions and rewards: ht = a1, o1, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' , at, ot, rt (2) 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='davidsilver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='uk/teaching/ 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='3 Observability A special case exists when the observation received by the agent ot is identical to the environment state st (such that there is no need to distinguish between the two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This is the assumption underlying the formalism of Markov decision processes covered in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An environment is partially observable if the agent cannot observe the full environment state, meaning that the con- ditional distribution for its next observation given its current observation does not satisfy the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This assumption underlies the formalism of a partially observable Markov decision process which we describe in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='4 Markov processes and Markov reward processes A Markov process (or Markov chain) is a sequence of random states with the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' It is defined in terms of the tuple ⟨S, P⟩ where S is a finite set of states and P : S × S → [0, 1] is the state transition probability kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A Markov Reward Process (MRP) ⟨S, P, r, γ⟩ extends the Markov process by including a reward function r : S × S → R for each state transition and a discount factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The immediate expected reward in a given state is defined as: r(s) = � s′ P(s, s′)r(s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The discount factor γ ∈ [0, 1] is used to determine the present value of future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Conventionally, a reward received k steps into the future is of worth γk times what it would be worth if received immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' As we will shortly see, the cumulative sum of discounted rewards is a quantity RL agents often seek to maximise, and so γ < 1 ensures that this sum is bounded (assuming r is bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='5 Markov decision processes Single-agent RL can be formalised in terms of Markov decision processes (MDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The idea of an MDP is to capture the key components available to the learning agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' the agent’s sensation of the state of its environment, the actions it takes which can affect the state, and the rewards associated with states and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An MDP extends the formalism of an MRP to include a finite set of actions on which both P and r depend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Discrete-time, infinite-horizon MDPs are described in terms of the 5-tuple ⟨S, A, P, r, γ⟩ where S is the set of states, A is the set of actions, P : S × A × S → [0, 1] is the state transition probability kernel, r : S×A×S → R is the immediate reward function and γ ∈ [0, 1) is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The expected immediate reward for a given state and action is defined as r(s, a) = � s′ P(s, a, s′)r(s, a, s′), which we use for convenience subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='6 Policies, values and models Common components of a reinforcement learning agent are a policy, value func- tion and a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The policy π : S ×A → [0, 1] is the agent’s behaviour function which denotes the probability of taking action a in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Agents may also act according to a deterministic policy µ : S → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' We will assume that policies are stochastic unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 3 Given an MDP and a policy π, the observed state sequence is a Markov process ⟨S, Pπ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Pπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' s′) = � a∈A π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)P(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' s′) (3) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' the state and reward sequence is a MRP ⟨S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Pπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' rπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' γ⟩ in which: rπ(s) = � a∈A π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) (4) Starting from any particular state s at time step t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' the value function vπ(s) is a prediction of the expected discounted future reward given that the agent starts in state s and follows policy π: vπ(s) = Eπ � ∞ � t=0 γtrt+1|s0 = s � (5) where rt+1 = r(st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' st+1) which is the solution of an associated Bellman expectation equation: vπ(s) = � a∈A π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) � r(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) + γ � s′∈S P(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' s′)vπ(s′) � (6) In matrix form the Bellman expectation equation can be expressed in terms of the induced MRP: vπ = rπ + γPπvπ = (I − γPπ)−1rπ (7) where vπ ∈ R|S| and rπ ∈ R|S| are the vector of values and expected imme- diate rewards respectively for each state under policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' We can also define a Bellman expectation backup operator: T π(v) = rπ + γPπv (8) which has a fixed point of vπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An action-value for a policy π can also be defined, which is the expected dis- counted future reward for executing action a and subsequently following policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' qπ(s, a) = r(s, a) + γ � s′∈S P(s, a, s′)vπ(s′) = r(s, a) + γ � s′∈S P(s, a, s′) � a′∈A π(s′, a′)qπ(s′, a′) (9) The process of estimating vπ or qπ is known as policy evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Policies can be evaluated without directly knowing or estimating a model, using instead the directly sampled experience of the environment, an approach which is known as ‘model-free’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However a ‘model-based’ approach is also possible in which a model is used to predict what the environment will do next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A key component of a model is an estimate of P(s, a, s′), the probability of the next state given the current state and action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Another is an estimate of r(s, a), the expected immediate reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 4 Policy evaluation enables a value function to be learned for a given policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, we often wish to learn the best possible policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The value function for this is known as the optimal value function and corresponds to the maximum value function over all policies: v∗(s) = max π vπ(s) (10) The definition of the optimal action-value function (which evaluates the im- mediate action a in state s) is similarly: q∗(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) = max π qπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) (11) A partial ordering over policies can be defined according to: π ≥ π′ if vπ(s) ≥ vπ′(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' ∀s (12) For any MDP there exists an optimal policy π∗ that is better than or equal to all other policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' All optimal policies achieve the optimal value function and optimal action-value function and there is always a deterministic optimal policy for any MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The latter is achieved by selecting: a = arg max a∈A q∗(s, a) (13) If there are many possible actions which satisfy this, any of these may be chosen to constitute an optimal policy (of which there may be many).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The optimal value and state-value functions satisfy Bellman optimality equations: v∗(s) = max a∈A q∗(s, a) v∗(s) = max a∈A � r(s, a) + γ � s′∈S P(s, a, s′)v∗(s′) � q∗(s, a) = r(s, a) + γ � s′∈S P(s, a, s′)max a′ q∗(s′, a′) (14) The Bellman optimality equation is non-linear with no closed form solution (in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Solving it therefore requires iterative solution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='7 Dynamic programming Dynamic programming (DP) (Bertsekas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 1995) refers to a collection of algorithms that can be used to compute optimal policies given a perfect model of the environment as an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In general, DP solves complex problems by breaking them down into subproblems and then combining the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' It is particularly useful for overlapping subproblems, the solutions to which reoccur many times when solving the overall problem, making it more computationally efficient to cache and reuse them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' When applied to MDPs, the recursive decomposition of DP corresponds to the Bellman equation and the cached solution to the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' DP assumes that the MDP is fully known and therefore does not address the full RL problem but instead addresses the problem of planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' By planning, the prediction problem can be addressed by finding the value function vπ of a given policy 5 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This can be evaluated by iterative application of the Bellman Expectation Backup (Equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This leads to convergence to a unique fixed point vπ, which can be shown using the contraction mapping theorem (also known as the Banach fixed-point theorem) (Banach, 1922).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' When a Bellman expectation backup operator T π is applied to two value functions u and v over states, we find that it is a γ- contraction: ||T π(u) − T π(v)||∞ = ||(rπ + γPπu) − (rπ + γPπv)||∞ = ||γPπ(u − v)||∞ ≤ ||γPπ1||u − v||∞||∞ ≤ γ||u − v||∞ (15) where 1 is a vector of ones and the infinity norm of a vector a is denoted ||a||∞ and is defined as the maximum value of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This contraction ensures that both u and v converge to the unique fixed point of T π which is vπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For control, DP can be used to find the optimal value function v∗ and in turn the optimal policy π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' One possibility is policy iteration in which the current policy π is first evaluated as described and then subsequently improved to π′ such that: π′(s) = arg max a∈A qπ(s, a) (16) This improves the value from any state s over one step: qπ(s, π′(s)) = max a∈A qπ(s, a) ≥ � a∈A π(s, a)qπ(s, a) = vπ(s) (17) It can be shown that this improves the value function such that that vπ′(s) ≥ vπ(s) (Silver, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This process is then repeated, with improvements ending when the Bellman optimality equation (14) has been satisfied and convergence to π∗ achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A generalisation of policy iteration is also possible in which, instead of waiting for policy evaluation to converge, only n steps of evaluation are taken before policy improvement occurs and the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' If n = 1 this is known as value iteration, as the policy is no longer explicit (being a direct consequence of the value function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Like policy iteration, value iteration is also guaranteed to converge to the optimal value function and policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This can be demonstrated using the contraction mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 3 Model-free approaches 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Prediction As has been outlined, dynamic programming can be used to solve known MDPs enabling optimal value functions and policies to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, in many cases the MDP is not directly known - instead an agent taking actions in the MDP must learn directly from its experiences, as it transitions from state to state and receives rewards accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' One approach, known as ‘model-free’, seeks to solve MDPs without learning transitions or rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For prediction, a 6 key quantity to estimate in this setting is the expected discounted future reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A sampled estimate of this, starting from state st, is known as the return: Rt = rt+1 + γrt+2 + γ2rt+3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' = ∞ � k=0 γkrt+k+1 (18) which depends on the actions sampled from the policy, and states from transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Monte-Carlo (MC) methods seek to estimate this directly using complete episodes of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Introducing a learning rate αt, the agent’s value function can therefore be updated according to2: v(st) ← v(st) + αt � Rt − v(st) � (19) The value function updated in this way will converge to a solution with min- imum mean-square error (best fit to the observed returns), assuming a suitable sequential decrease in the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Temporal-difference (TD) learning methods learn from incomplete episodes by bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For example, if learning occurs after a single step, this is known as TD(0), which has the following update: v(st) ← v(st) + αt � rt+1 + γv(st+1) − v(st) � (20) where rt+1 + γv(st+1) is known as the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This approximates the full-width Bellman expectation backup (Equation 8) in which every successor state and action is considered, with experiences instead being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' TD(0) will con- verge to the solution of the maximum likelihood Markov model which best fits the data (again assuming a suitable sequential decrease in the learning rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This solution may be different from the minimum mean-square error solution of MC methods, which do not assume the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Unlike MC methods, TD methods introduce bias into the estimated return as the currently estimated value function may be different from the true value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, they generally have reduced variance relative to MC meth- ods, as in MC the estimated return depends on a potentially long sequence of random actions, transitions and rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The distinction between MC and TD methods can be blurred by considering multi-step TD methods (rather than only TD(0)), in which rewards are sampled for a number of steps before the value function is used to compute an estimate of future rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The n-step return is defined as: R(n) t = rt+1 + γrt+2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' + γn−1rt+n + γnv(st+n) (21) As n → ∞ it tends towards the unbiased MC return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An algorithm may seek to find a good bias-variance tradeoff by estimating a weighted combination of n-step returns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' one popular method to do this is known as TD(λ): Rλ t = (1 − λ) ∞ � n=1 λn−1R(n) t (22) where λ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 2assuming a table-based representation rather than use of a function approximator 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Control with action-value functions Model free control concerns itself with optimising rather than evaluating the RL objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Policies may be evaluated according to various objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In the case of continuing environments, the objective can be the average value or the average reward per time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' We focus instead on episodic environments, assuming an initial distribution over starting states p0(s) : S → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The objective is thus: J(π) = Eπ � ∞ � t=0 γtrt+1|p0(s) � (23) Note that if the domain of the starting state distribution is only over a single starting state, the objective is simply the value function (Equation 5) in that starting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This objective can equivalently be expressed as: J(π) = Es∼ρπ,a∼π[r(s, a)] (24) where: ρπ(s) := � s′ ∞ � t=0 γtp(st = s|s′, π)p0(s′) (25) is the improper discounted state distribution induced by policy π starting from an initial state distribution p0(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='4 we describe policy gradient methods which seek to optimise this objective directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, we first consider model-free approaches which rely on an action- value function q(s, a) to achieve control (a value function v(s) alone is insufficient for model-free control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The optimal action-value function q∗(s, a) must be learned, with MC and TD methods both viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Once it has been learned, an optimal policy may be achieved by selecting the best action in each state (Equation 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, unlike dynamic programming, full-width backups are not used and so if actions are selected greedily (meaning those with highest action-values are always chosen) then certain states and actions may never be correctly evalu- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Model-free RL methods must therefore allow for enough exploration dur- ing learning before ultimately exploiting this learning to achieve near-optimal cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' One simple approach, known as ǫ-greedy is to take a random action with probability ǫ but otherwise act greedily according to the current estimate of the action-value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The value of ǫ can be decreased with the number of episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This can satisfy a condition known as greedy in the limit of infinite exploration in which all state-action pairs are explored infinitely many times and the policy converges to the greedy policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' One popular algorithm for model-free control is known as Q-learning (Watkins and Dayan, 1992), which seeks to learn the optimal action-value function whilst using a pol- icy which also takes exploratory actions (such as epsilon greedy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This learning is termed off-policy as the policy used to sample experience is different from the policy being learned (the optimal policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The resulting update is: q(st, at) ← q(st, at) + α � rt+1 + γmax a′∈A q(st+1, a′) − q(st, at) � (26) 8 An alternative to off-policy Q-learning is on-policy SARSA (Rummery and Niranjan, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This uses the sampled sampled state st, action at, reward rt+1, next state st+1, and next action at+1 for updates3: q(st, at) ← q(st, at) + α(rt+1 + γq(st+1, at+1) − q(st, at)) (27) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='3 Value function approximation So far we have assumed a tabular representation of states and actions such that each state is separately updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, in practice we would like value functions and policies to generalise to new states and actions, and so it is ben- eficial to use function approximators such as deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A common approach is to approximate the value function or action-value function: vw(s) = ˆv(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' w) ≈ vπ(s) qw(s, a) = ˆq(s, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' w) ≈ qπ(s, a) (28) where w are the parameters we wish to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' If we start by assuming we know the true value function vπ, we can define a mean square error between the approximate value function and the true function: L(w) = Eπ[(vπ(s) − vw(s))2] (29) Given a distribution of states s ∼ p(s)4, we can minimise this iteratively using stochastic gradient descent: w ← w + α(vπ(st) − vw(st))∇wvw(st) (30) In reality we can only use a better estimate of vπ provided by the sampled reward(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For example, if we use the TD(0) target the update is: w ← w + α(rt+1 + γvw(st+1) − vw(st))∇wvw(st) (31) Updates like this are known as ‘semi-gradient’ as the gradient of the value function used to define the target is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' If we use a linear function approximator vw(s) = x(s)T w (where features x(s) and w are vectors), then we find: w ← w + α(rt+1 + γvw(st+1) − vw(st))x(st) (32) indicating that the linear weights are updated in proportion to the activity of their corresponding features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Non-linear function approximators can also be used, but typically have weaker convergence guarantees than linear function approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Nevertheless, due to their flexibility such approximators have enabled impressive performance in a number of challenging domains, such as Atari games (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2015) and Go (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 3and also gives SARSA its name 4we later discuss a method for sampling states 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='4 Policy gradient methods Parameterised stochastic policies πθ may be improved using the policy gradient theorem (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This can be derived for any of the common RL objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' To demonstrate a derivation of this result we use a starting state objective J(θ) = vπθ(s0) with a single starting state s0: ∇θJ(θ) = ∇θvπ(s0) = ∇θ � a π(s0, a)qπ(s0, a) = � a ∇θπ(s0, a)qπ(s0, a) + π(s0, a)∇θqπ(s0, a) = � a ∇θπ(s0, a)qπ(s0, a) + π(s0, a)∇θ � r(s0, a) + � s′ γP(s0, a, s′)vπ(s′) � = � a ∇θπ(s0, a)qπ(s0, a) + π(s0, a) � s′ γP(s0, a, s′)∇θvπ(s′) (33) We note that we could continue to unroll ∇θvπ(s′) on the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='S in the same way as we have already done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Considering now transitions from starting state s0 to arbitrary state s we therefore find: ∇θvπ(s0) = � s ∞ � t=0 γtp(st = s|s0, π) � a ∇θπ(s, a)qπ(s, a) (34) where �∞ t=0 γtp(st = s|s0, π) is the discounted state distribution ρπ(s) from a fixed starting state s0 (Equation 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This derivation holds even when there is a distribution over starting states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' and gives us the policy gradient theorem: ∇θJ(θ) = � s ρπ(s) � a ∇θπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)qπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) (35) Using the likelihood ratio trick: ∇θπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) = π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)∇θπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) = π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)∇θ log π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) (36) this can be equivalently expressed as: ∇θJ(θ) = � s ρπ(s) � a π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)qπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)∇θ log π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a) = Eπ[qπ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)∇θ log π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' a)] (37) The policy gradient theorem result enables model-free learning as gradi- ents need only be determined for the policy rather than for properties of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' There are a variety of approaches for determining qπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' If qπ is ap- proximated using the sample return (Equation 18), this leads to the algorithm known as REINFORCE (Williams, 1992): θ ← θ + αRt∇θ log π(st, at) (38) 10 As there is no bootstrapping here, this is also known as MC policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An alternative approach is to separately approximate qπ with a ‘critic’ qw giving rise to what are commonly known as ‘actor-critic’ methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' These introduce two sets of parameter updates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' the critic parameters w are updated to approximate qπ, and the policy (actor) parameters θ are updated according to the policy gradient as indicated by the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The critic itself can be updated according to the TD error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An example of this approach is SARSA actor-critic: w ← w + α1(rt+1 + γqw(st+1, at+1) − qw(st, at))∇wqw(st, at) θ ← θ + α2qw(st, at)∇θ log π(st, at) (39) where different learning rates α1 and α2 may be used for the actor and the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='5 Baselines Whether we use REINFORCE or an actor-critic based approach to policy gradi- ents, it is possible to reduce the variance further by the introduction of baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' If this baseline depends only on the state s, then we find it introduces no bias: � s ρπ(s) � a ∇θπ(s, a)b(s) = � s ρπ(s)b(s)∇θ � a π(s, a) = � s ρπ(s)b(s)∇θ1 = 0 (40) A natural choice for the state-dependent baseline is the value function: ∇θJ(θ) = Eπ[(qπ(s, a) − vπ(s))∇θ log π(s, a)] = Eπ[Aπ(s, a)∇θ log π(s, a)] (41) where Aπ is known as the advantage, which may in some algorithms be approx- imated directly (rather than approximating both qπ and vπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='6 Compatible function approximation In the general case, our choice to approximate qπ with qw introduces bias such that there are no guarantees of convergence to a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, in the special case of a compatible function approximator we can introduce no bias and take steps in the direction of the true policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This becomes possible when the critic’s function approximator reaches a minimum in the mean-squared error: 0 = Eπ[∇w(qπ(s, a) − qw(s, a))2] = Eπ[(qπ(s, a) − qw(s, a))∇wqw(s, a)] (42) If we choose qw(s, a) such that ∇wqw(s, a) = ∇θ log π(s, a) we find: Eπ[qπ(s, a)∇θ log π(s, a)] = Eπ[qw(s, a)∇θ log π(s, a)] (43) 11 where the L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='S is equal to the true policy gradient and so our function ap- proximation has introduced no bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For example, if the policy is a Boltzmann policy with a linear combination of features, of the form: π(s, a) = eθT φ(s,a) � a′ eθT φ(s,a′) (44) then a compatible value function must be linear in the same features as the policy except normalised to zero mean for each state using a subtractive baseline (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' qw(s, a) = wT [φ(s, a) − � a′ φ(s, a′)π(s, a′)] (45) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='7 Deterministic policy gradients Rather than have a policy specify a probability for certain actions in certain states we can instead have it simply be a function mapping states to actions µθ : S → A and, in the case of continuous actions, seek to find the gradient of the objective with respect to the policy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An example of an al- gorithm which uses such an approach is Deterministic Policy Gradients (DPG) (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The DPG algorithm builds on the deterministic policy gradient theorem: ∇θJ(θ) = Es∼ρµ[∇θµθ(s)∇aqµ(s, a)|a=µθ(s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' (46) where the parameters of the policy are adjusted in an off-policy fashion using an exploratory behavioural policy (which is a noisy version of the deterministic policy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In practice qµ is approximated by the critic qw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' which is differentiable in the action and updated using Q-learning: δt = rt+1 + γqw(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' µθ(st+1)) − qw(st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' at) w ← w + α1δt∇wqw(st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' at) The parameters of the policy are then updated according to: θ ← θ + α2∇θµθ(st)∇aqw(st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' at)|a=µθ(st) (47) 4 Model-based Approaches In model-free RL agents learn to take actions directly from experiences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' without ever modelling transitions in the environment or reward functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' whereas in model-based RL the agent attempts to learn these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The key benefit is that if the agent can perfectly predict the environment ‘in its head’, then it no longer needs to interact directly with the environment in order to learn an optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Model Learning Recall that MDPs are defined in terms of the 5-tuple ⟨S, A, P, r, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Although models can be predictions about anything, a natural starting point is to ap- proximate the state transition function Pη ≈ P and immediate reward function 12 rη ≈ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' We can then use dynamic programming to learn the optimal policy for an approximate MDP ⟨S, A, Pη, rη, γ⟩, the performance of which may be worse than for the true MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Given a fixed set of experiences, a model can be learned using supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For predicting immediate expected scalar rewards, this is a regression problem whereas for predicting the distribution over next states this a density estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Given the simplicity of this framing, a range of function approximators may be employed, including neural networks and Gaussian pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Combining model-free and model-based approaches Once a model is learned it can be used for planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, in many situ- ations it is computationally infeasible to do the full-width backups of dynamic programming as the state space is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Instead, experiences can be sam- pled from the model and used as data by a model-free algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A well known architecture which combines model-based and model-free RL is the Dyna architecture (Sutton, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Dyna treats samples of simulated and real experience similarly, using both to learn a value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Simulated experience is generated by the model which is itself learned from real experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In Dyna, model-free based updates depend on the state the agent is currently in, whereas for the model-based component starting states can be sampled randomly and then rolled forwards using the model to update the value function using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' TD learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' One potential disadvantage of Dyna is that it does not preferentially treat the state the agent is currently in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In many cases, such as deciding on the next move in chess, it is useful to start all rollouts from the current state (the board position) when choosing the next move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This is known as forward search, where a search tree is built with the current state as the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Forward based search often uses sample based rollouts rather than full-width ones so as to be computationally tractable and this is known as simulation-based search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' An effective algorithm for simulation-based search is Monte-Carlo Tree search (Coulom, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' It uses the MC return to estimate the action-value function for all nodes in the search tree using the current policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' It then improves the pol- icy, for example by being ǫ-greedy with respect to the new action-value function (or more commonly handling exploration-exploitation using Upper Confidence Trees, see Kocsis and Szepesv´ari (2006) for a more detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' MC Tree Search is equivalent to MC control applied to simulated experience and therefore is guaranteed to converge on the optimal search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Instead of using MC control for search it is also possible to use TD-based control, which will increase bias but reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Model-based RL is a highly active area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Recent advances include MuZero (Schrittwieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2020), which extends model-based predictions to value functions and policies, and Dreamer which plans using latent variable models (Hafner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 13 5 Latent variables and partial observability 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Latent variable models Hidden or ‘latent’ variables correspond to variables which are not directly ob- served but nevertheless influence observed variables and thus may be inferred from observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In reinforcement learning, it can be beneficial for agents to infer latent variables as these often provide a simpler and more parsimonious description of the world, enabling better predictions of future states and thus more effective control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Latent variable models are common in the field of unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Given data p(x) we may describe a probability distribution over x according to: p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' θx|z, θz) = � dz p(x|z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' θx|z)p(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' θz) (48) where θx|z parameterises the conditional distribution x|z and θz parame- terises the distribution over z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Key aims in unsupervised learning include capturing high-dimensional cor- relations with fewer parameters (as in probabilistic principal components analy- sis), generating samples from a data distribution, describing an underlying gen- erative process z which describes causes of x, and flexibly modelling complex distributions even when the underlying components are simple (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' belonging to an exponential family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Partially observable Markov decision processes A partially observable Markov decision process (POMDP) (Kaelbling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 1998) is a generalisation of an MDP in which the agent cannot directly ob- serve the true state of the system, the dynamics of which is determined by an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Formally, a POMDP is a 7-tuple ⟨S, A, P, r, O, Ω, γ⟩ where S is the set of states, A is the set of actions, P : S × A × S → [0, 1] is the state transition probability kernel, r : S × A × S → R is the reward function, O is the set of observations, Ω : S × A × O → [0, 1] is the observation probability kernel and γ ∈ [0, 1) is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' As with MDPs, agents in POMDPs seek to learn a policy π(sα t ) which maximises some notion of cumulative reward, com- monly Eπ[�∞ t=0 γtrt+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This policy depends on the agent’s representation of state sα t = f(ht), which is a function of its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' One approach to solving POMDPs is by maintaining a belief state over the latent environment state - transitions for which satisfy the Markov property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Maintaining a belief over states only requires knowledge of the previous belief state, the action taken and the current observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Beliefs may then be updated according to: b′(s′) = ηΩ(o′|s′, a) � s∈S P(s′|s, a)b(s) (49) where η = 1/ � s′ Ω(o′|s′, a) � s∈S P(s′|s, a)b(s) is a normalising constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A Markovian belief state allows a POMDP to be formulated as an MDP where every belief is a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, in practice, maintaining belief states in POMDPs will be computationally intractable for any reasonably sized problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' In order to address this, approximate solutions may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Alternatively, 14 agents learning using function approximators which condition on the past can construct their own state representations, which may in turn enable relevant aspects of the state to be approximately Markov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 6 Deep reinforcement learning The policies and value functions used in reinforcement learning can be learned using artificial neural network function approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' When such networks have many layers they are conventionally denoted as ‘deep’, and are typically trained on large amounts of data using stochastic gradient descent (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The application of deep networks in model-free reinforcement learning garnered extensive attention when they were successfully used to learn a variety of Atari games from scratch (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' For the particular problem of learning from pixels a convolutional neural network architecture was used (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 1998), which are highly effective at extracting useful features from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' They have been extensively used on supervised image classification tasks due to their ability to scale to large and complex datasets (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A deep analysis of deep reinforcement learning (DRL) is beyond the scope of this summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However we review two key techniques used to overcome the technical challenge of stabilising training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='1 Experience replay As an agent interacts with its environment it receives experiences that can be used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' However, rather than using those experiences immediately, it is possible to store such experience in a ‘replay buffer’ and sample them at a later point in time for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' The benefits of such an approach were introduced by Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' (2013) for their ‘deep Q-learning’ algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' At each timestep, this method stores experiences et = (st, at, rt+1, st+1) in a replay buffer over many episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' After sufficient experience has been collected, Q-learning updates are then applied to randomly sampled experiences from the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' This breaks the correlation between samples, reducing the variance of updates and the potential to overfit to recent experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Further improvements to the method can be made by prioritised (as opposed to random) sampling of experiences according to their importance, determined using the temporal-difference error (Schaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content='2 Target networks When using temporal difference learning with deep function approximators a common challenge is stability of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' A source of instability arises when the same function approximator is used to evaluate both the value of the current state and the value of the target state for the temporal difference update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' After such updates, the approximated value of both current and target state change (unlike tabular methods), which can lead to a runaway target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' To address this, deep RL algorithms often make use of a separate target network that remains stable even whilst the standard network is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' As it is not desirable for the target network to diverge too far from the standard network’s improved 15 predictions, at fixed intervals the parameters of the standard network can be copied to the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Alternatively, this transition is made more slowly using Polyak averaging: φtarg ← ρφtarg + (1 − ρ)φ (50) where φ are the parameters of the standard network and ρ is a hyperparameter typically close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 16 References Sanjeevan Ahilan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' Structures for Sophisticated Behaviour: Feudal Hierarchies and World Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' PhD thesis, UCL (University College London), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} 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learning, 8(3-4):229–256, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htAzT4oBgHgl3EQfa_y7/content/2301.01379v1.pdf'} diff --git a/j9AyT4oBgHgl3EQf_Pqt/vector_store/index.pkl b/j9AyT4oBgHgl3EQf_Pqt/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a9d6f33f3a54b6b33a8465e240f3d7f9cee5601b --- /dev/null +++ b/j9AyT4oBgHgl3EQf_Pqt/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d2b7a95eed26227422c42895e923c9eb7a44d5a888f0fd285185f307293ea4bd +size 167053 diff --git a/j9E4T4oBgHgl3EQfsw0a/content/2301.05218v1.pdf b/j9E4T4oBgHgl3EQfsw0a/content/2301.05218v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..752d5c411cceb667713592a96d92b2698714d2ef --- /dev/null +++ b/j9E4T4oBgHgl3EQfsw0a/content/2301.05218v1.pdf @@ 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+size 72494 diff --git a/jtE0T4oBgHgl3EQf7QKQ/content/tmp_files/2301.02774v1.pdf.txt b/jtE0T4oBgHgl3EQf7QKQ/content/tmp_files/2301.02774v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e671496e49de2ee34289e05640ca689c95699e0d --- /dev/null +++ b/jtE0T4oBgHgl3EQf7QKQ/content/tmp_files/2301.02774v1.pdf.txt @@ -0,0 +1,3179 @@ +arXiv:2301.02774v1 [nlin.SI] 7 Jan 2023 +Second order Killing tensors related to symmetric spaces +E.O.Porubov, A.V.Tsiganov +St.Petersburg State University, St.Petersburg, Russia +e–mail: evg.porub@gmail.com, andrey.tsiganov@gmail.com +Abstract +We discuss the pairs of quadratic integrals of motion belonging to the n-dimensional +space of independent integrals of motion in involution, that provide integrability of the +corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisen- +hart theory, additional integrals of motion are polynomials of the fourth, sixth and other +orders in momenta. The main focus is on the second-order Killing tensors corresponding +to quadratic integrals of motion and relating to the special combinations of rotations and +translations in Euclidean space. +Keywords: Killing tensors, integrable systems, symmetric spaces +1 +Introduction +Let A and B be non-degenerate symmetric second-order tensor fields on Euclidean space Rn. +If the Schouten bracket between them is zero +[[A, B]] = 0 +and the eigenvalue problem +(A − λB)ψ = 0 +(1.1) +has n simple eigenvalues and normal eigenvectors, then A and B generate a n-dimensional linear +space of second-order tensor fields, all in involution and with common eigenvectors. It allows +us to calculate n independent functions on the cotangent bundle T ∗Rn +T1 = +� +ij +Aijpipj , +T2 = +� +ij +Bijpipj , +T3 = +� +ij +Kij +3 pipj, +. . . , +Tn = +� +ij +Kij +n pipj +in involution +{Ti, Tj} = 0 , +with respect to canonical Poisson brackets +{qi, qj} = 0 , +{pi, pj} = 0 , +{qi, pj} = δij , +i, j = 1, . . . , n . +By adding suitable potentials +H1 = T1 + V1(q1, . . . , qn) , +H2 = T2 + V2(q1, . . . , qn), +. . . , +Hn = Tn + Vn(q1, . . . , qn) +we obtain the n-dimensional space of first integrals in involution [5], see also [1, 10, 12, 13]. +Thus, second-order tensors A and B define the completely integrable system, if they satisfy +a set of conditions in Rn which can be verified without an explicit calculation of all the integrals +of motion. +In [22, 23, 24, 25] we found a few pairs of the second-order tensors A and B which also de- +fines integrable and superintegrable systems on T ∗Rn, but the corresponding eigenvalue problem +(1.1) has no simple eigenvalues and normal eigenvectors. Our main aim is to construct enough +1 + +examples to find the new criterion that two tensors A and B in Rn define a completely integrable +system on T ∗Rn. It is natural to say that these tensors A and B are completely integrable. +In this note, we consider tensors A and B related to the special linear combinations of +rotation and translations in Euclidean space Rn. They are associated with Newton’s equations +of motion +¨qα = +� +β,γ,δ +Rα +β,γ,−δqβqγqδ − ωαqα, +α, β, γ, δ = 1, . . . , N +(1.2) +and Hamiltonian +H = 1 +2 +� +α +gα,−αp2 +α − 1 +4 +� +α,β,γ,δ +R−α,β,γ,−δqαqβqγqδ + 1 +2 +� +α +ωα (qα)2 +(1.3) +which were studied in [7, 8, 18]. Here gα,−α and Rα +β,γ,−δ are metric and curvature tensors +(constant tensors), which correspond to classes A.III, BD.I, C.I and D.III of symmetric spaces +in the Cartan classification. +In this case, A = g is metric in Euclidean space and B = K is a Killing tensor, which +satisfies the Killing equation +∇iKjk + ∇jKki + ∇kKij = 0, +(1.4) +where ∇ is the Levi-Civita connection of g. +Relation of these Hamiltonian systems with the generalised multicomponent NLS hierarchy +gives the Lax matrix +L(µ) = µ2A + µ +� +α +qα� +eα − e−α +� +− 1 +a +� +α +gα,−αpα +� +eα + e−α +� ++ 1 +a +� +α,β +qαqβ[eα, e−β] + Λ (1.5) +and integrals of motion in the involution. Here matrix Λ depends on parameters ωi and describes +a shift of the orbit obtained in the framework of the Adler-Kostant-Symes theorem [18]. All +these results were reproduced in the various textbooks [17, 19, 20, 21], where readers can find +all the necessary definitions and details, see also [9] and references within. +When all ωα = 0, Hamiltonian H (1.3) commutes with a family of the noncommutative +linear integrals of motion associated with the various combinations of rotations. In this case, +the Lax matrix allows us to find a family of commuting integrals of motion, the numbers of +which do not permit us to talk about integrability by the Liouville theorem in the general case. +If ωα ̸= 0 the Lax matrix L(µ) (1.5) generates a necessary number of the integrals of +motion, which are polynomials in momenta of order two, four, six, eight, etc. The leading parts +of these polynomials , +H(2ℓ) +i += +2ℓ +� +jk...m +Kjk...m +i +pjpk · · · pm + · · · , +ℓ = 1, 2, . . . , +define the Killing tensors of valency 2ℓ in Euclidean space Rn +[[g, Ki]] = 0 , +where [[., .]] is a Schouten bracket. Below we will restrict ourselves to the study of second-order +Killing tensors in a few low-dimensional Euclidean spaces. +Proposition 1 The Hamiltonian H (1.3) is in involution with the polynomial of fourth order +in momenta +G = −1 +4 +� +α,β,γ,δ +R−α,β,γ,−δpαpβpγpδ + +� +α,β +Sα,β(q)pαpβ + W(q) , +whose leading part is defined by the curvature tensor R (1.3) . +2 + +We can prove this proposition using properties of the Cartan-Weil basis, definitions of the +Killing form, and metric and curvature tensors on symmetric spaces. It is beyond the scope of +this article to discuss these properties and fully prove this proposition. So, we only single out +this integral of motion from other coefficients of the equation defining the spectral curve of the +Lax matrix (1.5). +1.1 +Killing tensors of valency two +In Euclidean space, the generic Killing tensor of valency two is given by +K = +� +i,j +aijXi ◦ Xj + +� +i,j,k +bijkXi ◦ Xj,k + +� +i,j,k,m +cijkmXi,j ◦ Xk,m , +(1.6) +where +Xi = ∂i +Xi,j = qiXj − qjXi , +∂k = +∂ +∂qk +is a basis of translations and rotations, aij, bijk and cijm are parameters and ◦ denotes symmetric +product. +Dimension of vector space of the Killing tensors of valency m in n-dimensional Euclidean +space is given by the Delong-Takeuchi-Thompson formula +d = 1 +n +�n + m +m + 1 +��n + m − 1 +m +� += 1 +n +�n + 2 +3 +��n + 1 +2 +� += n(n + 2)(n + 1)2 +12 +, +where we put m = 2 calculating the number of parameters aij, bijk and cijkm. +Thus, we can find all the Killing tensors of valency two related to Hamiltonian H = T + V +(1.3) solving the equation +d (KdV ) = 0 , +(1.7) +which means that 1-form KdV is an exact. Here V is a function on Rn, canonically lifted to +T ∗Rn and KdV denotes the 1-form image of dV by K, interpreted as a linear endomorphism +over 1-forms, whose components are gα,βKβ,γ∂γV . +Substituting K (1.6) and potential +V = 1 +4 +� +α,β,γ,δ +R−α,β,γ,−δqαqβqγqδ − 1 +2 +� +α +ωα (qα)2 +into (1.7) we obtain a linear system of equations for coefficients aij, bijk and cijkm. Then we +can study the properties of the obtained solutions. Construction of the polynomials of higher +order in momenta commuting with the Hamiltonian H is an open question in this approach. +According to [1, 5, 12] Sylvester’s criterion can be used to verify that K has real and simple +eigenvalues with respect to metric g. The vanishing of the Haantjes torsion of K allows us to +prove that eigenvectors are normal, see historical remarks in [12]. +The Haantjes tensor (torsion) +HK(u, v) = K2NK(u, v) + NK(Ku, Kv) − K (NK(Ku, v) + NK(u, Kv)) , +is defined by using the Nijenhuis tensor (torsion) +NK(u, v) = K2[u, v] + [Ku, Kv] − K ([Ku, v] + [u, Kv]) , +where u, v are arbitrary vector fields and [., .] denotes the commutator of two vector fields [12]. +Below we prove that associated with Hamiltonian H (1.3) Killing tensors of valency two +have non-zero Haantjes torsion. Nevertheless, using Lax matrix L(µ) (1.5) we can get a set of +completely integrable Killing tensors related to the special values of parameters aij, bijk and +cijkm in (1.6). It could be useful to construct similar Killing tensors on non-Euclidean space +[6, 24, 25, 27]. +3 + +2 +Symmetric spaces of A.III type +Let us consider equations of motion (1.2) in Euclidean space Rmn and Hamiltonian H (1.3) +associated with the Riemannian pair +SU(m + n)/S +� +U(m) × U(n) +� +, +m ≤ n , +n + m ≥ 4 . +The typical representation of su(m + n) is a set of (m + n) × (m + n) matrices with an obvious +block-matrix structure associated with the following Cartan decomposition +g ≡ k ⊕ p, +k = s(u(m) ⊕ u(n)) . +Here k consists of block-diagonal matrices, while the linear space p is spanned by block-off- +diagonal matrices: +k ≃ +� +u(m) +0 +0 +u(n) +� +, +p ≃ +� +0 +P + iQ +P T − iQT +0 +� +. +The corresponding Cartan element A in (1.5) acts as −I on so(m) and as I on so(n). +Following to [7, 8] we choose a representation in which Lax matrix (1.5) reads as +L(µ) = + + +a − 2µ2Im +0 +0 +b + 2µ2In + + + + + +C +P − 2iµQ +P T + 2iµQT +¯C + + , +i = +√ +−1 +(2.1) +where Im and In are the m×m and n×n unit matrices, a and b are diagonal matrices depending +on m real numbers ak and n real numbers parameters bi +a = diagm(a1, . . . , am) , +b = diagn(b1, . . . , bn) , +ai, bi ∈ R , +and T means matrix transposition. +Matrices C and ¯C are symmetric m × m and n × n matrices with entries depending on +coordinates +Ci,j = − +n +� +k=1 +q(i−1)n+k q(j−1)n+k , +i, j = 1, . . . , m, +and +¯Ci,j = − +m−1 +� +k=0 +qkn+iqkn+j , +i, j = 1, . . . , n. +Matrices P and Q are m × n matrices depending linearly on p and q with entries +Pij = p(i−1)n+j , +and +Qij = q(i−1)n+j , +i = 1, . . . , m, +j = 1, . . . , n , +see examples below. +The Hamiltonian (1.3) is given by +H = 1 +4TraceL2 +���� +µ=0 +− 1 +4 +m +� +j=1 +a2 +j − 1 +4 +n +� +i=1 +b2 +i = 1 +2 +n +� +i=1 +p2 +i + 1 +2 +m−1 +� +j=0 +� n +� +i=1 +q2 +jn+i +�2 +(2.2) ++ +m−1 +� +k,j=0;k>j +� n +� +i=1 +qjn+iqkn+i +�2 ++ 1 +2 +m−1 +� +j=0 +aj+1 +� n +� +i=1 +q2 +jn+i +� +− 1 +2 +n +� +i=1 +bi + + +m−1 +� +j=0 +q2 +jn+i + + . +When ai ̸= 0 and bi ̸= 0, there are two basic sets of integrals of motion obtained from the +characteristic polynomial of the Lax matrix +τ(z, µ) = det +� +z I − L(µ) +� +, +which are associated with so(m) and so(n), respectively. Because +{τ(x, λ), τ(y, µ)} = 0 , +all these integrals of motion are in the involution for each other. +4 + +2.1 +First set of the independent integrals of motion +The m residues of the function +∆1(z, µ) = +τ(z, µ) +�m +i=1(z − ai + 2µ2) +(2.3) +at z = ai − 2µ2 generate mn independent integrals of motion h(2ℓ) +i +Residue ∆1(z, µ)|z=ai−2µ2 = +n−1 +� +k=0 +µ2kh +� +2(n−k) +� +i +, +i = 1, . . . , m, +which are polynomials of degree at most 2m since we take m ≤ n. So, there are +• m quadratic polynomials in momenta h(2) +1 , . . . , h(2) +m ; +• m quartic polynomials in momenta h(4) +1 , . . . , h(4) +m ; +• m sextic polynomials in momenta h(6) +1 , . . . , h(6) +m ; +• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +• m polynomials of 2m-order in momenta h(2m) +1 +, . . . , h(2m) +m +and m(n − m) remaining polynomials of 2m-order in momenta. +Polynomials of the second order in momenta have the following form +h(2) +i += − +m +� +k̸=i +M 2 +ik +ai − ak ++ ti(p) + vi(q) , +(2.4) +where functions +Mik = +n +� +Jjℓ , +Jjℓ = qjpℓ − qℓpj , +constitute realization of Lie algebra so∗(m) associated with compositions of n simple rotations +in Rmn. +Functions ti(p) correspond to compositions of the n translations +ti(p) = +n +� +p2 +ℓ , +and vi(q) are polynomials of the fourth order in coordinates qi. +2.2 +Second set of the independent integrals of motion +The n residues of the function +∆2(z, µ) = +τ(z, µ) +�n +i=1(z − bi − 2µ2) +(2.5) +at z = bi + 2µ2 generate mn independent integrals of motion H(2ℓ) +i +Residue ∆2(z, µ)|z=bi+2µ2 = +m−1 +� +k=0 +µ2kH +� +2(m−k) +� +i +, +i = 1, . . . , n +which are polynomials of order 2ℓ in momenta. So, there are +• n quadratic polynomials in momenta H(2) +1 , . . . , H(2) +n ; +5 + +• n quartic polynomials in momenta H(4) +1 , . . . , H(4) +n ; +• n sextic polynomials in momenta H(6) +1 , . . . , H(6) +n ; +• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +• n polynomials of 2m-order in momenta H(2m) +1 +, . . . , H(2m) +n +. +In this case polynomials of second order in momenta have the following form +H(2) +i += +n +� +k̸=i +N 2 +ik +bi − bk ++ Ti(p) + Ui(q) , +(2.6) +where functions +Nik = +m +� +Jjℓ , +Jjℓ = qjpℓ − qℓpj , +form realization of so∗(n) via compositions of m simple rotations in Rmn. +Functions Ti(p) correspond to compositions of the m translations +Ti(p) = +n +� +ℓ +p2 +ℓ , +and Ui(q) are polynomials of the fourth order in coordinates. +Summing up, we have n + m − 1 quadratic integrals of motion +f1 + · · · + fm = 2H = F1 + · · · Fn , +associated with the linear combinations of rotations, which realise so∗(m) and so∗(n), and with +the linear combinations of translations. +Proposition 2 Equations of motion (1.2) defined by H (2.2) have only n+ m− 1 independent +quadratic integrals of motion in involution. +We can directly prove this proposition for low-dimensional case R4 substituting generic solution +(1.6) of the Killing equation (1.4) into (1.7) and solving the resulting system of linear equations. +In the generic case, we have to calculate a number of unknown coefficients by the Delong- +Takeuchi-Thompson formula and compare this number with a rank of this system of linear +equations. +2.3 +Euclidean space Rn, case m = 1 +When m = 1 we have the so-called Garnier system and all the second-order Killing tensors +Ki = − +� +k̸=i +Xik · Xik +bi − bk +− Xi · Xi +consist of single rotation and single translation +Xi,k = qi∂k − qk∂i , +Xi = ∂i , +and their Haanties torsion is equal to zero. Thus, the Hamilton-Jacoby equation H = Ei admits +additive separation of variables. Separated variables are the standard elliptic coordinates in Rn +and, therefore, this integrable system constrained to an ellipsoid remains integrable [26]. +When m > 1 the corresponding Killing tensors of valency two have nontrivial Haantjes +torsion. It means that the Hamilton-Jacobi equation H = E does not admit the separation of +variables in the curvilinear orthogonal coordinates. Below we present a few examples of the +corresponding quadratic integrals of motion. +6 + +2.4 +Euclidean space R4, case m = n = 2 +Because so(4) ≃ so(2) × so(2) there appear double rotations or Clifford displacements in R4, +which can be associated with the left- and right-multiplication by a unit quaternion. It is a +classical object in the geometry of the fourth-dimensional Euclidean space [2, 15, 16]. +The 4 × 4 Lax matrix (2.1) is equal to +L(µ) = + + + + +q2 +1+q2 +2+a1−2µ2 +q1q3+q2q4 +p1−2iµq1 +p2−2iµq2 +q1q3+q2q4 +q2 +3+q2 +4+a2−2µ2 +p3−2iµq3 +p4−2iµq4 +p1−2iµq1 +p3−2iµq3 +b1−q2 +1−q2 +3+2µ2 +−q1q2−q3q4 +p2−2iµq2 +p4−2iµq4 +−q1q2−q3q4 +b2−q2 +2−q2 +4+2µ2 + + + + , +(2.7) +so Hamiltonian H (2.2) has the form +H =p2 +1 +2 + p2 +2 +2 + p2 +3 +2 + p2 +4 +2 + 1 +2(q2 +1 + q2 +2)2 + 1 +2(q2 +3 + q2 +4)2 + (q1q3 + q2q4)2 +(2.8) ++a1 − b1 +2 +q2 +1 + a1 − b2 +2 +q2 +2 + a2 − b1 +2 +q2 +3 + a2 − b2 +2 +q2 +4 . +Because +1 +2(q2 +1 + q2 +2)2 + 1 +2(q2 +3 + q2 +4)2 + (q1q3 + q2q4)2 = (q2 +1 + q2 +2 + q2 +3 + q2 +4)2 +2 +− (q1q4 − q2q3)2 +this Hamiltonian coincides with (13b) case from the paper [8] after permutation of indexes. +Spectral curve of the Lax matrix L(µ) (2.7) is a non-hyperelliptic curve defined by char- +acteristic equation +C : +det +� +zI − L(µ) +� += 0. +In generic case its genus is equal to five g = 5, when a1 = a2 or b1 = b2 genus of this non- +hyperelliptic curve C is equal to four g = 4. +First set of integrals of motion +Two residues of the function ∆(z, µ) (2.3) +∆(z, µ) = +det +� +zI − L(µ) +� +(z − a1 + 2µ2)(z − a2 + 2µ2) +are equal to +Res|z=ai−2µ2 ∆(z, µ) = 4µ2fi + gi , +i = 1, 2. +where f1,2 and g1,2 are the second and fourth-order polynomials in momenta, respectively. +Third residue in infinity +Res|z=∞ ∆(z, µ) = −4µ2(f1 + f2) − (g1 + g2) , +give rise to integrals of motions f1 + f2 = 2H and g1 + g2 = f3 which are polynomials of the +second order in momenta. +Quadratic integrals of motion f1,2 have the following form +f1 = − +M 2 +12 +a1 − a2 ++ p2 +1 + p2 +2 + v1 +and +f2 = +M 2 +12 +a1 − a2 ++ p2 +3 + p2 +4 + v2, +(2.9) +where +v1 =(q2 +1 + q2 +2 + q2 +3 + a1 − b1)q2 +1 + (q2 +1 + q2 +2 + q2 +4 + a1 − b2)q2 +2 + 2q1q2q3q4 , +v2 =(q2 +1 + q2 +3 + q2 +4 + a2 − b1)q2 +3 + (q2 +2 + q2 +3 + q2 +4 + a2 − b2)q2 +4 + 2q1q2q3q4 . +7 + +Here M12 is a function associated a double rotation in R4 +M12 = J1,3 + J2,4 = (q1p3 − q3p1) + (q2p4 − q4p2) . +It commutes with terms in f1,2 associated with translations +{M12, p2 +1 + p2 +2} = {M12, p2 +3 + p2 +4} = 0 , +and with function associated with the second independent double rotation in R4 +N12 = J1,2 + J3,4 = (q1p2 − q2p1) + (q3p4 − p3q4) , +so that +{M12, N12} = 0 . +The second independent rotation appears in the following linear combination of the integrals +of motion +f3 =(b1 + b2)H − g1 − g2 − a1f1 − a2f2 +=N 2 +12 − +(b1−b2)((q2 +1+q2 +2+q2 +3+q2 +4)(q2 +1 −q2 +2 +q2 +3−q2 +4 )+(q2 +1 −q2 +2)a1+(q2 +3−q2 +4)a2−(q2 +1 +q2 +3)b1+(q2 +2 +q2 +4)b2) +2 +. +When b1 = b2 linear integral of motion N12 is a function on f1,2 and g1,2. +The leading term in quartic polynomials in momenta is the perfect square +(a1 − a2)g1 = (p1p4 − p2p3)2 + · · · +and +(a2 − a1)g1 = (p1p4 − p2p3)2 + · · · . +There is also a combination of the integrals of motion +g3 = 2H2 − a1g1 − a2g2 +with the leading part defined by the curvature tensor R (1.3, 2.8) +g3 = −1 +4 +� +α,β,γ,δ +R−α,β,γ,−δpαpβpγpδ + · · · = −1 +2(p2 +1 + p2 +2 + p2 +3 + p2 +4)2 − (p1p4 − p2p3)2 + · · · . +Second set of integrals of motion +Residues of the function ∆(z, µ) (2.5) +∆(z, µ) = +det +� +zI − L(µ) +� +(z − b1 − 2µ2)(z − b2 − 2µ2) +are equal to +Res|z=bi+2µ2 ∆(z, µ) = −4µ2Fi + Gi , +i = 1, 2. +Res|z=∞ ∆(z, µ) = 8µ2H − (G1 + G2) , +F1 + F2 − 2H = 0 , +where polynomials of second order in momenta F1,2 and G1 + G2 are independent for each +other. +Quadratic integrals of motion F1,2 are equal to +F1 = +N 2 +12 +b1 − b2 ++ p2 +1 + p2 +3 + V1 +and +F2 = − N 2 +12 +b1 − b2 ++ p2 +2 + p2 +4 + V2 , +8 + +where +V1 += +(q2 +1 + q2 +2 + q2 +3 + a1 − b1)q2 +1 + (q2 +1 + q2 +3 + q2 +4 + a2 − b1)q2 +3 + 2q1q2q3q4 , +V2 += +(q2 +1 + q2 +2 + q2 +4 + a1 − b2)q2 +2 + (q2 +2 + q2 +3 + q2 +4 + a2 − b2)q2 +4 + 2q1q2q3q4 . +Here N12 is a function associated with the double rotation in R4: +N12 = J1,2 + J3,4 = (q1p2 − q2p1) + (q3p4 − p3q4) . +This function commutes with terms in F1,2 associated with translations +{N12, p2 +1 + p2 +3} = {N12, p2 +2 + p2 +4} = 0 , +and with function associated with the independent second double rotation +M12 = J1,3 + J2,4 = (q1p3 − q3p1) + (q2p4 − p2q4) , +which was included in the definition of f1,2 (2.9) from the first set of integrals of motion. +This function also appears in the following linear combination of integral of motion from +the second set of integrals of motion +F3 = G1 + G2 − b1F1 − b2F2 − (a1 + a2)H += M 2 +12 + +(a1−a2) +� +p2 +3+p2 +4−p2 +1−p2 +2+(q2 +1 −q2 +3)b1+(q2 +2−q2 +4 )b2−(q2 +1+q2 +2)a1+(q2 +3 +q2 +4)a2−(q2 +1+q2 +2)2+(q2 +3 +q2 +4)2� +2 +. +When a1 = a2 linear integral of motion N13 is a function on F1,2 and G1,2. +The leading term in the quartic invariants is a perfect square +(b1 − b2)G1 = (p1p4 − p2p3)2 + . . . +and +(b2 − b1)G2 = (p1p4 − p2p3)2 + . . . , . +As above, there are quartic invariant +G3 = 2H2 − b1G1 − b2G2 +with leading term defined by the curvature tensor R (1.3, 2.8) +G3 = −1 +4 +� +α,β,γ,δ +R−α,β,γ,−δpαpβpγpδ + · · · = 1 +2(p2 +1 + p2 +2 + p2 +3 + p2 +4)2 − (p1p4 − p2p3)2 + · · · . +Summing up, there are only m + n − 1 = 3 independent quadratic integrals of motion +among f1, f2, f3 and F1, F2, F3 in R4. The corresponding Killing tensors of valency two have +non-zero Haantjes torsion. +2.5 +Euclidean space R6, case m = 2 and n = 3 +The 5 × 5 Lax matrix (2.1) reads as +L(µ) = + + + + + + +q2 +1+q2 +2+q2 +3+a1−2µ2 +q1q4+q2q5+q3q6 +p1−2iµq1 +p2−2iµq2 +p3−2iµq3 +q1q4+q2q5+q3q6 +q2 +4+q2 +5+q2 +6+a2−2µ2 +p4−2iµq4 +p5−2iµq5 +p6−2iµq6 +p1+2iµq1 +p4+2iµq4 +b1−q2 +1−q2 +4+2µ2 +−q1q2−q4q5 +−q1q3−q4q6 +p2+2iµq2 +p5+2iµq5 +−q1q2−q4q5 +b2−q2 +2−q2 +5+2µ2 +−q2q3−q5q6 +p3+2iµq3 +p6+2iµq6 +−q1q3−q4q6 +−q2q3−q5q6 +b3−q2 +3−q2 +6+2µ2 + + + + + + +, +so Hamiltonian H (1.3,2.2) reads as +H =1 +2 +6 +� +i=1 +p2 +i + (q2 +1 + q2 +2 + q2 +3)2 +2 ++ (q2 +4 + q2 +5 + q2 +6)2 +2 ++ (q1q4 + q2q5 + q3q6)2 +(2.10) +−q2 +1 + q2 +4 +2 +b1 − q2 +2 + q2 +5 +2 +b2 − q2 +3 + q2 +6 +2 +b3 + q2 +1 + q2 +2 + q2 +3 +2 +a1 + q2 +4 + q2 +5 + q2 +6 +2 +a2 . +9 + +When ai = 0 and bi = 0 this Hamiltonian commutes with the following linear integrals of +motion +M12 =(q1p4 − p4q1) + (q2p5 − p2q5) + (q3p6 − p3q6) , +N12 = (q1p2 − p1q2) + (q4p5 − p4q5) , +N13 =(q1p3 − p1q3) + (q4p6 − p4q6) , +N23 = (q2p3 − p2q3) + (q5p6 − p5q6) , +associated with rotations in R5. +The equation for the spectral curve of the Lax matrix contains only five commuting func- +tions H, F1, F2 and G1, G2 +τ(z, µ) =z5 − 2µ2z4 − 2(4µ4 + H)z3 + (16µ6 + 4Hµ2 + F1)z2 ++(16µ8 + 8Hµ4 − 4F 2 +2 µ2 + G1)z − 32µ10 − 16Hµ6 + (8F 2 +2 − 4F1)µ4 − 2G1µ2 + G2 , +where +F1 = M 2 +12 − N 2 +12 − N 2 +13 − N 2 +23 , +F2 = M 2 +12 . +Thus, we must find the missing integral of motion using other properties of the Lax matrix +similar to the full Toda lattice [3]. +In the generic case ai ̸= 0 and bi ̸= 0 spectral curve of the Lax matrix L(µ) is a genus +six non-hyperelliptic curve, that allows us to get six independent integrals of motion in the +involution. +First set of integrals of motion +Two residues of the function ∆(z, µ) (2.3) +∆(z, µ) = +det +� +zI − L(µ) +� +(z − a1 + 2µ2)(z − a2 + 2µ2) +are equal to +Res|z=ai−2µ2 ∆(z, µ) = −16µ4fi + µ2gi + wi , +i = 1, 2. +where f1,2 are polynomials of second order in momenta. Because 2m = 4 other integrals of +motion g1,2 and w1,2 are polynomials of fourth order in momenta. +Residue at infinity is equal to +Res|z=∞ ∆(z, µ) = 32µ4H − µ2(g1 + g2) − (w1 + w2) , +f1 + f2 − 2H = 0 , +Integrals of motion f1,2 are polynomials of second order in momenta +f1 = − M 2 +12 +b1 − b2 ++ p2 +1 + p2 +2 + p2 +3 + v1 , +and +f2 = +M 2 +12 +b1 − b2 ++ p2 +4 + p2 +5 + p2 +6 + v2 +(2.11) +where +v1 =(q2 +1 + q2 +2 + q2 +3 + q2 +4 + a1 − b1)q2 +1 + (q2 +1 + q2 +2 + q2 +3 + q2 +5 + a1 − b2)q2 +2 ++(q2 +1 + q2 +2 + q2 +3 + q2 +6 + a1 − b3)q2 +3 + 2q1q2q4q5 + 2q1q3q4q6 + 2q2q3q5q6 , +v2 =(q2 +1 + q2 +4 + q2 +5 + q2 +6 + a2 − b1)q2 +4 + (q2 +2 + q2 +4 + q2 +5 + q2 +6 + a2 − b2)q2 +5 ++(q2 +3 + q2 +4 + q2 +5 + q2 +6 + a2 − b3)q2 +6 + 2q1q2q4q5 + 2q1q3q4q6 + 2q2q3q5q6 . +Here M12 is a function associated with the triple rotation in R6: +M12 = J14 + J25 + J36 = (q1p4 − p4q1) + (q2p5 − p2q5) + (q3p6 − p3q6) . +(2.12) +10 + +Linear combinations of other integrals of motion are associated with double rotations in R6. +For instance, the polynomial of second order in momenta +f3 = 2(b1 + b2 + b3)H + g1 + g2 +4 +− 2a1f1 − 2a2f2 +is equal to +f3 = N 2 +12 + N 2 +13 + N 2 +23 + (p2 +1 + p2 +4)b1 + (p2 +2 + p2 +5)b2 + (p2 +3 + p2 +6)b3 + v3 , +where +N12 =J12 + J45 = (q1p2 − p1q2) + (q4p5 − p4q5) , +N13 =J13 + J46 = (q1p3 − p1q3) + (q4p6 − p4q6) , +(2.13) +N23 =J23 + J56 = (q2p3 − p2q3) + (q5p6 − p5q6) , +and +v3 = (q4 +1 + q2 +1q2 +2 + q2 +1q2 +3 + 2q2 +1q2 +4 + 2q1q2q4q5 + 2q1q3q4q6 + q4 +4 + q2 +4q2 +5 + q2 +4q2 +6 + a1q2 +1 + a2q2 +4)b1 ++ (q2 +1q2 +2 + 2q1q2q4q5 + q4 +2 + q2 +2q2 +3 + 2q2 +2q2 +5 + 2q2q3q5q6 + q2 +4q2 +5 + q4 +5 + q2 +5q2 +6 + a1q2 +2 + a2q2 +5)b2 ++ (q2 +1q2 +3 + 2q1q3q4q6 + q2 +2q2 +3 + 2q2q3q5q6 + q4 +3 + 2q2 +3q2 +6 + q2 +4q2 +6 + q2 +5q2 +6 + q4 +6 + a1q2 +3 + a2q2 +6)b3 +− (q2 +1 + q2 +4)b2 +1 − (q2 +2 + q2 +5)b2 +2 − (q2 +3 + q2 +6)b2 +3 . +We will omit such explicit expressions for the bulky potentials below for brevity. +Second set of integrals of motion +Three residues of the function ∆(z, µ) (2.5) +∆(z, µ) = +det +� +zI − L(µ) +� +(z − b1 − 2µ2)(z − b2 − 2µ2)(z − b3 − 2µ2) +are equal to +Res|z=bi+2µ2 ∆(z, µ) = 4µ2Fi + Gi , +i = 1, 2, 3. +where Fi and Gi are the second and fourth-order polynomials in momenta. +Residue in infinity reads as +Res|z=∞ ∆(z, µ) = 8µ2H − (G1 + G2 + G3) , +2H + F1 + F2 + F3 = 0 . +Polynomials of second order in momenta are defined by double rotations and double translations +11 + +(2.6) +F1 = − N 2 +12 +b1 − b2 +− +N 2 +13 +b1 − b3 +− p2 +1 − p2 +4 − (q2 +1 + q2 +2 + q2 +3 + 2q2 +4 + a1 − b1)q2 +1 +− (q2 +4 + q2 +5 + q2 +6 + a2 − b1)q2 +4 − 2(q2q5 + q3q6)q1q4 , +F2 = − N 2 +21 +b2 − b1 +− +N 2 +23 +b2 − b3 +− p2 +2 − p2 +5 − (q2 +1 + q2 +2 + q2 +3 + 2q2 +5 + a1 − b2)q2 +2 +− (q2 +4 + q2 +5 + q2 +6 + a2 − b2)q2 +5 − 2(q1q4 + q3q6)q2q5 , +F3 = − N 2 +31 +b3 − b1 +− +N 2 +32 +b3 − b2 +− p2 +3 − p2 +5 − (q2 +1 + q2 +2 + q2 +3 − 2q2 +6 + a1 − b3)q2 +3 +− (q2 +4 + q2 +5 + q2 +6 + a2 − b3)q2 +6 − 2(q1q4 + q2q5)q3q6 . +Functions Nij = −Nji (2.13) are associated with double rotations in R6 and realisations of +so(3) algebra with the brackets +{N12, N13} = N23 , +{N13, N23} = N12 , +{N23, N12} = N13 . +Leading term of the independent on F1, F2 and F3 polynomial of second order in momenta +F4 = G1 + G2 + G3 − b1F1 − b2F2 − b3F3 − (a1 + a2)H += M 2 +12 − a1 − a2 +2 +� +p2 +1 + p2 +2 + p2 +3 − p2 +4 − p2 +5 − p2 +6 + V4 +� +includes function M12 (2.12) associated with the triple rotation in R6. +The corresponding +potential V4 is equal to +V4 = (q2 +1 + q2 +2 + q2 +3 + q2 +4 + q2 +5 + q2 +6)(q2 +1 + q2 +2 + q2 +3 − q2 +4 − q2 +5 − q2 +6) ++ (q2 +1 + q2 +2 + q2 +3)a1 − (q2 +4 + q2 +5 + q2 +6)a2 − (q2 +1 − q2 +4)b1 − (q2 +2 − q2 +5)b2 − (q2 +3 − q2 +6)b3 . +When a1 = a2 linear polynomial M12 commutes with all the integrals of motion H, Fk and Gk. +The following combination of quartic integrals of motion +G4 = 2H2 − b1G1 − b2G2 − b3G3 +has the leading term defined by the curvature tensor R (1.3, 2.10) +G4 += +− 1 +4 +� +α,β,γ,δ R−α,β,γ,−δpαpβpγpδ + · · · += +1 +2(p2 +1 + p2 +2 + p2 +3)2 + 1 +2(p2 +4 + p2 +5 + p2 +6)2 + (p1p4 + p2p5 + p3p6)2 + · · · . +Summing up, there are only m + n − 1 = 4 independent quadratic integrals of motion among +f1, f2, f3 and F1, F2, F3, F4 in R6. The corresponding Killing tensors of valency two have non- +trivial Haantjes torsion. +12 + +2.6 +Euclidean space R9, case m = n = 3 +The 6 × 6 Lax matrix (2.1) is +L(µ) = +� +L11 +L12 +L21 +L22 +� +(2.14) +where +L11 += +� +−2µ2+q2 +1+q2 +2+q2 +3+a1 +q1q4+q2q5+q3q6 +q1q7+q2q8+q3q9 +q1q4+q2q5+q3q6 +−2µ2+q2 +4+q2 +5+q2 +6+a2 +q4q7+q5q8+q6q9 +q1q7+q2q8+q3q9 +q4q7+q5q8+q6q9 +−2µ2+q2 +7+q2 +8+q2 +9+a3 +� +, +L22 += +� +2µ2−q2 +1−q2 +4−q2 +7+b1 +−q1q2−q4q5−q7q8 +−q1q3−q4q6−q7q9 +−q1q2−q4q5−q7q8 +2µ2−q2 +2−q2 +5−q2 +8+b2 +−q2q3−q5q6−q8q9 +−q1q3−q4q6−q7q9 +−q2q3−q5q6−q8q9 +2µ2−q2 +3−q2 +6−q2 +9+b3 +� +, +L12 += +� p1−2iµq1 p2−2iµq2 p3−2iµq3 +p4−2iµq4 p5−2iµq5 p6−2iµq6 +p7−2iµq7 p8−2iµq8 p9−2iµq9 +� +, +L21 = +� p1+2iµq1 p4+2iµq4 p7+2iµq7 +p2+2iµq2 p5+2iµq5 p8+2iµq8 +p3+2iµq3 p6+2iµq6 p9+2iµq9 +� +. +Hamiltonian H (2.2) is given by +H =1 +2 +9 +� +i=1 +p2 +i + (q2 +1 + q2 +2 + q2 +3)2 +2 ++ (q2 +4 + q2 +5 + q2 +6)2 +2 ++ (q2 +7 + q2 +8 + q2 +9)2 +2 ++(q1q4 + q2q5 + q3q6)2 + (q1q7 + q2q8 + q3q9)2 + (q4q7 + q5q8 + q6q9)2 +−q2 +1 + q2 +4 + q2 +7 +2 +b1 − q2 +2 + q2 +5 + q2 +8 +2 +b2 − q2 +3 + q2 +6 + q2 +9 +2 +b3 ++q2 +1 + q2 +2 + q2 +3 +2 +a1 + q2 +4 + q2 +5 + q2 +6 +2 +a2 + q2 +7 + q2 +8 + q2 +9 +2 +a3 . +First set of integrals of motion +Residues of the function ∆(z, µ) (2.3) +∆(z, µ) = +det +� +zI − L(µ) +� +(z − a1 + 2µ2)(z − a2 + 2µ2)(z − a3 + 2µ2) +are equal to +Res|z=ai+2µ2 ∆(z, µ) = 16µ4fi + µ2gi + si , +i = 1, 2, 3. +Res|z=∞ ∆(z, µ) = 32Hµ4 − (g1 + g2 + g3)µ2 − (s1 + s2 + s3) . +13 + +where second-order polynomials in momenta are equal to +f1 = +M 2 +12 +a1 − a2 ++ +M 2 +13 +a1 − a3 +− p2 +1 − p2 +2 − p2 +3 − (2q2 +2 + 2q3 +2 + q4 +2 + q7 +2 + a1 − b1)q1 +2 +−(2q32 − q52 − q82 − a1 + b2)q22 − (q32 + q62 + q92 + a1 − b3)q32 +−2q2q3(q5q6 + q8q9) − 2q1q2(q4q5 + q7q8) − 2q1q3(q4q6 + q7q9) − q14 − q24 , +f2 = +M 2 +21 +a2 − a1 ++ +M 2 +23 +a2 − a3 +− p2 +4 − p2 +5 − p2 +6 − (q1 +2 + 2q5 +2 + 2q6 +2 + q7 +2 + a2 − b1)q4 +2 +−(q22 + 2q62 + q82 + a2 − b2)q52 − (q2 +3 + q2 +6 + q2 +9 + a2 − b3)q2 +6 +−q5q4(2q1q2 + 2q7q8) − 2q6q4(q1q3 + q7q9) − 2q5q6(q2q3 + q8q9) − q4 +4 − q4 +5 +and +f3 = +M 2 +31 +a3 − a1 ++ +M 2 +32 +a3 − a2 +− p2 +7 − p2 +8 − p2 +9 − (q2 +1 + q2 +4 + 2q2 +8 + 2q2 +9 + a3 − b1)q2 +7 +−(q2 +2 + q2 +5 + 2q2 +9 + a3 − b2)q2 +8 − (q2 +3 + q2 +6 + q2 +9 + a3 − b3)q2 +9 +−2q7q8(q1q2 + q4q5) − 2q7q9(q1q3 + q4q6) − 2q8q9(q2q3 + q5q6) − q4 +7 − q4 +8 , +Here Mij are given by +M12 =J14 + J25 + J36 = (q1p4 − p1q4) + (q2p5 − p2q5) + (q3p6 − p3q6) , +M13 =J17 + J28 + J39 = (q1p7 − p1q7) + (q2p8 − p2q8) + (q3p9 − p3q9) , +(2.15) +M23 =J47 + J58 + J69 = (q4p7 − p4q7) + (q5p8 − p5q8) + (q6p9 − p6q9) . +The following combination of integrals of motion is also a quadratic polynomial in momenta +f4 = g1 + g2 + g2 +4 ++ 2a1f1 + 2a2f2 + 2a3f3 , +which has the form +f4 = − + + +n +� +j=1 +bj + + +� nm +� +i=1 +p2 +i +� ++ +n +� +j=1 +bj +�m−1 +� +i=0 +p2 +j+im +� ++ N 2 +12 + N 2 +23 + N 2 +31 + u4(q) . +Here Nij: +N12 =J12 + J45 + J78 = (q1p2 − p1q2) + (q4p5 − p4q5) + (q7p8 − p7q8) , +N13 =J13 + J46 + J79 = (q1p3 − p1q3) + (q4p6 − p4q6) + (q7p9 − p7q9) , +(2.16) +N23 =J23 + J56 + J89 = (q2p3 − p2q3) + (q5p6 − p5q6) + (q8p9 − p8q9) . +14 + +Functions Mij (2.15) and Nij (2.16) are associated with two realizations of so∗(3) by using +independent triple rotations in R9. The Lie-Poisson brackets are +{M12, M13} = M23 , +{M13, M23} = M12 , +{M23, M12} = M13 . +and +{N12, N13} = N23 , +{N13, N23} = N12 , +{N23, N12} = N13 , +so that +{Nij, Mkl} = 0 . +Leading terms in polynomials of six order in momenta are +si = +1 +(ai − aj)(ai − ak) +� +p1p5p9 + p2p6p7 + p3p4p8 − p1p6p8 − p2p4p9 − p3p5p7 +�2 + · · · , +and, therefore, the sum of these polynomials is a polynomial of the fourth order in momenta +which is independent of g1, g2 and g3. +Second set of the integrals of motion +Residues of the function ∆(z, µ) (2.5) +∆(z, µ) = +det +� +zI − L(µ) +� +(z − b1 − 2µ2)(z − b2 − 2µ2)(z − b3 − 2µ2) +are equal to +Res|z=bi+2µ2 ∆(z, µ) = 16µ4Fi + µ2Gi + Si , +i = 1, 2, 3. +Res|z=∞ ∆(z, µ) = 32Hµ4 − (G1 + G2 + G3)µ2 − (S1 + S2 + S3) . +Second-order polynomials in momenta have the following form +F1 = − N 2 +12 +b1 − b2 +− +N 2 +13 +b1 − b3 +− p2 +1 − p2 +4 − p2 +7 − (q2 +1 + q2 +2 + q2 +3 + q2 +4 + q2 +7 + a1 − b1)q2 +1 +− (q2 +1 + q2 +4 + q2 +5 + q2 +6 + q2 +7 + a2 − b1)q2 +4 − (q2 +1 + q2 +4 + q2 +7 + q2 +8 + q2 +9 + a3 − b1)q2 +7 +− 2q1q4(q2q5 + q3q6) − 2q1q7(q2q8 + q3q9) − 2q4q7(q5q8 + q6q9) , +F2 = − N 2 +21 +b2 − b1 +− +N 2 +23 +b2 − b3 +− p2 +2 − p2 +5 − p2 +8 − (q2 +1 + q2 +2 + q2 +3 + q2 +5 + q2 +8 + a1 − b2)q2 +2 +− (q2 +2 + q2 +4 + q2 +5 + q2 +6 + q2 +8 + a2 − b2)q2 +5 − (q2 +2 + q2 +5 + q2 +7 + q2 +8 + q2 +9 + a3 − b2)q2 +8 +− 2q2q5(q1q4 + q3q6) − 2q2q8(q1q7 + q3q9) − 2q5q8(q4q7 + q6q9) +and +F3 = − N 2 +31 +b3 − b1 +− +N 2 +32 +b3 − b2 +− p2 +3 − p2 +6 − p2 +9 − (q2 +1 + q2 +2 + q2 +3 + q2 +6 + q2 +9 + a1 − b3)q2 +3 +− (q2 +3 + q2 +4 + q2 +5 + q2 +6 + q2 +9 + a2 − b3)q2 +6 − (q2 +3 + q2 +6 + q2 +7 + q2 +8 + q2 +9 + a3 − b3)q2 +9 +− 2q3q6(q1q4 + q2q5) − 2q3q9(q1q7 + q2q8) − 2q6q9(q4q7 + q5q8) , +15 + +Functions Mij, Nkl are given by 2.15,2.16. +The following combination of integrals of motion is also second order polynomial in mo- +menta +F4 = 1 +8(G1 + G2 + G3) − b1F1 − b2F2 − b3F3 +which is independent on F1, F2 and F3. It has the form +F4 = 1 +2 + + +m +� +j=1 +aj + + +� n +� +i=1 +p2 +i +� +− 1 +2 +m−1 +� +j=0 +aj+1 +� n +� +i=1 +p2 +jn+i +� ++ M 2 +12 +2 ++ M 2 +13 +2 ++ M 2 +23 +2 ++ U4(q) . +Leading terms in polynomials of six order in momenta are +Si = +1 +(bi − bj)(bi − bk) +� +p1p5p9 + p2p6p7 + p3p4p8 − p1p6p8 − p2p4p9 − p3p5p7 +�2 + · · · , +and, therefore, the sum of these polynomials is a polynomial of the fourth order in momenta +G4 = S1 + S2 + S3 , +which is independent on g1, g2 and g3. +Summing up, we have integrals of motion of second, fourth-order and sixth order in mo- +menta. Among them, there are five independent quadratic integrals of motion because +f1 + f2 + f3 = 2H = F1 + F2 + F3 +3 +Symmetric space of C.I type +The compact group Sp(n) of 2n × 2n matrices which are both symplectic and unitary is asso- +ciated with the root space Cn. Because +Sp(n) +U(n) ⊂ +SU(2n) +S (U(n) × U(n)) +we can get the Lax matrices starting with Lax matrices (2.1). Roughly speaking we have to +make n × n matrices Q and P symmetric, divide off-diagonal entries of P by two and impose +suitable restrictions on parameters ai and bi. +Below we present these Lax matrices at n = 2 and n = 3 and discuss the corresponding +quadratic integrals of motion. +3.1 +Euclidean space R3, case n = 2 +The 4 × 4 Lax matrix is equal to +L(µ) = + + + + +−2µ2+q2 +1+q2 +2+a1 +q1q2+q2q3 +p1−2iµq1 +p2 +2 −2iµq2 +q1q2+q2q3 +−2µ2+q2 +2+q2 +3+a2 +p2 +2 −2iµq2 +p3−2iµq3 +p1+2iµq1 +p2 +2 +2iµq2 +2µ2−q2 +1−q2 +2+b1 +−q1q2−q2q3 +p2 +2 +2iµq2 +p3+2iµq3 +−q1q2−q2q3 +2µ2−q2 +2−q2 +3+b2 + + + + , +(3.1) +where +a2 − a1 = b1 − b2 . +(3.2) +The Hamiltonian is given by +H += +p2 +1 +2 + p2 +2 +4 + p2 +3 +2 + (q2 +1 + 2q2 +2 + q2 +3)2 +2 +− +(q1q3 − q2 +2)2 + a1 − b1 +2 +q2 +1 + (a1 − b2) q2 +2 + a1 + b1 − 2b2 +2 +q2 +3 . +(3.3) +16 + +It coincides with the (13c) case from the paper [8]. +After canonical change of variables p2 → +√ +2p2 and q2 → q2/ +√ +2 we obtain standard +metric g = diag(1, 1, 1) in Euclidean space and integrable three-dimensional quartic potential +at bi = ai = 0 +V = 1 +2(q2 +1 + q2 +2 + q2 +3)2 − (2q1q3 − q2 +2)2 +4 +(3.4) +This potential is missing in the classification based on the Ziglin and Yoshida methods [4], since +authors studied only potentials in the following form +˜V = q4 +1 + aq2 +1q2 +2 + bq2 +1q2 +3 + cq4 +2 + dq2 +2q2 +3 + eq4 +3 , +whereas (3.4) involves linear in q1 and q3 term q1q3q2 +2. +Residues of the functions +∆(z, µ) = +det +� +Iz − L(µ) +� +(z + 2µ2 − a1)(z + 2µ2 − a2) +and +∆(z, µ) = +det +� +Iz − L(µ) +� +(z − 2µ2 − b1)(z − 2µ2 − b2) +coincide for each other up to the sign and replacement a1 − a2 = −(b1 − b2) which corresponds +to (3.2). +Let us consider residues +Res|z=bi+2µ2 ∆(z, µ) = −4µ2Fi + Gi , +i = 1, 2. +Res|z=∞ ∆(z, µ) = 8µ2H − (G1 + G2) . +Because +F1 + F2 − 2H = 0 +and +G1 + G2 + 2(b2 − a1)H = 0 . +there are two polynomials of the second order in momenta F1,2 and only one polynomial of the +fourth order in momenta G1 or G2 which are independent of each other. +In [8] authors argue that three integrals of motion F1, F2 and G1 + G2 are quadratic +polynomials in momenta, thus suggesting the existence of the point transformation to new +variables in which equations of motion (1.2) can be separated. Unfortunately, the authors did +not notice that these integrals of motion are functionally dependent; therefore, their statement +is incorrect. +Let us present these integrals explicitly +F1 = p2 +1 + p2 +2 +4 + +M 2 +12 +b1 − b2 ++ (q2 +1 + 2q2 +2 + a1 − b1)q2 +1 + (q2 +1 + q2 +2 + q2 +3 + 2q1q3 + a1 − b2)q2 +2 , +F2 = p2 +3 + p2 +2 +4 + +M 2 +12 +b2 − b1 ++ (2q2 +2 + q2 +3 + a2 − b2)q2 +3 + (q2 +1 + q2 +2 + q2 +3 + 2q1q3 + a2 − b1)q2 +2 . +where +M12 = 1 +2(q1p2 − 2q2p1 − q3p2 + 2q2p3) . +At b1 = b2 we have a linear integral of motion M12 associated with a double rotation. After re- +duction by the corresponding Noether’s symmetry, we obtain new quadratic-linear Hamiltonian +H commuting with the quartic invariant G1,2 in T ∗R2 +For Euclidean space R3 generic solution K (1.6) of the Killing equation (1.4) depends on +20 parameters. Using modern computer software we can directly prove that there are only two +independent solutions to the equation +d(KdV ) = 0 +associated with the integrals of motion F1,2. +17 + +3.2 +Euclidean space R6, case n = 3 +The 6 × 6 Lax matrix (2.14) generates three quadratic invariants Fi, three quartic Gi and three +sextic invariants Si. After reduction, we have to get only six independent invariants. +The Lax matrix (2.14) after reduction looks like +L(µ) = +�ˆL11 +ˆL12 +ˆL21 +ˆL22 +� +where +ˆL11 += +� +−2µ2+q2 +1+q2 +2+q2 +3+a1 +q1q2+q2q4+q3q5 +q1q3+q2q5+q3q6 +q1q2+q2q4+q3q5 +−2µ2+q2 +2+q2 +4+q2 +5+b1−b2+a1 +q2q3+q4q5+q5q6 +q1q3+q2q5+q3q6 +q2q3+q4q5+q5q6 +−2µ2+q2 +3+q2 +5+q2 +6+b1−b3+a1 +� +, +ˆL22 += +� +2µ2−q2 +1−q2 +2−q2 +3+b1 +−q1q2−q2q4−q3q5 +−q1q3−q2q5−q3q6 +−q1q2−q2q4−q3q5 +2µ2−q2 +2−q2 +4−q2 +5+b2 +−q2q3−q4q5−q5q6 +−q1q3−q2q5−q3q6 +−q2q3−q4q5−q5q6 +2µ2−q2 +3−q2 +5−q2 +6+b3 +� +, +ˆL12 += + + +p1−2iµq1 +p2 +2 −2iµq2 +p3 +2 −2iµq3 +p2 +2 −2iµq2 p4−2iµq4 +p5 +2 −2iµq5 +p3 +2 −2iµq3 +p5 +2 −2iµq5 p6−2iµq6 + + , +ˆL21 = + + +p1+2iµq1 +p2 +2 +2iµq2 +p3 +2 +2iµq3 +p2 +2 +2iµq2 p4+2iµq4 +p5 +2 +2iµq5 +p3 +2 +2iµq3 +p5 +2 +2iµq5 p6+2iµq6 + + . +Here we impose the following restrictions on arbitrary parameters in (2.14) +a3 = b1 − b3 + a1 , +a2 = b1 − b2 + a1 . +Calculating integrals of motion using three residues of the function +∆ = +det +� +Iz − L(µ) +� +(z − 2µ2 − b1)(z − 2µ2 − b2)(z − 2µ2 − b3) +at z = bi + 2µ2 +Res|z=bi+2µ2 ∆(z, µ) = −16µ4Fi + µ2Gi + Si +we obtain the following quadratic integrals of motion +F1 = M 2 +12 +b1 − b2 ++ +M 2 +13 +b1 − b3 ++ T1 + V1 , +T1 = p2 +1 + p2 +2 +4 + p2 +3 +4 , +F2 = M 2 +21 +b2 − b1 ++ +M 2 +23 +b2 − b3 ++ T2 + V2 , +T2 = p2 +2 +4 + p2 +4 + p2 +5 +4 , +F3 = M 2 +31 +b3 − b1 ++ +M 2 +32 +b3 − b2 ++ T3 + V3 , +T3 = p2 +3 +4 + p2 +5 +4 + p2 +6 , +where functions associated with the triple rotations are equal to +M12 = −M21 = 1 +2(q1p2 − 2p1q2 + 2q2p4 − p2q4 + q3p5 − p3q5) , +M13 = −M31 = 1 +2(q1p3 − 2p1q3 + q2p5 − p2q5 + 2q3p6 − p3q6) , +M23 = −M32 = 1 +2(q2p3 − p2q3 + q4p5 − 2p4q5 + 2q5p6 − p5q6) . +For brevity, we omit explicit expressions for the potentials Vk. +18 + +Residue at infinity gives rise to a relation between quadratic integrals +F1 + F2 + F3 − 2H = 0 +and relations between other integrals of motion +1 +4 (G1 + G2 + G3) + b1F1 + b2F2 + b3F3 − 2(b1 − b2 − b3 + 2a1)H = 0 +and +S1 + S2 + S3 +− +1 +4(b1G1 + b2G2 + b3G3) +− +(b2 +1 − a2a3)F1 − (b2 +2 − a1a3)F2 − (b2 +3 − a1a2)F3 = 0 . +From nine dependent integrals of motion Fi, Gi and Si we have to choose six independent, for +instance, we can take three quadratic integrals of motion, two integrals of motion of fourth +order and one integral of sixth order in momenta. +4 +Symmetric space of D.III type +This is another reduction of the A.III case +SO(n) +U(n) ⊂ +SU(2n) +S (U(n) × U(n)) +associated with the root space Dn. In this case we have to take Lax matrices (2.1), make n × n +matrices Q and P antisymmetric, and impose suitable restrictions on parameters ai and bi. +Isomorphism D3 ∼= A3 yields a correspondence +SO(6) +U(3) +∼= +SU(4) +S(U(1) × U(3)) +so that we have the well-known Garnier system in R3 and the corresponding Hamilton-Jacobi +equation H = E is separable in the elliptic coordinates. +Following [7] we restrict ourselves by calculation of the quadratic integrals of motion for +the D4 case. +4.1 +Euclidean space R6, case n = 4 +The 8 × 8 Lax matrix (2.1) after reduction has the following form +L(µ) = +�¯L11 +¯L12 +¯L21 +¯L22 +� +. +There are two symmetric matrices +¯L11 = + + +q2 +1+q2 +2+q2 +3+a1−2µ2 +q2q4+q3q5 +−q1q4+q3q6 +−q1q5−q2q6 +q2q4+q3q5 +q2 +1+q2 +4+q2 +5+a2−2µ2 +q1q2+q5q6 +q1q3−q4q6 +−q1q4+q3q6 +q1q2+q5q6 +q2 +2+q2 +4+q2 +6+a3−2µ2 +q2q3+q4q5 +−q1q5−q2q6 +q1q3−q4q6 +q2q3+q4q5 +q2 +3+q2 +5+q2 +6+a4−2µ2 + + , +¯L22 = + + +2µ2−q2 +1−q2 +2−q2 +3+b1 +−q2q4−q3q5 +q1q4−q3q6 +q1q5+q2q6 +−q2q4−q3q5 +2µ2−q2 +1−q2 +4−q2 +5+b2 +−q1q2−q5q6 +−q1q3+q4q6 +q1q4−q3q6 +−q1q2−q5q6 +2µ2−q2 +2−q2 +4−q2 +6+b3 +−q2q3−q4q5 +q1q5+q2q6 +−q1q3+q4q6 +−q2q3−q4q5 +2µ2−q2 +3−q2 +5−q2 +6+b4 + + +and two antisymmetric matrices +¯L12 = +� +0 +p1−2iµq1 +p2−2iµq2 +p3−2iµq3 +−p1+2iµq1 +0 +p4−2iµq4 +p5−2iµq5 +−p2+2iµq2 −p4+2iµq4 +0 +p6−2iµq6 +−p3+2iµq3 −p5+2iµq5 −p6+2iµq6 +0 +� +, +19 + +¯L21 = +� +0 +−p1−2iµq1 −p2−2iµq2 −p3−2iµq3 +p1+2iµq1 +0 +−p4−2iµq4 −p5−2iµq5 +p2+2iµq2 +p4+2iµq4 +0 +−p6−2iµq6 +p3+2iµq3 +p5+2iµq5 +p6+2iµq6 +0 +� +. +The parameters must satisfy the following constraints +a2 − a1 = b1 − b2 , +a3 − a1 = b1 − b3 , +a4 − a1 = b1 − b4 . +Four residues of the function +∆ = +det +� +Iz − L(µ) +� +(z − 2µ2 − b1)(z − 2µ2 − b2)(z − 2µ2 − b3)(z − 2µ2 − b4) +at z = bi + 2µ2 are polynomials of sixth order in momenta +Res|z=bi+2µ2 ∆(z, µ) = −64µ6Fi + µ4Gi + µ2Si + Wi , +where Fi, Gi, Si and Wi are the second, fourth, sixth and eighth-order polynomials in momenta. +As a result, we have 16 dependent integrals of motion and residue at infinity yields various +relations between these polynomials, for instance +F1 + F2 + F3 + F4 − 2H = 0 . +We show only the leading part of the quadratic integrals of motion and omit explicit expressions +for the potentials Vk +F1 = +M 2 +12 +b1 − b2 ++ +M 2 +13 +b1 − b2 ++ +M 2 +14 +b1 − b4 ++ T1 + V1 , +F2 = +M 2 +21 +b2 − b1 ++ +M 2 +23 +b2 − b3 ++ +M 2 +24 +b2 − b4 ++ T2 + V2 , +F3 = +M 2 +31 +b3 − b1 ++ +M 2 +32 +b3 − b2 ++ +M 2 +34 +b3 − b4 ++ T3 + V3 , +F4 = +M 2 +41 +b4 − b1 ++ +M 2 +42 +b4 − b2 ++ +M 2 +43 +b4 − b3 ++ T4 + V4 . +where functions +M12 =(q2p4 − p2q4) + (q3p5 − p3q5) , +M13 = (q1p4 − p1q4) + (q6p3 − p6q3) , +M14 =(q1p5 − p1q5) + (q2p6 − p2q6) , +M23 = (q1p2 − p1q2) + (q5p6 − p5q6) , +M24 =(q1p3 − p1q3) + (q6p4 − p6q4) , +M34 = (q2p3 − p2q3) + (q4p5 − p4q5) , +are related to double rotations in R6, whereas functions +T1 = p2 +1 + p2 +2 + p2 +3 , +T2 = p2 +1 + p2 +4 + p2 +5 , +T3 = p2 +2 + p2 +4 + p2 +6 , +T4 = p2 +3 + p2 +5 + p2 +6 , +are defined by triple translations. The direct calculations show that the Haantjes torsion of the +corresponding Killing tensors is not zero. +From the sixteen integrals of motion Fi, Gi, Si and Wi we have to choose six independent +integrals, four of which are quadratic polynomials in momenta. +5 +Symmetric spaces of BD.I type +Symmetric space +SO(m + n) +SO(m) × SO(n) +is only Hermitian when m = 2 since in general so(m) + so(n) has no centre. When m = 2 the +so(2) subalgebra is the centre and depending upon whether q is odd or even this symmetric +space is associated with either B(n+1)/2 or D(n+2)/2 root systems. +The simplest nontrivial example is associated with D3, and, similar to [7], we present the +Lax matrix for this system even though D3 ∼= A3. +20 + +5.1 +Euclidean space R4, case m = n = 2 +We present 6 × 6 Lax matrix (1.5) using the same Cartan-Weil basis as in [7] +L(µ) = +� +�L11 +�L12 +�L21 +�L22 +� +. +where +�L11 = + + +−2µ2+2q2 +1+2q2 +2+2q2 +3+2q2 +4+a1 +p1−2iµq1 +p2−2iµq2 +p1+2iµq1 +−2q2 +1+2q2 +3+a2 −2q1q2+2q3q4 +p2+2iµq2 +−2q1q2+2q3q4 −2q2 +2+2q2 +4+a3 + + +�L22 = + + +2µ2−2q2 +1−2q2 +2−2q2 +3−2q2 +4+b1 +−p1−2iµq1 +−p2−2iµq2 +−p1+2iµq1 +2q2 +1−2q2 +3+b2 2q1q2−2q3q4 +−p2+2iµq2 +2q1q2−2q3q4 2q2 +2−2q2 +4+b3 + + +and +�L12 = +� +0 +p3−2iµq3 +p4−2iµq4 +−p3+2iµq3 +0 +−2q1q4+2q2q3 +−p4+2iµq4 2q1q4−2q2q3 +0 +� +, +�L21 = +� +0 +−p3−2iµq3 +−p4−2iµq4 +p3+2iµq3 +0 +2q1q4−2q2q3 +p4+2iµq4 −2q1q4+2q2q3 +0 +� +. +Parameters satisfy the following relations +a2 = a1 + b1 − b2 , +a3 = a1 + b1 − b3 . +In this case Hamiltonian H (2.2) has the form +H = p2 +1 + p2 +2 + p2 +3 + p2 +4 + 4(q2 +1 + q2 +2 + q2 +3 + q2 +4)2 − 8(q1q3 + q2q4)2 ++ 2(b2 − b1)q2 +1 + 2(b3 − b1)q2 +2 + 2(a1 − b2)q2 +3 + 2(a1 − b3)q2 +4 +Because D3 ∼= A3 this Hamiltonian coincides with (2.8) up to rescaling and canonical transfor- +mation qi → −qi and pi → −pi of one of the coordinates and momenta. +The corresponding second-order Killing tensors are discussed in Section 2. +5.2 +Euclidean space R2n−1, m = 2. +Let us consider representation of the Lie algebra so(2n + 1) by (2n + 1) × (2n + 1) matrices X +[11], which satisfy +X + SXTS−1 = 0 , +S = +2n+1 +� +k=1 +(−1)k+1Ek,2n+2−k , +where Eij are matrices whose only non-zero entry is a unit in row i and column j. +In this case, Cartan involution is related to the following element A = E1,1 −E2n+1,2n+1 of +the Cartan subalgebra. In this representation Lax matrix (1.5) has the following block structure +L(µ) = + + + + + + +2µ2 +⃗xT +0 +⃗y +0 +s · ⃗x +0 +⃗yT · s +−2µ2 + + + + + + ++ C + Λ , +where the central block of zeroes has dimensionality (2n − 1) × (2n − 1), the column vectors x +and y have the following entries +⃗xi = pi − 2iqi , +⃗yi = pi + 2iqi , +i = 1, . . . , 2n − 1 , +21 + +and s is (2n − 1) × (2n − 1) matrix +s = +2n−1 +� +k=1 +(−1)kEk,2n−k . +Matrix Λ is a numerical matrix which satisfies Λ + SΛT S−1 = 0 and, following [18], which +determines a shift of the orbit. +5.3 +Euclidean space R3, case n = 2 +For symmetric space +SO(5) +SO(2)×SO(2) we have the following 5 × 5 Lax matrix +L(µ) = + + +2µ2 +p1−2iµq1 +p2−2iµq2 +p3−2iµq3 +0 +p1+2iµq1 +0 +0 +0 +−p3+2iµq3 +p2+2iµq2 +0 +0 +0 +p2−2iµq2 +p3+2iµq3 +0 +0 +0 +−p1+2iµq1 +0 +−p3−2iµq3 p2+2iµq2 −p1−2iµq1 +−2µ2 + + + 2C + 2Λ +where +C = + + + +−q2 +1−q2 +2−q2 +3 +0 +0 +0 +0 +0 +q2 +1−q2 +3 +(q1+q3)q2 +0 +0 +0 +(q1+q3)q2 +0 +(q1+q3)q2 +0 +0 +0 +(q1+q3)q2 +−q2 +1+q2 +3 +0 +0 +0 +0 +0 +q2 +1+q2 +2+q2 +3 + + + , +Λ = + + +a1 0 +0 +0 +0 +0 a2 a3 +0 +0 +0 a3 0 +a3 +0 +0 +0 a3 −a2 +0 +0 +0 +0 +0 +−a1 + + . +The Hamiltonian (1.3) looks like +H = 1 +4trL2 +���� +µ=0 +− 2a2 +1 − 2a2 +2 − 4a2 +3 = p2 +1 + p2 +2 + p2 +3 + 4(q2 +1 + q2 +2 + q2 +3)2 − 2(2q1q3 − q2 +2)2 +− 4(a1 − a2)q2 +1 − 4(a1 + a2)q2 +3 − 4q2 (a1q2 − 2a3(q1 + q3)) . +The quadratic integral of motion +F = (q1p2 − p1q2 + q2p3 − q3p2)2 − (p1 + p3)(a2(p1 − p3) + 2a3p2) + U +where +U = 4(q1 + q3) +� +a2(q1 − q3) + 2a3q2 +� +(a1 − q2 +1 − q2 +2 − q2 +3) − 4(a2 +2 + a2 +3)(q2 +1 + q2 +3) +− 8q2(q1 − q3)a2a3 − 8(q1q3 + q2 +2)a2 +3 , +defines second-order Killing tensor with non-zero torsion. +The spectral curve of the Lax matrix is defined through the equation +z5 − 2(2µ4 + 4a1µ2 + 2a2 +1 + 2a2 +2 + 4a2 +3 + H)z3 ++ +� +16(a2 +2 + 2a2 +3)µ4 + 8 +� +F + 4a1(a2 +2 + 2a2 +3) +� +µ2 + G1/2 − H2� +z = 0 . +The leading term of the polynomial of fourth order in momenta G is defined by the curvatures +tensor R +G = − 1 +4 +� +α,β,γ,δ +R−α,β,γ,−δqαqβqγqδ + · · · += 4(p2 +1 + p2 +2 + p2 +3)2 − 2(2p1p3 − p2 +2)2 + · · · . +22 + +When ai = 0 we have Hamiltonian (3.3-3.4) up to canonical transformation +H = 1 +4trL2 +���� +µ=0 += p2 +1 + p2 +2 + p2 +3 + 4(q2 +1 + q2 +2 + q2 +3)2 − 2(2q1q3 − q2 +2)2 , +that follows from the equivalence of the symmetric spaces, see [11]. +5.4 +Euclidean space R5, case n = 3 +The 7 × 7 Lax matrix reads as +L(µ) = + + + + + +2µ2 +p1−2iµq1 +p2−2iµq2 +p3−2iµq3 +p4−2iµq4 +p5−2iµq5 +0 +p1+2iµq1 +0 +0 +0 +0 +0 +−p5+2iµq5 +p2+2iµq2 +0 +0 +0 +0 +0 +p4−2iµq4 +p3+2iµq3 +0 +0 +0 +0 +0 +−p3+2iµq3 +p4+2iµq4 +0 +0 +0 +0 +0 +p2−2iµq2 +p5+2iµq5 +0 +0 +0 +0 +0 +−p1+2iµq1 +0 +−p5−2iµq5 p4+2iµq4 −p3−2iµq3 p2+2iµq2 −p1−2iµq1 +−2µ2 + + + + + +(5.1) ++ 2 + + + + + + + +− �5 +k=1 q2 +k +0 +0 +0 +0 +0 +0 +0 +q2 +1−q2 +5 +q1q2+q4q5 +q3(q1−q5) +q1q4+q2q5 +0 +0 +0 +q1q2+q4q5 +q2 +2−q2 +4 +q3(q2+q4) +0 +q1q4+q2q5 +0 +0 +q3(q1−q5) q3(q2+q4) +0 +q3(q2+q4) −q3(q1−q5) +0 +0 +q1q4+q2q5 +0 +q3(q2+q4) +−q2 +2+q2 +4 +q1q2+q4q5 +0 +0 +0 +q1q4+q2q5 −q3(q1−q5) q1q2+q4q5 +−q2 +1+q2 +5 +0 +0 +0 +0 +0 +0 +0 +�5 +k=1 q2 +k + + + + + + + +. +The corresponding Hamiltonian +H = 1 +4trL2 +���� +µ=0 += +5 +� +k=1 +p2 +k + 4 +� 5 +� +k=1 +q2 +k +�2 +− 2(2q1q5 − 2q2q4 + q2 +3)2 +(5.2) +commutes with the four linear integrals of motion +r1 = (q1p2 − p1q2) + (q4p5 − p4q5) , +r2 = (q2p3 − p2q3) + (q3p4 − p3q4) , +r3 = (q1p3 − p1q3) + (q5p3 − p5q3) , +r4 = (q1p4 − p1q4) + (q2p5 − p2q5) , +so that +{r1, r2} = −r3 , +{r1, r3} = r2 , +{r1, r4} = 0 , +{r4, r2} = r3 , +{r4, r3} = −r2 , +{r2, r3} = r4 − r1 . +The spectral curve of the Lax matrix +det(z · I − L(µ)) = z7 − 2(2µ4 + H)z5 + (8F1µ2 + G1)z3 − 4G2z = 0 +gives rise to four independent integrals of motion in involution H, G1 and +F1 = (r2 +1 + r2 +2 + r2 +3 + r2 +4) , +G2 = (r1 + r4)2� +(r1 − r4)2 + 2(r2 +2 + r2 +3) +� +. +Using Hamiltonian and fourth order polynomial G1 we can get the integral of motion defined +by a curvature tensor R (1.3, 2.8) +G3 = 2G1 + 2H2 = −1 +4 +� +α,β,γ,δ +R−α,β,γ,−δpαpβpγpδ + · · · += −4(p2 +1 + p2 +2 + p3 +3 + p2 +4 + p2 +5)2 + 2(2p1p5 − 2p2p4 + p2 +3)2 + · · · . +23 + +which is independent on H and rk. +Because {rk, G1} = 0 we have a completely integrable system with the five independent +integrals of motion in involution, for instance +r1, r4, r2 +2 + r2 +3 , H, G1 . +Nevertheless, the Lax matrix (5.1) generates only four of them, similar to the generalized Toda +lattice [3]. +By adding a constant matrix Λ to L(µ) (5.1), where +Λ = + + + + + + + + + + +a1 +0 +0 +0 +0 +0 +0 +0 +a2 +0 +0 +0 +0 +0 +0 +0 +a3 +a4 +0 +0 +0 +0 +0 +a4 +0 +a4 +0 +0 +0 +0 +0 +a4 +−a3 +0 +0 +0 +0 +0 +0 +0 +−a2 +0 +0 +0 +0 +0 +0 +0 +−a1 + + + + + + + + + + +, +we have +H = 1 +4trL2 +���� +µ=0 +− 2a2 +1 − 2a2 +2 − 2a2 +3 − 4a2 +4 = +5 +� +k=1 +p2 +k + 4 +� 5 +� +k=1 +q2 +k +�2 +− 2(2q1q5 − 2q2q4 + q2 +3)2 ++(a1 − a2)q2 +1 + (a1 − a3)q2 +2 + q3(a1q3 − 2a4q2 − 2a4q4) + (a1 + a3)q2 +4 + (a2 + a1)q2 +5 . +In this case equation for the spectral curve +z7 − 4(µ4 − 2µ2a1 + a2 +1 + a2 +2 + a2 +3 + 2a2 +4 + H/2)z5+ +� +16(a2 +2 + a2 +3 + 2a2 +4)µ4 + F1µ2 + G1 +� +z3 +− +� +64a2 +2(a2 +3 + 2a2 +4)µ4 + F2µ2 + G2 +� +z = 0 +contains a sufficient number of integrals of motion for integrability by the Liouville theorem. +There are three polynomials H, F1, F2 of the second order in momenta and two polynomials G1 +and G2 of the fourth order in momenta. +6 +Reductive Homogeneous Spaces +According to [7] in previous sections, we consider symmetric spaces which are reductive homoge- +neous spaces on which the canonical connections have zero torsion. In this section, we consider +one example associated with reductive homogeneous spaces which have non-zero torsion. +Following [7] let us consider symmetric space +SU(3) +S(U(1) × U(1) × U(1)) +and 3 × 3 Lax matrix +L = + + + + + + +q2 +1 + q2 +2 +2iµq1 − p1 +2iµq2 − p2 +−2iµq1 − p1 +4µ2 − q2 +1 − 2ω1 +−q2q1 +−2iµq2 − p2 +−q2q1 +4µ2 − q2 +2 − 2ω2 + + + + + + +. +24 + +The corresponding Hamiltonian of the anharmonic oscillator +H = 1 +4trL2 +���� +µ=0 +− ω2 +1 − ω2 +2 = p2 +1 + p2 +2 +2 ++ (q2 +1 + q2 +2)2 +2 ++ ω1q2 +1 + ω2q2 +2 +commutes with the following quadratic integral of motion +f = −(q1p2 − p1q2)2 − 2ω1p2 +2 − 2ω2p2 +1 − 2(ω1q2 +2 + ω2q2 +1 + 2ω1ω2)(q2 +1 + q2 +2) , +associated with a simple rotation in R3. Other Lax matrices associated with the three-wave +interaction system are discussed in [14]. +Using N-wave hierarchies we can get quadratic integrals of motion associated with N − 2 +rotations in the corresponding Euclidean space. Here we do not discuss the examples of these +calculations in detail. +7 +Conclusion +We present examples of the Killing tensors of valency two generating quadratic integrals of +motion for the integrable systems having additional integrals of motion which are polynomials +of higher order in momenta. +All these Killing tensors are related to the special combinations of rotations and trans- +lations. It will be interesting to find criteria which allow us to extract these special Killing +tensors from the generic solution of the Killing equation on Euclidean, Riemannian and pseudo- +Riemannian spaces of constant curvature. +The work was supported by the Russian Science Foundation (project 21-11-00141). +The second author (AVT) gratefully acknowledges the kind hospitality provided by Yanqi +Lake Beijing Institute of Mathematical Sciences and Applications during his stay in Fall 2022 +when work on this text was finished. +References +[1] Benenti S., Separability in Riemannian manifolds, SIGMA v. 12, 013, 21 pages, 2016. +[2] Conway J.H., Smith D.A., On Quaternions and Octonions: Their Geometry, Arithmetic, +and Symmetry, A. K. Peters, Ltd., Natick, MA, 2003. +[3] Deift P., Li L. C., Nanda T., C. Tomei C., The Toda flow on a generic orbit is integrable, +Comm.on Pure and Applied Math., v.39, n.2, pp.183 - 232, 1986. +[4] Dorizzi B., Grammaticos B., Hietarinta J., Ramani A., Schwarz F., New integrable three- +dimensional quartic potentials, Phys. Lett. A, v.116, n.9, pp.432-436, 1986. +[5] Eisenhart L.P., Separable systems of St¨ackel, Ann. Math., v.35, pp. 284-305, 1934. +[6] Eisenhart L.P., St¨ackel systems in conformal Euclidean space, Ann. Math., v. 36, n. 1, pp. +57-70, 1935. +[7] Fordy A., Kulish P.P., Nonlinear Schr¨odinger equations and simple Lie algebras, Comm. +Math. Phys., v.89, pp.427-443,1983. +[8] Fordy A., Woiciechowski S., Marshall I., A family of integrable quartic potentials related +to symmetric spaces, Phys. Lett. A., v.113, n.6, pp.395-400, 1986. +[9] Gerdjikov V.S., Ivanov R.I., Multicomponent Fokas-Lenells equations on Hermitian sym- +metric spaces, Nonlinearity, v.34, n.2, 939, 2021. +25 + +[10] Grigorev Yu. A.,Tsiganov A.V., Symbolic software for separation of variables in the Hamil- +ton–Jacobi equation for the L-systems, Regul. Chaotic Dyn., v.10:4, pp.413–422, 2005. +[11] Helgason S., +Differential geometry, Lie groups and symmetric spaces, (Graduate studies +in Mathematics, vol.34), AMS, Providence, Rhode Island, 2001. +[12] Horwood J., McLenaghan R., Smirnov R., Invariant classification of orthogonally separable +Hamiltonian systems in Euclidean space, Commun. Math. Phys., v.259, pp.679-709, 2005. +[13] Kalnins E.G, Miller Jr. W., Killing tensors and variable separation for Hamilton-Jacobi +and Helmholtz equations, SIAM J. Math. Anal., v.11, pp.1011-1026, 1980. +[14] Kostov N. A., Tsiganov A. V., New Lax pair for restricted multiple three wave interaction +system, quasiperiodic solutions and bi-Hamiltonian structure, v. 13, no. 6, pp. 593-601, +2008. +[15] Lounesto P., Clifford algebras and spinors, Cambridge University Press, 2001. +[16] Manning H.P., Geometry of four dimensions, New York, The Macmillan Company, 1914. +https://archive.org/details/geometryoffourdi033495mbp +[17] Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Springer Basel +AG, 1989. +[18] Reyman A.G., Interpretation of integrable systems of the anharmonic oscillator type via +the method of orbits, Zap. Nauchn. Sem. LOMI, v.155, pp. 187-189, 1986 and J. Math. +Sci., v. 41:2, pp. 999-1001, 1988. +[19] Reyman A.G., Semenov-Tian-Shansky M.A., +Group-Theoretical Methods in the Theory +of Finite-Dimensional Integrable Systems, In: Dynamical Systems VII (Eds.: V. I. Arnold, +S. P. Novikov), Springer, 1994. +[20] Reyman A.G., Semenov-Tian-Shansky M.A., Integrable Systems, The Computer Research +Institute Publishing, Moscow-Izhvek, 2003. +[21] Trofimov, V. V., Fomenko, A. T., Geometric and algebraic mechanisms of the integrability +of Hamiltonian systems on homogeneous spaces and Lie algebras, In: Dynamical Systems +VII (Eds.: V. I. Arnold, S. P. Novikov), Springer, 1994. +[22] Tsiganov A.V., Killing tensors with nonvanishing Haantjes torsion and integrable systems, +Regular and Chaotic Dynamics, v.20, pp. 463-475, 2015. +[23] Tsiganov A.V., +Two integrable systems with integrals of motion of degree four, Theor. +Math. Phys., v. 186, n.3, pp. 383-394, 2016. +[24] Tsiganov A.V., On integrable systems outside Nijenhuis and Haantjes geometry, J. Geom. +Phys, v.178, 104571, 2022. +[25] Tsiganov A.V., +On Killing tensors in three-dimensional Euclidean space, Theoret. and +Math. Phys., v.212, n.1, 1019-1032, 2022. +[26] Wojciechowski S., Integrability of one particle in a perturbed central quartic potential, Phys. +Scripta, v.31, pp.433-438, 1985. +[27] Walker M., Penrose R., On quadratic first integrals of the geodesic equations for type 22 +spacetimes, Comm. Math. Phys., v.18, n.4, pp.265-274, 1970. +26 + diff --git a/jtE0T4oBgHgl3EQf7QKQ/content/tmp_files/load_file.txt b/jtE0T4oBgHgl3EQf7QKQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98bb43700dd7b2de547a7e41f29a5f9d02db308b --- /dev/null +++ b/jtE0T4oBgHgl3EQf7QKQ/content/tmp_files/load_file.txt @@ -0,0 +1,906 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf,len=905 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='02774v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='SI] 7 Jan 2023 Second order Killing tensors related to symmetric spaces E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='Porubov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='Tsiganov St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='Petersburg State University, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='Petersburg, Russia e–mail: evg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='porub@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='com, andrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='tsiganov@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='com Abstract We discuss the pairs of quadratic integrals of motion belonging to the n-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In contrast to the Eisen- hart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Keywords: Killing tensors, integrable systems, symmetric spaces 1 Introduction Let A and B be non-degenerate symmetric second-order tensor fields on Euclidean space Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' If the Schouten bracket between them is zero [[A, B]] = 0 and the eigenvalue problem (A − λB)ψ = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) has n simple eigenvalues and normal eigenvectors, then A and B generate a n-dimensional linear space of second-order tensor fields, all in involution and with common eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It allows us to calculate n independent functions on the cotangent bundle T ∗Rn T1 = � ij Aijpipj , T2 = � ij Bijpipj , T3 = � ij Kij 3 pipj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , Tn = � ij Kij n pipj in involution {Ti, Tj} = 0 , with respect to canonical Poisson brackets {qi, qj} = 0 , {pi, pj} = 0 , {qi, pj} = δij , i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' By adding suitable potentials H1 = T1 + V1(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , qn) , H2 = T2 + V2(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , qn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , Hn = Tn + Vn(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , qn) we obtain the n-dimensional space of first integrals in involution [5], see also [1, 10, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Thus, second-order tensors A and B define the completely integrable system, if they satisfy a set of conditions in Rn which can be verified without an explicit calculation of all the integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In [22, 23, 24, 25] we found a few pairs of the second-order tensors A and B which also de- fines integrable and superintegrable systems on T ∗Rn, but the corresponding eigenvalue problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) has no simple eigenvalues and normal eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Our main aim is to construct enough 1 examples to find the new criterion that two tensors A and B in Rn define a completely integrable system on T ∗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It is natural to say that these tensors A and B are completely integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this note, we consider tensors A and B related to the special linear combinations of rotation and translations in Euclidean space Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' They are associated with Newton’s equations of motion ¨qα = � β,γ,δ Rα β,γ,−δqβqγqδ − ωαqα, α, β, γ, δ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , N (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) and Hamiltonian H = 1 2 � α gα,−αp2 α − 1 4 � α,β,γ,δ R−α,β,γ,−δqαqβqγqδ + 1 2 � α ωα (qα)2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) which were studied in [7, 8, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here gα,−α and Rα β,γ,−δ are metric and curvature tensors (constant tensors), which correspond to classes A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='III, BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='I, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='I and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='III of symmetric spaces in the Cartan classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case, A = g is metric in Euclidean space and B = K is a Killing tensor, which satisfies the Killing equation ∇iKjk + ∇jKki + ∇kKij = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) where ∇ is the Levi-Civita connection of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Relation of these Hamiltonian systems with the generalised multicomponent NLS hierarchy gives the Lax matrix L(µ) = µ2A + µ � α qα� eα − e−α � − 1 a � α gα,−αpα � eα + e−α � + 1 a � α,β qαqβ[eα, e−β] + Λ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) and integrals of motion in the involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here matrix Λ depends on parameters ωi and describes a shift of the orbit obtained in the framework of the Adler-Kostant-Symes theorem [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' All these results were reproduced in the various textbooks [17, 19, 20, 21], where readers can find all the necessary definitions and details, see also [9] and references within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When all ωα = 0, Hamiltonian H (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) commutes with a family of the noncommutative linear integrals of motion associated with the various combinations of rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case, the Lax matrix allows us to find a family of commuting integrals of motion, the numbers of which do not permit us to talk about integrability by the Liouville theorem in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' If ωα ̸= 0 the Lax matrix L(µ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) generates a necessary number of the integrals of motion, which are polynomials in momenta of order two, four, six, eight, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The leading parts of these polynomials , H(2ℓ) i = 2ℓ � jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='m Kjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='m i pjpk · · · pm + · · · , ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , define the Killing tensors of valency 2ℓ in Euclidean space Rn [[g, Ki]] = 0 , where [[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=']] is a Schouten bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Below we will restrict ourselves to the study of second-order Killing tensors in a few low-dimensional Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Proposition 1 The Hamiltonian H (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) is in involution with the polynomial of fourth order in momenta G = −1 4 � α,β,γ,δ R−α,β,γ,−δpαpβpγpδ + � α,β Sα,β(q)pαpβ + W(q) , whose leading part is defined by the curvature tensor R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 2 We can prove this proposition using properties of the Cartan-Weil basis, definitions of the Killing form, and metric and curvature tensors on symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It is beyond the scope of this article to discuss these properties and fully prove this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' So, we only single out this integral of motion from other coefficients of the equation defining the spectral curve of the Lax matrix (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 Killing tensors of valency two In Euclidean space, the generic Killing tensor of valency two is given by K = � i,j aijXi ◦ Xj + � i,j,k bijkXi ◦ Xj,k + � i,j,k,m cijkmXi,j ◦ Xk,m , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6) where Xi = ∂i Xi,j = qiXj − qjXi , ∂k = ∂ ∂qk is a basis of translations and rotations, aij, bijk and cijm are parameters and ◦ denotes symmetric product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Dimension of vector space of the Killing tensors of valency m in n-dimensional Euclidean space is given by the Delong-Takeuchi-Thompson formula d = 1 n �n + m m + 1 ��n + m − 1 m � = 1 n �n + 2 3 ��n + 1 2 � = n(n + 2)(n + 1)2 12 , where we put m = 2 calculating the number of parameters aij, bijk and cijkm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Thus, we can find all the Killing tensors of valency two related to Hamiltonian H = T + V (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) solving the equation d (KdV ) = 0 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7) which means that 1-form KdV is an exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here V is a function on Rn, canonically lifted to T ∗Rn and KdV denotes the 1-form image of dV by K, interpreted as a linear endomorphism over 1-forms, whose components are gα,βKβ,γ∂γV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Substituting K (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6) and potential V = 1 4 � α,β,γ,δ R−α,β,γ,−δqαqβqγqδ − 1 2 � α ωα (qα)2 into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7) we obtain a linear system of equations for coefficients aij, bijk and cijkm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Then we can study the properties of the obtained solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Construction of the polynomials of higher order in momenta commuting with the Hamiltonian H is an open question in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' According to [1, 5, 12] Sylvester’s criterion can be used to verify that K has real and simple eigenvalues with respect to metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The vanishing of the Haantjes torsion of K allows us to prove that eigenvectors are normal, see historical remarks in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The Haantjes tensor (torsion) HK(u, v) = K2NK(u, v) + NK(Ku, Kv) − K (NK(Ku, v) + NK(u, Kv)) , is defined by using the Nijenhuis tensor (torsion) NK(u, v) = K2[u, v] + [Ku, Kv] − K ([Ku, v] + [u, Kv]) , where u, v are arbitrary vector fields and [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='] denotes the commutator of two vector fields [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Below we prove that associated with Hamiltonian H (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) Killing tensors of valency two have non-zero Haantjes torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Nevertheless, using Lax matrix L(µ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) we can get a set of completely integrable Killing tensors related to the special values of parameters aij, bijk and cijkm in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It could be useful to construct similar Killing tensors on non-Euclidean space [6, 24, 25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 3 2 Symmetric spaces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='III type Let us consider equations of motion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) in Euclidean space Rmn and Hamiltonian H (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) associated with the Riemannian pair SU(m + n)/S � U(m) × U(n) � , m ≤ n , n + m ≥ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The typical representation of su(m + n) is a set of (m + n) × (m + n) matrices with an obvious block-matrix structure associated with the following Cartan decomposition g ≡ k ⊕ p, k = s(u(m) ⊕ u(n)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here k consists of block-diagonal matrices, while the linear space p is spanned by block-off- diagonal matrices: k ≃ � u(m) 0 0 u(n) � , p ≃ � 0 P + iQ P T − iQT 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The corresponding Cartan element A in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) acts as −I on so(m) and as I on so(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Following to [7, 8] we choose a representation in which Lax matrix (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) reads as L(µ) = \uf8eb \uf8ed a − 2µ2Im 0 0 b + 2µ2In \uf8f6 \uf8f8 + \uf8eb \uf8ed C P − 2iµQ P T + 2iµQT ¯C \uf8f6 \uf8f8 , i = √ −1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) where Im and In are the m×m and n×n unit matrices, a and b are diagonal matrices depending on m real numbers ak and n real numbers parameters bi a = diagm(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , am) , b = diagn(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , bn) , ai, bi ∈ R , and T means matrix transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Matrices C and ¯C are symmetric m × m and n × n matrices with entries depending on coordinates Ci,j = − n � k=1 q(i−1)n+k q(j−1)n+k , i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , m, and ¯Ci,j = − m−1 � k=0 qkn+iqkn+j , i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Matrices P and Q are m × n matrices depending linearly on p and q with entries Pij = p(i−1)n+j , and Qij = q(i−1)n+j , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , m, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , n , see examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The Hamiltonian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) is given by H = 1 4TraceL2 ���� µ=0 − 1 4 m � j=1 a2 j − 1 4 n � i=1 b2 i = 1 2 n � i=1 p2 i + 1 2 m−1 � j=0 � n � i=1 q2 jn+i �2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) + m−1 � k,j=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='k>j � n � i=1 qjn+iqkn+i �2 + 1 2 m−1 � j=0 aj+1 � n � i=1 q2 jn+i � − 1 2 n � i=1 bi \uf8eb \uf8ed m−1 � j=0 q2 jn+i \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When ai ̸= 0 and bi ̸= 0, there are two basic sets of integrals of motion obtained from the characteristic polynomial of the Lax matrix τ(z, µ) = det � z I − L(µ) � , which are associated with so(m) and so(n), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Because {τ(x, λ), τ(y, µ)} = 0 , all these integrals of motion are in the involution for each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 First set of the independent integrals of motion The m residues of the function ∆1(z, µ) = τ(z, µ) �m i=1(z − ai + 2µ2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) at z = ai − 2µ2 generate mn independent integrals of motion h(2ℓ) i Residue ∆1(z, µ)|z=ai−2µ2 = n−1 � k=0 µ2kh � 2(n−k) � i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , m, which are polynomials of degree at most 2m since we take m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' So, there are m quadratic polynomials in momenta h(2) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , h(2) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' m quartic polynomials in momenta h(4) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , h(4) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' m sextic polynomials in momenta h(6) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , h(6) m ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' m polynomials of 2m-order in momenta h(2m) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , h(2m) m and m(n − m) remaining polynomials of 2m-order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Polynomials of the second order in momenta have the following form h(2) i = − m � k̸=i M 2 ik ai − ak + ti(p) + vi(q) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) where functions Mik = n � Jjℓ , Jjℓ = qjpℓ − qℓpj , constitute realization of Lie algebra so∗(m) associated with compositions of n simple rotations in Rmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Functions ti(p) correspond to compositions of the n translations ti(p) = n � p2 ℓ , and vi(q) are polynomials of the fourth order in coordinates qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 Second set of the independent integrals of motion The n residues of the function ∆2(z, µ) = τ(z, µ) �n i=1(z − bi − 2µ2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) at z = bi + 2µ2 generate mn independent integrals of motion H(2ℓ) i Residue ∆2(z, µ)|z=bi+2µ2 = m−1 � k=0 µ2kH � 2(m−k) � i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , n which are polynomials of order 2ℓ in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' So, there are n quadratic polynomials in momenta H(2) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , H(2) n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 5 n quartic polynomials in momenta H(4) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , H(4) n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' n sextic polynomials in momenta H(6) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , H(6) n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' n polynomials of 2m-order in momenta H(2m) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , H(2m) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case polynomials of second order in momenta have the following form H(2) i = n � k̸=i N 2 ik bi − bk + Ti(p) + Ui(q) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6) where functions Nik = m � Jjℓ , Jjℓ = qjpℓ − qℓpj , form realization of so∗(n) via compositions of m simple rotations in Rmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Functions Ti(p) correspond to compositions of the m translations Ti(p) = n � ℓ p2 ℓ , and Ui(q) are polynomials of the fourth order in coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Summing up, we have n + m − 1 quadratic integrals of motion f1 + · · · + fm = 2H = F1 + · · · Fn , associated with the linear combinations of rotations, which realise so∗(m) and so∗(n), and with the linear combinations of translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Proposition 2 Equations of motion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) defined by H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) have only n+ m− 1 independent quadratic integrals of motion in involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' We can directly prove this proposition for low-dimensional case R4 substituting generic solution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6) of the Killing equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7) and solving the resulting system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In the generic case, we have to calculate a number of unknown coefficients by the Delong- Takeuchi-Thompson formula and compare this number with a rank of this system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 Euclidean space Rn, case m = 1 When m = 1 we have the so-called Garnier system and all the second-order Killing tensors Ki = − � k̸=i Xik · Xik bi − bk − Xi · Xi consist of single rotation and single translation Xi,k = qi∂k − qk∂i , Xi = ∂i , and their Haanties torsion is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Thus, the Hamilton-Jacoby equation H = Ei admits additive separation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Separated variables are the standard elliptic coordinates in Rn and, therefore, this integrable system constrained to an ellipsoid remains integrable [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When m > 1 the corresponding Killing tensors of valency two have nontrivial Haantjes torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It means that the Hamilton-Jacobi equation H = E does not admit the separation of variables in the curvilinear orthogonal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Below we present a few examples of the corresponding quadratic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 Euclidean space R4, case m = n = 2 Because so(4) ≃ so(2) × so(2) there appear double rotations or Clifford displacements in R4, which can be associated with the left- and right-multiplication by a unit quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It is a classical object in the geometry of the fourth-dimensional Euclidean space [2, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The 4 × 4 Lax matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) is equal to L(µ) = \uf8eb \uf8ec \uf8ec \uf8ed q2 1+q2 2+a1−2µ2 q1q3+q2q4 p1−2iµq1 p2−2iµq2 q1q3+q2q4 q2 3+q2 4+a2−2µ2 p3−2iµq3 p4−2iµq4 p1−2iµq1 p3−2iµq3 b1−q2 1−q2 3+2µ2 −q1q2−q3q4 p2−2iµq2 p4−2iµq4 −q1q2−q3q4 b2−q2 2−q2 4+2µ2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7) so Hamiltonian H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) has the form H =p2 1 2 + p2 2 2 + p2 3 2 + p2 4 2 + 1 2(q2 1 + q2 2)2 + 1 2(q2 3 + q2 4)2 + (q1q3 + q2q4)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8) +a1 − b1 2 q2 1 + a1 − b2 2 q2 2 + a2 − b1 2 q2 3 + a2 − b2 2 q2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Because 1 2(q2 1 + q2 2)2 + 1 2(q2 3 + q2 4)2 + (q1q3 + q2q4)2 = (q2 1 + q2 2 + q2 3 + q2 4)2 2 − (q1q4 − q2q3)2 this Hamiltonian coincides with (13b) case from the paper [8] after permutation of indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Spectral curve of the Lax matrix L(µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7) is a non-hyperelliptic curve defined by char- acteristic equation C : det � zI − L(µ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In generic case its genus is equal to five g = 5, when a1 = a2 or b1 = b2 genus of this non- hyperelliptic curve C is equal to four g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' First set of integrals of motion Two residues of the function ∆(z, µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) ∆(z, µ) = det � zI − L(µ) � (z − a1 + 2µ2)(z − a2 + 2µ2) are equal to Res|z=ai−2µ2 ∆(z, µ) = 4µ2fi + gi , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where f1,2 and g1,2 are the second and fourth-order polynomials in momenta, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Third residue in infinity Res|z=∞ ∆(z, µ) = −4µ2(f1 + f2) − (g1 + g2) , give rise to integrals of motions f1 + f2 = 2H and g1 + g2 = f3 which are polynomials of the second order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Quadratic integrals of motion f1,2 have the following form f1 = − M 2 12 a1 − a2 + p2 1 + p2 2 + v1 and f2 = M 2 12 a1 − a2 + p2 3 + p2 4 + v2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9) where v1 =(q2 1 + q2 2 + q2 3 + a1 − b1)q2 1 + (q2 1 + q2 2 + q2 4 + a1 − b2)q2 2 + 2q1q2q3q4 , v2 =(q2 1 + q2 3 + q2 4 + a2 − b1)q2 3 + (q2 2 + q2 3 + q2 4 + a2 − b2)q2 4 + 2q1q2q3q4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 7 Here M12 is a function associated a double rotation in R4 M12 = J1,3 + J2,4 = (q1p3 − q3p1) + (q2p4 − q4p2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It commutes with terms in f1,2 associated with translations {M12, p2 1 + p2 2} = {M12, p2 3 + p2 4} = 0 , and with function associated with the second independent double rotation in R4 N12 = J1,2 + J3,4 = (q1p2 − q2p1) + (q3p4 − p3q4) , so that {M12, N12} = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The second independent rotation appears in the following linear combination of the integrals of motion f3 =(b1 + b2)H − g1 − g2 − a1f1 − a2f2 =N 2 12 − (b1−b2)((q2 1+q2 2+q2 3+q2 4)(q2 1 −q2 2 +q2 3−q2 4 )+(q2 1 −q2 2)a1+(q2 3−q2 4)a2−(q2 1 +q2 3)b1+(q2 2 +q2 4)b2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When b1 = b2 linear integral of motion N12 is a function on f1,2 and g1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The leading term in quartic polynomials in momenta is the perfect square (a1 − a2)g1 = (p1p4 − p2p3)2 + · · · and (a2 − a1)g1 = (p1p4 − p2p3)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' There is also a combination of the integrals of motion g3 = 2H2 − a1g1 − a2g2 with the leading part defined by the curvature tensor R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8) g3 = −1 4 � α,β,γ,δ R−α,β,γ,−δpαpβpγpδ + · · · = −1 2(p2 1 + p2 2 + p2 3 + p2 4)2 − (p1p4 − p2p3)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Second set of integrals of motion Residues of the function ∆(z, µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) ∆(z, µ) = det � zI − L(µ) � (z − b1 − 2µ2)(z − b2 − 2µ2) are equal to Res|z=bi+2µ2 ∆(z, µ) = −4µ2Fi + Gi , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Res|z=∞ ∆(z, µ) = 8µ2H − (G1 + G2) , F1 + F2 − 2H = 0 , where polynomials of second order in momenta F1,2 and G1 + G2 are independent for each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Quadratic integrals of motion F1,2 are equal to F1 = N 2 12 b1 − b2 + p2 1 + p2 3 + V1 and F2 = − N 2 12 b1 − b2 + p2 2 + p2 4 + V2 , 8 where V1 = (q2 1 + q2 2 + q2 3 + a1 − b1)q2 1 + (q2 1 + q2 3 + q2 4 + a2 − b1)q2 3 + 2q1q2q3q4 , V2 = (q2 1 + q2 2 + q2 4 + a1 − b2)q2 2 + (q2 2 + q2 3 + q2 4 + a2 − b2)q2 4 + 2q1q2q3q4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here N12 is a function associated with the double rotation in R4: N12 = J1,2 + J3,4 = (q1p2 − q2p1) + (q3p4 − p3q4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' This function commutes with terms in F1,2 associated with translations {N12, p2 1 + p2 3} = {N12, p2 2 + p2 4} = 0 , and with function associated with the independent second double rotation M12 = J1,3 + J2,4 = (q1p3 − q3p1) + (q2p4 − p2q4) , which was included in the definition of f1,2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9) from the first set of integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' This function also appears in the following linear combination of integral of motion from the second set of integrals of motion F3 = G1 + G2 − b1F1 − b2F2 − (a1 + a2)H = M 2 12 + (a1−a2) � p2 3+p2 4−p2 1−p2 2+(q2 1 −q2 3)b1+(q2 2−q2 4 )b2−(q2 1+q2 2)a1+(q2 3 +q2 4)a2−(q2 1+q2 2)2+(q2 3 +q2 4)2� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When a1 = a2 linear integral of motion N13 is a function on F1,2 and G1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The leading term in the quartic invariants is a perfect square (b1 − b2)G1 = (p1p4 − p2p3)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' and (b2 − b1)G2 = (p1p4 − p2p3)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' As above, there are quartic invariant G3 = 2H2 − b1G1 − b2G2 with leading term defined by the curvature tensor R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8) G3 = −1 4 � α,β,γ,δ R−α,β,γ,−δpαpβpγpδ + · · · = 1 2(p2 1 + p2 2 + p2 3 + p2 4)2 − (p1p4 − p2p3)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Summing up, there are only m + n − 1 = 3 independent quadratic integrals of motion among f1, f2, f3 and F1, F2, F3 in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The corresponding Killing tensors of valency two have non-zero Haantjes torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 Euclidean space R6, case m = 2 and n = 3 The 5 × 5 Lax matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) reads as L(µ) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed q2 1+q2 2+q2 3+a1−2µ2 q1q4+q2q5+q3q6 p1−2iµq1 p2−2iµq2 p3−2iµq3 q1q4+q2q5+q3q6 q2 4+q2 5+q2 6+a2−2µ2 p4−2iµq4 p5−2iµq5 p6−2iµq6 p1+2iµq1 p4+2iµq4 b1−q2 1−q2 4+2µ2 −q1q2−q4q5 −q1q3−q4q6 p2+2iµq2 p5+2iµq5 −q1q2−q4q5 b2−q2 2−q2 5+2µ2 −q2q3−q5q6 p3+2iµq3 p6+2iµq6 −q1q3−q4q6 −q2q3−q5q6 b3−q2 3−q2 6+2µ2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , so Hamiltonian H (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) reads as H =1 2 6 � i=1 p2 i + (q2 1 + q2 2 + q2 3)2 2 + (q2 4 + q2 5 + q2 6)2 2 + (q1q4 + q2q5 + q3q6)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='10) −q2 1 + q2 4 2 b1 − q2 2 + q2 5 2 b2 − q2 3 + q2 6 2 b3 + q2 1 + q2 2 + q2 3 2 a1 + q2 4 + q2 5 + q2 6 2 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 9 When ai = 0 and bi = 0 this Hamiltonian commutes with the following linear integrals of motion M12 =(q1p4 − p4q1) + (q2p5 − p2q5) + (q3p6 − p3q6) , N12 = (q1p2 − p1q2) + (q4p5 − p4q5) , N13 =(q1p3 − p1q3) + (q4p6 − p4q6) , N23 = (q2p3 − p2q3) + (q5p6 − p5q6) , associated with rotations in R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The equation for the spectral curve of the Lax matrix contains only five commuting func- tions H, F1, F2 and G1, G2 τ(z, µ) =z5 − 2µ2z4 − 2(4µ4 + H)z3 + (16µ6 + 4Hµ2 + F1)z2 +(16µ8 + 8Hµ4 − 4F 2 2 µ2 + G1)z − 32µ10 − 16Hµ6 + (8F 2 2 − 4F1)µ4 − 2G1µ2 + G2 , where F1 = M 2 12 − N 2 12 − N 2 13 − N 2 23 , F2 = M 2 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Thus, we must find the missing integral of motion using other properties of the Lax matrix similar to the full Toda lattice [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In the generic case ai ̸= 0 and bi ̸= 0 spectral curve of the Lax matrix L(µ) is a genus six non-hyperelliptic curve, that allows us to get six independent integrals of motion in the involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' First set of integrals of motion Two residues of the function ∆(z, µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) ∆(z, µ) = det � zI − L(µ) � (z − a1 + 2µ2)(z − a2 + 2µ2) are equal to Res|z=ai−2µ2 ∆(z, µ) = −16µ4fi + µ2gi + wi , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where f1,2 are polynomials of second order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Because 2m = 4 other integrals of motion g1,2 and w1,2 are polynomials of fourth order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Residue at infinity is equal to Res|z=∞ ∆(z, µ) = 32µ4H − µ2(g1 + g2) − (w1 + w2) , f1 + f2 − 2H = 0 , Integrals of motion f1,2 are polynomials of second order in momenta f1 = − M 2 12 b1 − b2 + p2 1 + p2 2 + p2 3 + v1 , and f2 = M 2 12 b1 − b2 + p2 4 + p2 5 + p2 6 + v2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='11) where v1 =(q2 1 + q2 2 + q2 3 + q2 4 + a1 − b1)q2 1 + (q2 1 + q2 2 + q2 3 + q2 5 + a1 − b2)q2 2 +(q2 1 + q2 2 + q2 3 + q2 6 + a1 − b3)q2 3 + 2q1q2q4q5 + 2q1q3q4q6 + 2q2q3q5q6 , v2 =(q2 1 + q2 4 + q2 5 + q2 6 + a2 − b1)q2 4 + (q2 2 + q2 4 + q2 5 + q2 6 + a2 − b2)q2 5 +(q2 3 + q2 4 + q2 5 + q2 6 + a2 − b3)q2 6 + 2q1q2q4q5 + 2q1q3q4q6 + 2q2q3q5q6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here M12 is a function associated with the triple rotation in R6: M12 = J14 + J25 + J36 = (q1p4 − p4q1) + (q2p5 − p2q5) + (q3p6 − p3q6) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='12) 10 Linear combinations of other integrals of motion are associated with double rotations in R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' For instance, the polynomial of second order in momenta f3 = 2(b1 + b2 + b3)H + g1 + g2 4 − 2a1f1 − 2a2f2 is equal to f3 = N 2 12 + N 2 13 + N 2 23 + (p2 1 + p2 4)b1 + (p2 2 + p2 5)b2 + (p2 3 + p2 6)b3 + v3 , where N12 =J12 + J45 = (q1p2 − p1q2) + (q4p5 − p4q5) , N13 =J13 + J46 = (q1p3 − p1q3) + (q4p6 − p4q6) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='13) N23 =J23 + J56 = (q2p3 − p2q3) + (q5p6 − p5q6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' and v3 = (q4 1 + q2 1q2 2 + q2 1q2 3 + 2q2 1q2 4 + 2q1q2q4q5 + 2q1q3q4q6 + q4 4 + q2 4q2 5 + q2 4q2 6 + a1q2 1 + a2q2 4)b1 + (q2 1q2 2 + 2q1q2q4q5 + q4 2 + q2 2q2 3 + 2q2 2q2 5 + 2q2q3q5q6 + q2 4q2 5 + q4 5 + q2 5q2 6 + a1q2 2 + a2q2 5)b2 + (q2 1q2 3 + 2q1q3q4q6 + q2 2q2 3 + 2q2q3q5q6 + q4 3 + 2q2 3q2 6 + q2 4q2 6 + q2 5q2 6 + q4 6 + a1q2 3 + a2q2 6)b3 − (q2 1 + q2 4)b2 1 − (q2 2 + q2 5)b2 2 − (q2 3 + q2 6)b2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' We will omit such explicit expressions for the bulky potentials below for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Second set of integrals of motion Three residues of the function ∆(z, µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) ∆(z, µ) = det � zI − L(µ) � (z − b1 − 2µ2)(z − b2 − 2µ2)(z − b3 − 2µ2) are equal to Res|z=bi+2µ2 ∆(z, µ) = 4µ2Fi + Gi , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where Fi and Gi are the second and fourth-order polynomials in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Residue in infinity reads as Res|z=∞ ∆(z, µ) = 8µ2H − (G1 + G2 + G3) , 2H + F1 + F2 + F3 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Polynomials of second order in momenta are defined by double rotations and double translations 11 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6) F1 = − N 2 12 b1 − b2 − N 2 13 b1 − b3 − p2 1 − p2 4 − (q2 1 + q2 2 + q2 3 + 2q2 4 + a1 − b1)q2 1 − (q2 4 + q2 5 + q2 6 + a2 − b1)q2 4 − 2(q2q5 + q3q6)q1q4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' F2 = − N 2 21 b2 − b1 − N 2 23 b2 − b3 − p2 2 − p2 5 − (q2 1 + q2 2 + q2 3 + 2q2 5 + a1 − b2)q2 2 − (q2 4 + q2 5 + q2 6 + a2 − b2)q2 5 − 2(q1q4 + q3q6)q2q5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' F3 = − N 2 31 b3 − b1 − N 2 32 b3 − b2 − p2 3 − p2 5 − (q2 1 + q2 2 + q2 3 − 2q2 6 + a1 − b3)q2 3 − (q2 4 + q2 5 + q2 6 + a2 − b3)q2 6 − 2(q1q4 + q2q5)q3q6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Functions Nij = −Nji (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='13) are associated with double rotations in R6 and realisations of so(3) algebra with the brackets {N12, N13} = N23 , {N13, N23} = N12 , {N23, N12} = N13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Leading term of the independent on F1, F2 and F3 polynomial of second order in momenta F4 = G1 + G2 + G3 − b1F1 − b2F2 − b3F3 − (a1 + a2)H = M 2 12 − a1 − a2 2 � p2 1 + p2 2 + p2 3 − p2 4 − p2 5 − p2 6 + V4 � includes function M12 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='12) associated with the triple rotation in R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The corresponding potential V4 is equal to V4 = (q2 1 + q2 2 + q2 3 + q2 4 + q2 5 + q2 6)(q2 1 + q2 2 + q2 3 − q2 4 − q2 5 − q2 6) + (q2 1 + q2 2 + q2 3)a1 − (q2 4 + q2 5 + q2 6)a2 − (q2 1 − q2 4)b1 − (q2 2 − q2 5)b2 − (q2 3 − q2 6)b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When a1 = a2 linear polynomial M12 commutes with all the integrals of motion H, Fk and Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The following combination of quartic integrals of motion G4 = 2H2 − b1G1 − b2G2 − b3G3 has the leading term defined by the curvature tensor R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='10) G4 = − 1 4 � α,β,γ,δ R−α,β,γ,−δpαpβpγpδ + · · · = 1 2(p2 1 + p2 2 + p2 3)2 + 1 2(p2 4 + p2 5 + p2 6)2 + (p1p4 + p2p5 + p3p6)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Summing up, there are only m + n − 1 = 4 independent quadratic integrals of motion among f1, f2, f3 and F1, F2, F3, F4 in R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The corresponding Killing tensors of valency two have non- trivial Haantjes torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 Euclidean space R9, case m = n = 3 The 6 × 6 Lax matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) is L(µ) = � L11 L12 L21 L22 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='14) where L11 = � −2µ2+q2 1+q2 2+q2 3+a1 q1q4+q2q5+q3q6 q1q7+q2q8+q3q9 q1q4+q2q5+q3q6 −2µ2+q2 4+q2 5+q2 6+a2 q4q7+q5q8+q6q9 q1q7+q2q8+q3q9 q4q7+q5q8+q6q9 −2µ2+q2 7+q2 8+q2 9+a3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' L22 = � 2µ2−q2 1−q2 4−q2 7+b1 −q1q2−q4q5−q7q8 −q1q3−q4q6−q7q9 −q1q2−q4q5−q7q8 2µ2−q2 2−q2 5−q2 8+b2 −q2q3−q5q6−q8q9 −q1q3−q4q6−q7q9 −q2q3−q5q6−q8q9 2µ2−q2 3−q2 6−q2 9+b3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' L12 = � p1−2iµq1 p2−2iµq2 p3−2iµq3 p4−2iµq4 p5−2iµq5 p6−2iµq6 p7−2iµq7 p8−2iµq8 p9−2iµq9 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' L21 = � p1+2iµq1 p4+2iµq4 p7+2iµq7 p2+2iµq2 p5+2iµq5 p8+2iµq8 p3+2iµq3 p6+2iµq6 p9+2iµq9 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Hamiltonian H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) is given by H =1 2 9 � i=1 p2 i + (q2 1 + q2 2 + q2 3)2 2 + (q2 4 + q2 5 + q2 6)2 2 + (q2 7 + q2 8 + q2 9)2 2 +(q1q4 + q2q5 + q3q6)2 + (q1q7 + q2q8 + q3q9)2 + (q4q7 + q5q8 + q6q9)2 −q2 1 + q2 4 + q2 7 2 b1 − q2 2 + q2 5 + q2 8 2 b2 − q2 3 + q2 6 + q2 9 2 b3 +q2 1 + q2 2 + q2 3 2 a1 + q2 4 + q2 5 + q2 6 2 a2 + q2 7 + q2 8 + q2 9 2 a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' First set of integrals of motion Residues of the function ∆(z, µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) ∆(z, µ) = det � zI − L(µ) � (z − a1 + 2µ2)(z − a2 + 2µ2)(z − a3 + 2µ2) are equal to Res|z=ai+2µ2 ∆(z, µ) = 16µ4fi + µ2gi + si , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Res|z=∞ ∆(z, µ) = 32Hµ4 − (g1 + g2 + g3)µ2 − (s1 + s2 + s3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 13 where second-order polynomials in momenta are equal to f1 = M 2 12 a1 − a2 + M 2 13 a1 − a3 − p2 1 − p2 2 − p2 3 − (2q2 2 + 2q3 2 + q4 2 + q7 2 + a1 − b1)q1 2 −(2q32 − q52 − q82 − a1 + b2)q22 − (q32 + q62 + q92 + a1 − b3)q32 −2q2q3(q5q6 + q8q9) − 2q1q2(q4q5 + q7q8) − 2q1q3(q4q6 + q7q9) − q14 − q24 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='f2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='a2 − a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='a2 − a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 − (q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + 2q5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + 2q6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + q7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + a2 − b1)q4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='−(q22 + 2q62 + q82 + a2 − b2)q52 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a2 − b3)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='−q5q4(2q1q2 + 2q7q8) − 2q6q4(q1q3 + q7q9) − 2q5q6(q2q3 + q8q9) − q4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 − q4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='f3 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='a3 − a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='M 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='a3 − a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 + 2q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 + 2q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a3 − b1)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='−(q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 + 2q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a3 − b2)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a3 − b3)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='−2q7q8(q1q2 + q4q5) − 2q7q9(q1q3 + q4q6) − 2q8q9(q2q3 + q5q6) − q4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7 − q4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here Mij are given by M12 =J14 + J25 + J36 = (q1p4 − p1q4) + (q2p5 − p2q5) + (q3p6 − p3q6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' M13 =J17 + J28 + J39 = (q1p7 − p1q7) + (q2p8 − p2q8) + (q3p9 − p3q9) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='15) M23 =J47 + J58 + J69 = (q4p7 − p4q7) + (q5p8 − p5q8) + (q6p9 − p6q9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The following combination of integrals of motion is also a quadratic polynomial in momenta f4 = g1 + g2 + g2 4 + 2a1f1 + 2a2f2 + 2a3f3 , which has the form f4 = − \uf8eb \uf8ed n � j=1 bj \uf8f6 \uf8f8 � nm � i=1 p2 i � + n � j=1 bj �m−1 � i=0 p2 j+im � + N 2 12 + N 2 23 + N 2 31 + u4(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here Nij: N12 =J12 + J45 + J78 = (q1p2 − p1q2) + (q4p5 − p4q5) + (q7p8 − p7q8) , N13 =J13 + J46 + J79 = (q1p3 − p1q3) + (q4p6 − p4q6) + (q7p9 − p7q9) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='16) N23 =J23 + J56 + J89 = (q2p3 − p2q3) + (q5p6 − p5q6) + (q8p9 − p8q9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 14 Functions Mij (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='15) and Nij (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='16) are associated with two realizations of so∗(3) by using independent triple rotations in R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The Lie-Poisson brackets are {M12, M13} = M23 , {M13, M23} = M12 , {M23, M12} = M13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' and {N12, N13} = N23 , {N13, N23} = N12 , {N23, N12} = N13 , so that {Nij, Mkl} = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Leading terms in polynomials of six order in momenta are si = 1 (ai − aj)(ai − ak) � p1p5p9 + p2p6p7 + p3p4p8 − p1p6p8 − p2p4p9 − p3p5p7 �2 + · · · , and, therefore, the sum of these polynomials is a polynomial of the fourth order in momenta which is independent of g1, g2 and g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Second set of the integrals of motion Residues of the function ∆(z, µ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) ∆(z, µ) = det � zI − L(µ) � (z − b1 − 2µ2)(z − b2 − 2µ2)(z − b3 − 2µ2) are equal to Res|z=bi+2µ2 ∆(z, µ) = 16µ4Fi + µ2Gi + Si , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Res|z=∞ ∆(z, µ) = 32Hµ4 − (G1 + G2 + G3)µ2 − (S1 + S2 + S3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Second-order polynomials in momenta have the following form F1 = − N 2 12 b1 − b2 − N 2 13 b1 − b3 − p2 1 − p2 4 − p2 7 − (q2 1 + q2 2 + q2 3 + q2 4 + q2 7 + a1 − b1)q2 1 − (q2 1 + q2 4 + q2 5 + q2 6 + q2 7 + a2 − b1)q2 4 − (q2 1 + q2 4 + q2 7 + q2 8 + q2 9 + a3 − b1)q2 7 − 2q1q4(q2q5 + q3q6) − 2q1q7(q2q8 + q3q9) − 2q4q7(q5q8 + q6q9) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='F2 = − N 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='b2 − b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='N 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='b2 − b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 + a1 − b2)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 + a2 − b2)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a3 − b2)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− 2q2q5(q1q4 + q3q6) − 2q2q8(q1q7 + q3q9) − 2q5q8(q4q7 + q6q9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='F3 = − N 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='b3 − b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='N 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='b3 − b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 − p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a1 − b3)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a2 − b3)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 − (q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='7 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8 + q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 + a3 − b3)q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='− 2q3q6(q1q4 + q2q5) − 2q3q9(q1q7 + q2q8) − 2q6q9(q4q7 + q5q8) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 15 Functions Mij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Nkl are given by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='15,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The following combination of integrals of motion is also second order polynomial in mo- menta F4 = 1 8(G1 + G2 + G3) − b1F1 − b2F2 − b3F3 which is independent on F1, F2 and F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It has the form F4 = 1 2 \uf8eb \uf8ed m � j=1 aj \uf8f6 \uf8f8 � n � i=1 p2 i � − 1 2 m−1 � j=0 aj+1 � n � i=1 p2 jn+i � + M 2 12 2 + M 2 13 2 + M 2 23 2 + U4(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Leading terms in polynomials of six order in momenta are Si = 1 (bi − bj)(bi − bk) � p1p5p9 + p2p6p7 + p3p4p8 − p1p6p8 − p2p4p9 − p3p5p7 �2 + · · · , and, therefore, the sum of these polynomials is a polynomial of the fourth order in momenta G4 = S1 + S2 + S3 , which is independent on g1, g2 and g3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Summing up, we have integrals of motion of second, fourth-order and sixth order in mo- menta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Among them, there are five independent quadratic integrals of motion because f1 + f2 + f3 = 2H = F1 + F2 + F3 3 Symmetric space of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='I type The compact group Sp(n) of 2n × 2n matrices which are both symplectic and unitary is asso- ciated with the root space Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Because Sp(n) U(n) ⊂ SU(2n) S (U(n) × U(n)) we can get the Lax matrices starting with Lax matrices (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Roughly speaking we have to make n × n matrices Q and P symmetric, divide off-diagonal entries of P by two and impose suitable restrictions on parameters ai and bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Below we present these Lax matrices at n = 2 and n = 3 and discuss the corresponding quadratic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 Euclidean space R3, case n = 2 The 4 × 4 Lax matrix is equal to L(µ) = \uf8eb \uf8ec \uf8ec \uf8ed −2µ2+q2 1+q2 2+a1 q1q2+q2q3 p1−2iµq1 p2 2 −2iµq2 q1q2+q2q3 −2µ2+q2 2+q2 3+a2 p2 2 −2iµq2 p3−2iµq3 p1+2iµq1 p2 2 +2iµq2 2µ2−q2 1−q2 2+b1 −q1q2−q2q3 p2 2 +2iµq2 p3+2iµq3 −q1q2−q2q3 2µ2−q2 2−q2 3+b2 \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) where a2 − a1 = b1 − b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) The Hamiltonian is given by H = p2 1 2 + p2 2 4 + p2 3 2 + (q2 1 + 2q2 2 + q2 3)2 2 − (q1q3 − q2 2)2 + a1 − b1 2 q2 1 + (a1 − b2) q2 2 + a1 + b1 − 2b2 2 q2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) 16 It coincides with the (13c) case from the paper [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' After canonical change of variables p2 → √ 2p2 and q2 → q2/ √ 2 we obtain standard metric g = diag(1, 1, 1) in Euclidean space and integrable three-dimensional quartic potential at bi = ai = 0 V = 1 2(q2 1 + q2 2 + q2 3)2 − (2q1q3 − q2 2)2 4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) This potential is missing in the classification based on the Ziglin and Yoshida methods [4], since authors studied only potentials in the following form ˜V = q4 1 + aq2 1q2 2 + bq2 1q2 3 + cq4 2 + dq2 2q2 3 + eq4 3 , whereas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) involves linear in q1 and q3 term q1q3q2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Residues of the functions ∆(z, µ) = det � Iz − L(µ) � (z + 2µ2 − a1)(z + 2µ2 − a2) and ∆(z, µ) = det � Iz − L(µ) � (z − 2µ2 − b1)(z − 2µ2 − b2) coincide for each other up to the sign and replacement a1 − a2 = −(b1 − b2) which corresponds to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Let us consider residues Res|z=bi+2µ2 ∆(z, µ) = −4µ2Fi + Gi , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Res|z=∞ ∆(z, µ) = 8µ2H − (G1 + G2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Because F1 + F2 − 2H = 0 and G1 + G2 + 2(b2 − a1)H = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' there are two polynomials of the second order in momenta F1,2 and only one polynomial of the fourth order in momenta G1 or G2 which are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In [8] authors argue that three integrals of motion F1, F2 and G1 + G2 are quadratic polynomials in momenta, thus suggesting the existence of the point transformation to new variables in which equations of motion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) can be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Unfortunately, the authors did not notice that these integrals of motion are functionally dependent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' therefore, their statement is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Let us present these integrals explicitly F1 = p2 1 + p2 2 4 + M 2 12 b1 − b2 + (q2 1 + 2q2 2 + a1 − b1)q2 1 + (q2 1 + q2 2 + q2 3 + 2q1q3 + a1 − b2)q2 2 , F2 = p2 3 + p2 2 4 + M 2 12 b2 − b1 + (2q2 2 + q2 3 + a2 − b2)q2 3 + (q2 1 + q2 2 + q2 3 + 2q1q3 + a2 − b1)q2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where M12 = 1 2(q1p2 − 2q2p1 − q3p2 + 2q2p3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' At b1 = b2 we have a linear integral of motion M12 associated with a double rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' After re- duction by the corresponding Noether’s symmetry, we obtain new quadratic-linear Hamiltonian H commuting with the quartic invariant G1,2 in T ∗R2 For Euclidean space R3 generic solution K (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='6) of the Killing equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) depends on 20 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Using modern computer software we can directly prove that there are only two independent solutions to the equation d(KdV ) = 0 associated with the integrals of motion F1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 Euclidean space R6, case n = 3 The 6 × 6 Lax matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='14) generates three quadratic invariants Fi, three quartic Gi and three sextic invariants Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' After reduction, we have to get only six independent invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The Lax matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='14) after reduction looks like L(µ) = �ˆL11 ˆL12 ˆL21 ˆL22 � where ˆL11 = � −2µ2+q2 1+q2 2+q2 3+a1 q1q2+q2q4+q3q5 q1q3+q2q5+q3q6 q1q2+q2q4+q3q5 −2µ2+q2 2+q2 4+q2 5+b1−b2+a1 q2q3+q4q5+q5q6 q1q3+q2q5+q3q6 q2q3+q4q5+q5q6 −2µ2+q2 3+q2 5+q2 6+b1−b3+a1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' ˆL22 = � 2µ2−q2 1−q2 2−q2 3+b1 −q1q2−q2q4−q3q5 −q1q3−q2q5−q3q6 −q1q2−q2q4−q3q5 2µ2−q2 2−q2 4−q2 5+b2 −q2q3−q4q5−q5q6 −q1q3−q2q5−q3q6 −q2q3−q4q5−q5q6 2µ2−q2 3−q2 5−q2 6+b3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' ˆL12 = \uf8eb \uf8ed p1−2iµq1 p2 2 −2iµq2 p3 2 −2iµq3 p2 2 −2iµq2 p4−2iµq4 p5 2 −2iµq5 p3 2 −2iµq3 p5 2 −2iµq5 p6−2iµq6 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' ˆL21 = \uf8eb \uf8ed p1+2iµq1 p2 2 +2iµq2 p3 2 +2iµq3 p2 2 +2iµq2 p4+2iµq4 p5 2 +2iµq5 p3 2 +2iµq3 p5 2 +2iµq5 p6+2iµq6 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here we impose the following restrictions on arbitrary parameters in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='14) a3 = b1 − b3 + a1 , a2 = b1 − b2 + a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Calculating integrals of motion using three residues of the function ∆ = det � Iz − L(µ) � (z − 2µ2 − b1)(z − 2µ2 − b2)(z − 2µ2 − b3) at z = bi + 2µ2 Res|z=bi+2µ2 ∆(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' µ) = −16µ4Fi + µ2Gi + Si we obtain the following quadratic integrals of motion F1 = M 2 12 b1 − b2 + M 2 13 b1 − b3 + T1 + V1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' T1 = p2 1 + p2 2 4 + p2 3 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' F2 = M 2 21 b2 − b1 + M 2 23 b2 − b3 + T2 + V2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' T2 = p2 2 4 + p2 4 + p2 5 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' F3 = M 2 31 b3 − b1 + M 2 32 b3 − b2 + T3 + V3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' T3 = p2 3 4 + p2 5 4 + p2 6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where functions associated with the triple rotations are equal to M12 = −M21 = 1 2(q1p2 − 2p1q2 + 2q2p4 − p2q4 + q3p5 − p3q5) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' M13 = −M31 = 1 2(q1p3 − 2p1q3 + q2p5 − p2q5 + 2q3p6 − p3q6) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' M23 = −M32 = 1 2(q2p3 − p2q3 + q4p5 − 2p4q5 + 2q5p6 − p5q6) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' For brevity, we omit explicit expressions for the potentials Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 18 Residue at infinity gives rise to a relation between quadratic integrals F1 + F2 + F3 − 2H = 0 and relations between other integrals of motion 1 4 (G1 + G2 + G3) + b1F1 + b2F2 + b3F3 − 2(b1 − b2 − b3 + 2a1)H = 0 and S1 + S2 + S3 − 1 4(b1G1 + b2G2 + b3G3) − (b2 1 − a2a3)F1 − (b2 2 − a1a3)F2 − (b2 3 − a1a2)F3 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' From nine dependent integrals of motion Fi, Gi and Si we have to choose six independent, for instance, we can take three quadratic integrals of motion, two integrals of motion of fourth order and one integral of sixth order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 4 Symmetric space of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='III type This is another reduction of the A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='III case SO(n) U(n) ⊂ SU(2n) S (U(n) × U(n)) associated with the root space Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case we have to take Lax matrices (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1), make n × n matrices Q and P antisymmetric, and impose suitable restrictions on parameters ai and bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Isomorphism D3 ∼= A3 yields a correspondence SO(6) U(3) ∼= SU(4) S(U(1) × U(3)) so that we have the well-known Garnier system in R3 and the corresponding Hamilton-Jacobi equation H = E is separable in the elliptic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Following [7] we restrict ourselves by calculation of the quadratic integrals of motion for the D4 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 Euclidean space R6, case n = 4 The 8 × 8 Lax matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) after reduction has the following form L(µ) = �¯L11 ¯L12 ¯L21 ¯L22 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' There are two symmetric matrices ¯L11 = \uf8eb \uf8ed q2 1+q2 2+q2 3+a1−2µ2 q2q4+q3q5 −q1q4+q3q6 −q1q5−q2q6 q2q4+q3q5 q2 1+q2 4+q2 5+a2−2µ2 q1q2+q5q6 q1q3−q4q6 −q1q4+q3q6 q1q2+q5q6 q2 2+q2 4+q2 6+a3−2µ2 q2q3+q4q5 −q1q5−q2q6 q1q3−q4q6 q2q3+q4q5 q2 3+q2 5+q2 6+a4−2µ2 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' ¯L22 = \uf8eb \uf8ed 2µ2−q2 1−q2 2−q2 3+b1 −q2q4−q3q5 q1q4−q3q6 q1q5+q2q6 −q2q4−q3q5 2µ2−q2 1−q2 4−q2 5+b2 −q1q2−q5q6 −q1q3+q4q6 q1q4−q3q6 −q1q2−q5q6 2µ2−q2 2−q2 4−q2 6+b3 −q2q3−q4q5 q1q5+q2q6 −q1q3+q4q6 −q2q3−q4q5 2µ2−q2 3−q2 5−q2 6+b4 \uf8f6 \uf8f8 and two antisymmetric matrices ¯L12 = � 0 p1−2iµq1 p2−2iµq2 p3−2iµq3 −p1+2iµq1 0 p4−2iµq4 p5−2iµq5 −p2+2iµq2 −p4+2iµq4 0 p6−2iµq6 −p3+2iµq3 −p5+2iµq5 −p6+2iµq6 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 19 ¯L21 = � 0 −p1−2iµq1 −p2−2iµq2 −p3−2iµq3 p1+2iµq1 0 −p4−2iµq4 −p5−2iµq5 p2+2iµq2 p4+2iµq4 0 −p6−2iµq6 p3+2iµq3 p5+2iµq5 p6+2iµq6 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The parameters must satisfy the following constraints a2 − a1 = b1 − b2 , a3 − a1 = b1 − b3 , a4 − a1 = b1 − b4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Four residues of the function ∆ = det � Iz − L(µ) � (z − 2µ2 − b1)(z − 2µ2 − b2)(z − 2µ2 − b3)(z − 2µ2 − b4) at z = bi + 2µ2 are polynomials of sixth order in momenta Res|z=bi+2µ2 ∆(z, µ) = −64µ6Fi + µ4Gi + µ2Si + Wi , where Fi, Gi, Si and Wi are the second, fourth, sixth and eighth-order polynomials in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' As a result, we have 16 dependent integrals of motion and residue at infinity yields various relations between these polynomials, for instance F1 + F2 + F3 + F4 − 2H = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' We show only the leading part of the quadratic integrals of motion and omit explicit expressions for the potentials Vk F1 = M 2 12 b1 − b2 + M 2 13 b1 − b2 + M 2 14 b1 − b4 + T1 + V1 , F2 = M 2 21 b2 − b1 + M 2 23 b2 − b3 + M 2 24 b2 − b4 + T2 + V2 , F3 = M 2 31 b3 − b1 + M 2 32 b3 − b2 + M 2 34 b3 − b4 + T3 + V3 , F4 = M 2 41 b4 − b1 + M 2 42 b4 − b2 + M 2 43 b4 − b3 + T4 + V4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where functions M12 =(q2p4 − p2q4) + (q3p5 − p3q5) , M13 = (q1p4 − p1q4) + (q6p3 − p6q3) , M14 =(q1p5 − p1q5) + (q2p6 − p2q6) , M23 = (q1p2 − p1q2) + (q5p6 − p5q6) , M24 =(q1p3 − p1q3) + (q6p4 − p6q4) , M34 = (q2p3 − p2q3) + (q4p5 − p4q5) , are related to double rotations in R6, whereas functions T1 = p2 1 + p2 2 + p2 3 , T2 = p2 1 + p2 4 + p2 5 , T3 = p2 2 + p2 4 + p2 6 , T4 = p2 3 + p2 5 + p2 6 , are defined by triple translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The direct calculations show that the Haantjes torsion of the corresponding Killing tensors is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' From the sixteen integrals of motion Fi, Gi, Si and Wi we have to choose six independent integrals, four of which are quadratic polynomials in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 5 Symmetric spaces of BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='I type Symmetric space SO(m + n) SO(m) × SO(n) is only Hermitian when m = 2 since in general so(m) + so(n) has no centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' When m = 2 the so(2) subalgebra is the centre and depending upon whether q is odd or even this symmetric space is associated with either B(n+1)/2 or D(n+2)/2 root systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The simplest nontrivial example is associated with D3, and, similar to [7], we present the Lax matrix for this system even though D3 ∼= A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1 Euclidean space R4, case m = n = 2 We present 6 × 6 Lax matrix (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) using the same Cartan-Weil basis as in [7] L(µ) = � �L11 �L12 �L21 �L22 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' where �L11 = \uf8eb \uf8ed −2µ2+2q2 1+2q2 2+2q2 3+2q2 4+a1 p1−2iµq1 p2−2iµq2 p1+2iµq1 −2q2 1+2q2 3+a2 −2q1q2+2q3q4 p2+2iµq2 −2q1q2+2q3q4 −2q2 2+2q2 4+a3 \uf8f6 \uf8f8 �L22 = \uf8eb \uf8ed 2µ2−2q2 1−2q2 2−2q2 3−2q2 4+b1 −p1−2iµq1 −p2−2iµq2 −p1+2iµq1 2q2 1−2q2 3+b2 2q1q2−2q3q4 −p2+2iµq2 2q1q2−2q3q4 2q2 2−2q2 4+b3 \uf8f6 \uf8f8 and �L12 = � 0 p3−2iµq3 p4−2iµq4 −p3+2iµq3 0 −2q1q4+2q2q3 −p4+2iµq4 2q1q4−2q2q3 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' �L21 = � 0 −p3−2iµq3 −p4−2iµq4 p3+2iµq3 0 2q1q4−2q2q3 p4+2iµq4 −2q1q4+2q2q3 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Parameters satisfy the following relations a2 = a1 + b1 − b2 , a3 = a1 + b1 − b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case Hamiltonian H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) has the form H = p2 1 + p2 2 + p2 3 + p2 4 + 4(q2 1 + q2 2 + q2 3 + q2 4)2 − 8(q1q3 + q2q4)2 + 2(b2 − b1)q2 1 + 2(b3 − b1)q2 2 + 2(a1 − b2)q2 3 + 2(a1 − b3)q2 4 Because D3 ∼= A3 this Hamiltonian coincides with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8) up to rescaling and canonical transfor- mation qi → −qi and pi → −pi of one of the coordinates and momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The corresponding second-order Killing tensors are discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2 Euclidean space R2n−1, m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Let us consider representation of the Lie algebra so(2n + 1) by (2n + 1) × (2n + 1) matrices X [11], which satisfy X + SXTS−1 = 0 , S = 2n+1 � k=1 (−1)k+1Ek,2n+2−k , where Eij are matrices whose only non-zero entry is a unit in row i and column j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case, Cartan involution is related to the following element A = E1,1 −E2n+1,2n+1 of the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this representation Lax matrix (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='5) has the following block structure L(µ) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 2µ2 ⃗xT 0 ⃗y 0 s · ⃗x 0 ⃗yT · s −2µ2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + C + Λ , where the central block of zeroes has dimensionality (2n − 1) × (2n − 1), the column vectors x and y have the following entries ⃗xi = pi − 2iqi , ⃗yi = pi + 2iqi , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' , 2n − 1 , 21 and s is (2n − 1) × (2n − 1) matrix s = 2n−1 � k=1 (−1)kEk,2n−k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Matrix Λ is a numerical matrix which satisfies Λ + SΛT S−1 = 0 and, following [18], which determines a shift of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3 Euclidean space R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' case n = 2 For symmetric space SO(5) SO(2)×SO(2) we have the following 5 × 5 Lax matrix L(µ) = \uf8eb \uf8ed 2µ2 p1−2iµq1 p2−2iµq2 p3−2iµq3 0 p1+2iµq1 0 0 0 −p3+2iµq3 p2+2iµq2 0 0 0 p2−2iµq2 p3+2iµq3 0 0 0 −p1+2iµq1 0 −p3−2iµq3 p2+2iµq2 −p1−2iµq1 −2µ2 \uf8f6 \uf8f8 + 2C + 2Λ where C = \uf8eb \uf8ec \uf8ed −q2 1−q2 2−q2 3 0 0 0 0 0 q2 1−q2 3 (q1+q3)q2 0 0 0 (q1+q3)q2 0 (q1+q3)q2 0 0 0 (q1+q3)q2 −q2 1+q2 3 0 0 0 0 0 q2 1+q2 2+q2 3 \uf8f6 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Λ = \uf8eb \uf8ed a1 0 0 0 0 0 a2 a3 0 0 0 a3 0 a3 0 0 0 a3 −a2 0 0 0 0 0 −a1 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The Hamiltonian (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3) looks like H = 1 4trL2 ���� µ=0 − 2a2 1 − 2a2 2 − 4a2 3 = p2 1 + p2 2 + p2 3 + 4(q2 1 + q2 2 + q2 3)2 − 2(2q1q3 − q2 2)2 − 4(a1 − a2)q2 1 − 4(a1 + a2)q2 3 − 4q2 (a1q2 − 2a3(q1 + q3)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The quadratic integral of motion F = (q1p2 − p1q2 + q2p3 − q3p2)2 − (p1 + p3)(a2(p1 − p3) + 2a3p2) + U where U = 4(q1 + q3) � a2(q1 − q3) + 2a3q2 � (a1 − q2 1 − q2 2 − q2 3) − 4(a2 2 + a2 3)(q2 1 + q2 3) − 8q2(q1 − q3)a2a3 − 8(q1q3 + q2 2)a2 3 , defines second-order Killing tensor with non-zero torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The spectral curve of the Lax matrix is defined through the equation z5 − 2(2µ4 + 4a1µ2 + 2a2 1 + 2a2 2 + 4a2 3 + H)z3 + � 16(a2 2 + 2a2 3)µ4 + 8 � F + 4a1(a2 2 + 2a2 3) � µ2 + G1/2 − H2� z = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The leading term of the polynomial of fourth order in momenta G is defined by the curvatures tensor R G = − 1 4 � α,β,γ,δ R−α,β,γ,−δqαqβqγqδ + · · · = 4(p2 1 + p2 2 + p2 3)2 − 2(2p1p3 − p2 2)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 22 When ai = 0 we have Hamiltonian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4) up to canonical transformation H = 1 4trL2 ���� µ=0 = p2 1 + p2 2 + p2 3 + 4(q2 1 + q2 2 + q2 3)2 − 2(2q1q3 − q2 2)2 , that follows from the equivalence of the symmetric spaces, see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4 Euclidean space R5, case n = 3 The 7 × 7 Lax matrix reads as L(µ) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 2µ2 p1−2iµq1 p2−2iµq2 p3−2iµq3 p4−2iµq4 p5−2iµq5 0 p1+2iµq1 0 0 0 0 0 −p5+2iµq5 p2+2iµq2 0 0 0 0 0 p4−2iµq4 p3+2iµq3 0 0 0 0 0 −p3+2iµq3 p4+2iµq4 0 0 0 0 0 p2−2iµq2 p5+2iµq5 0 0 0 0 0 −p1+2iµq1 0 −p5−2iµq5 p4+2iµq4 −p3−2iµq3 p2+2iµq2 −p1−2iµq1 −2µ2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) + 2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed − �5 k=1 q2 k 0 0 0 0 0 0 0 q2 1−q2 5 q1q2+q4q5 q3(q1−q5) q1q4+q2q5 0 0 0 q1q2+q4q5 q2 2−q2 4 q3(q2+q4) 0 q1q4+q2q5 0 0 q3(q1−q5) q3(q2+q4) 0 q3(q2+q4) −q3(q1−q5) 0 0 q1q4+q2q5 0 q3(q2+q4) −q2 2+q2 4 q1q2+q4q5 0 0 0 q1q4+q2q5 −q3(q1−q5) q1q2+q4q5 −q2 1+q2 5 0 0 0 0 0 0 0 �5 k=1 q2 k \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The corresponding Hamiltonian H = 1 4trL2 ���� µ=0 = 5 � k=1 p2 k + 4 � 5 � k=1 q2 k �2 − 2(2q1q5 − 2q2q4 + q2 3)2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='2) commutes with the four linear integrals of motion r1 = (q1p2 − p1q2) + (q4p5 − p4q5) , r2 = (q2p3 − p2q3) + (q3p4 − p3q4) , r3 = (q1p3 − p1q3) + (q5p3 − p5q3) , r4 = (q1p4 − p1q4) + (q2p5 − p2q5) , so that {r1, r2} = −r3 , {r1, r3} = r2 , {r1, r4} = 0 , {r4, r2} = r3 , {r4, r3} = −r2 , {r2, r3} = r4 − r1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The spectral curve of the Lax matrix det(z · I − L(µ)) = z7 − 2(2µ4 + H)z5 + (8F1µ2 + G1)z3 − 4G2z = 0 gives rise to four independent integrals of motion in involution H, G1 and F1 = (r2 1 + r2 2 + r2 3 + r2 4) , G2 = (r1 + r4)2� (r1 − r4)2 + 2(r2 2 + r2 3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Using Hamiltonian and fourth order polynomial G1 we can get the integral of motion defined by a curvature tensor R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='8) G3 = 2G1 + 2H2 = −1 4 � α,β,γ,δ R−α,β,γ,−δpαpβpγpδ + · · · = −4(p2 1 + p2 2 + p3 3 + p2 4 + p2 5)2 + 2(2p1p5 − 2p2p4 + p2 3)2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 23 which is independent on H and rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Because {rk, G1} = 0 we have a completely integrable system with the five independent integrals of motion in involution, for instance r1, r4, r2 2 + r2 3 , H, G1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Nevertheless, the Lax matrix (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1) generates only four of them, similar to the generalized Toda lattice [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' By adding a constant matrix Λ to L(µ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='1), where Λ = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed a1 0 0 0 0 0 0 0 a2 0 0 0 0 0 0 0 a3 a4 0 0 0 0 0 a4 0 a4 0 0 0 0 0 a4 −a3 0 0 0 0 0 0 0 −a2 0 0 0 0 0 0 0 −a1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , we have H = 1 4trL2 ���� µ=0 − 2a2 1 − 2a2 2 − 2a2 3 − 4a2 4 = 5 � k=1 p2 k + 4 � 5 � k=1 q2 k �2 − 2(2q1q5 − 2q2q4 + q2 3)2 +(a1 − a2)q2 1 + (a1 − a3)q2 2 + q3(a1q3 − 2a4q2 − 2a4q4) + (a1 + a3)q2 4 + (a2 + a1)q2 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this case equation for the spectral curve z7 − 4(µ4 − 2µ2a1 + a2 1 + a2 2 + a2 3 + 2a2 4 + H/2)z5+ � 16(a2 2 + a2 3 + 2a2 4)µ4 + F1µ2 + G1 � z3 − � 64a2 2(a2 3 + 2a2 4)µ4 + F2µ2 + G2 � z = 0 contains a sufficient number of integrals of motion for integrability by the Liouville theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' There are three polynomials H, F1, F2 of the second order in momenta and two polynomials G1 and G2 of the fourth order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 6 Reductive Homogeneous Spaces According to [7] in previous sections, we consider symmetric spaces which are reductive homoge- neous spaces on which the canonical connections have zero torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' In this section, we consider one example associated with reductive homogeneous spaces which have non-zero torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Following [7] let us consider symmetric space SU(3) S(U(1) × U(1) × U(1)) and 3 × 3 Lax matrix L = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed q2 1 + q2 2 2iµq1 − p1 2iµq2 − p2 −2iµq1 − p1 4µ2 − q2 1 − 2ω1 −q2q1 −2iµq2 − p2 −q2q1 4µ2 − q2 2 − 2ω2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 24 The corresponding Hamiltonian of the anharmonic oscillator H = 1 4trL2 ���� µ=0 − ω2 1 − ω2 2 = p2 1 + p2 2 2 + (q2 1 + q2 2)2 2 + ω1q2 1 + ω2q2 2 commutes with the following quadratic integral of motion f = −(q1p2 − p1q2)2 − 2ω1p2 2 − 2ω2p2 1 − 2(ω1q2 2 + ω2q2 1 + 2ω1ω2)(q2 1 + q2 2) , associated with a simple rotation in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Other Lax matrices associated with the three-wave interaction system are discussed in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Using N-wave hierarchies we can get quadratic integrals of motion associated with N − 2 rotations in the corresponding Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Here we do not discuss the examples of these calculations in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 7 Conclusion We present examples of the Killing tensors of valency two generating quadratic integrals of motion for the integrable systems having additional integrals of motion which are polynomials of higher order in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' All these Killing tensors are related to the special combinations of rotations and trans- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' It will be interesting to find criteria which allow us to extract these special Killing tensors from the generic solution of the Killing equation on Euclidean, Riemannian and pseudo- Riemannian spaces of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The work was supported by the Russian Science Foundation (project 21-11-00141).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' The second author (AVT) gratefully acknowledges the kind hospitality provided by 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=', Penrose R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=', On quadratic first integrals of the geodesic equations for type 22 spacetimes, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=', v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='18, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content='265-274, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtE0T4oBgHgl3EQf7QKQ/content/2301.02774v1.pdf'} diff --git a/k9E5T4oBgHgl3EQfHA4j/content/tmp_files/2301.05435v1.pdf.txt b/k9E5T4oBgHgl3EQfHA4j/content/tmp_files/2301.05435v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f87c0382c6faf7f92dddbf3439e7cdb584010749 --- /dev/null +++ b/k9E5T4oBgHgl3EQfHA4j/content/tmp_files/2301.05435v1.pdf.txt @@ -0,0 +1,1605 @@ +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +MARIAN BITTNER 1,2,3,*,†, WEI-TSE YANG 2,†, XUCONG ZHANG 2, AJAY SETH 3, +JAN VAN GEMERT 2, AND FRANS C. T. VAN DER HELM 3 +Abstract. Markerless estimation of 3D Kinematics has the great potential +to clinically diagnose and monitor movement disorders without referrals to +expensive motion capture labs; however, current approaches are limited by +performing multiple de-coupled steps to estimate the kinematics of a person +from videos. Most current techniques work in a multi-step approach by first +detecting the pose of the body and then fitting a musculoskeletal model to +the data for accurate kinematic estimation. +Errors in training data of the +pose detection algorithms, model scaling, as well the requirement of multiple +cameras limit the use of these techniques in a clinical setting. +Our goal is +to pave the way toward fast, easily applicable and accurate 3D kinematic +estimation . To this end, we propose a novel approach for direct 3D human +kinematic estimation D3KE from videos using deep neural networks. +Our +experiments demonstrate that the proposed end-to-end training is robust and +outperforms 2D and 3D markerless motion capture based kinematic estimation +pipelines in terms of joint angles error by a large margin (35% from 5.44 to +3.54 degrees). We show that D3KE is superior to the multi-step approach and +can run at video framerate speeds. This technology shows the potential for +clinical analysis from mobile devices in the future. +1. Introduction +3D Human kinematics involves measuring joint angles between body segments, +which is essential in the day-to-day practice of experts. Skilled physicians could +judge, just by looking at a specific motion of their patient, whether it is healthy +or abnormal. Skilled sports coaches can help their coachees achieve better perfor- +mance and lower injury risk by evaluating their movements through observation. +However, these visual examinations of human kinematics remain inherently subjec- +tive, leading to variation between and within human observers. Modern systems +and sensors could reduce these variations through more objective observations. Yet, +these systems make the measurement of human motion more costly and more time- +consuming. A system with the availability and ease of use of visual estimation +would help physicians and coaches make more objective observations more often, +ultimately raising their own and their subjects quality of life. +Digital cameras +have made the estimation of human kinematics more accessible but come at the +1 +Vicarious Perception Technologies (VicarVision), +1015 AH Amsterdam, +The +Netherlands +2 +Computer Vision Lab, Delft University of Technology, 2628 XE Delft, The +Netherlands +3 +Biomechanical Engineering, Delft University of Technology, 2628 CN Delft, The +Netherlands +*Corresponding author: mbittner.work@gmail.com +†These authors contributed equally to this work. +1 +arXiv:2301.05435v1 [cs.CV] 13 Jan 2023 + +2 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +cost of reduced accuracy. Compared to the more traditional Optical Motion cap- +ture (OMC) systems, markerless motion capture (MMC) systems do not require +specialized cameras and markers attached to the subject being monitored, but use +normal RGB cameras in combination with image-based automatic pose estimation +algorithms. Instead of specific markers, pose estimation algorithms detect the cen- +ters of major joints of the human body, such as the shoulders, hips, and knees. +These detected centers are usually referred to as key points. +Multiple commonly used markerless motion capture methods rely on 2D pose +estimation methods [27, 45, 46, 26]. Often these methods still need more than one +camera to generate a good estimation of the keypoints in 3D, which again requires +additional cameras to be set up. On the other hand, an increasing number of meth- +ods are using single-view (monocular) 3D pose estimation methods [21, 32, 43], +which allow to estimate a 3D pose just by using a single camera. This makes MMC +systems faster and more accessible as they do not require the additional time and +expertise to place markers on the subject or calibrate multiple cameras. However, +MMC systems assume that current pose estimation algorithms can accurately re- +place markerless motion capture systems for, e.g., biomechanical applications [52, +13, 60]. +Commonly used pose estimation algorithms introduce mistakes in kinematic es- +timation pipelines due to systematic errors in their predictions. +To detect key +points, most pose estimation methods are trained on a combination of images of a +person and ground truth annotations which map pixels in the image to their corre- +sponding joint center.These ground truth annotations are often manually conducted +by non-expert annotators, leading to errors caused by personal biases for training +and inaccuracies in the pose estimations [13]. For example, Needham et al. [41] +compared three often used pose estimation algorithms OpenPose [10], DeepLab- +Cut [38] and AlphaPose [17] algorithm against an OMC system and showed errors +in the estimation of joint centers of 30 mm to 50 mm with variations in 12 mm +to 25 mm in marker placement. Cronin [13] provides an overview of additional +problems with 2D pose estimation for kinematic analysis. These differences are +most likely due to a difference between the application that pose estimation algo- +rithms are often developed for and their application to, e.g., the biomedical do- +main, which has different accuracy requirements [52]. Wade et al. [60] proposed +to solve this problem by re-annotating existing large-scale datasets, this, however, +is a time-consuming process, when for example considering the COCO-keypoint +dataset https://cocodataset.org/#keypoints-2020 (accessed on 2 December +2022 ) consists of more than 250.000 labeled poses. For the evaluation of pose es- +timation algorithms, these labeling errors will just appear as a baseline error that +all algorithms training on the same data will have. However, for applications in the +biomedical domain and in situations such as kinematic estimation, where the pose +is just an intermediate step errors can propagate to subsequent tasks. +Errors in the estimated pose cannot not be corrected by most kinematic es- +timation pipelines because they all roughly follow a ‘multi-step’ approach. The +‘multi-step’ approach consists of +• Detection of the 3D pose (in one or more steps); +• (Optional) modeling of the pose with a (musculo)skeletal model. +• Calculation of kinematics and/or downstream tasks such as gait parameters +or dynamics. + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +3 +For example, Kidzinski et al. [27] used OpenPose to first predict key points from +a video and then trained a convolutional neural network (CNN) to predict the walk- +ing parameters of patients with cerebral palsy. Liao et al. [32] first model the 2D +pose in OpenPose then create a 3D pose using data-driven matching and finally +estimate 3D gait parameters. Noteboom et al. [43] first used VideoPose3D [49] to +estimate a 3D pose, followed by modeling in OpenSim [54] for the estimation of dy- +namics from a single camera. Pagnon et al. [45, 46] developed the handy Pose2Sim +tool, which first combines 2D OpenPose pose estimations from multiple cameras +into a 3D pose then models it in OpenSim. Because the pose estimation step is +de-coupled from kinematic estimation, errors in pose estimation propagate through +to the estimation of kinematics. Uchida and Seth [57] showed that 20 mm of marker +uncertainty leads to a variation of 15.9◦ in peak ankle plantarflexion angle and im- +pacts downstream tasks such as joint moment estimation. Della Croce et al. [14] +showed precision variation 13 mm to 25 mm, which leads to differences in estimated +joint angles up to 10°. Fonseca et al. [18] showed that misplacement of markers up +to 10 mm can lead to errors of 7◦ depending on the marker. With estimation errors +of 30 mm to 50 mm in keypoint estimation [41], it is to be expected that these errors +will substantially influence kinematic estimation from markerless motion capture. +Low-pass filter [40, 45] or bi-directional Kalman-filter [40] has been applied to com- +pensate for noisy key point estimations, but cannot correct for faults in keypoint +detection. Subsequent modeling and kinematic calculation steps can only compen- +sate for these inaccuracies. This ‘multi-step’ approach is probably inspired by the +steps of a traditional OMC method, as in the traditional OMC systems the pose +detection step is done using a different system and is thus isolated from the other +steps. In camera-based kinematic estimation pipelines, however, the de-coupling of +individual steps is no longer necessary. +Deep neural networks have often demonstrated their ability to outperform multi- +step systems, by implicitly learning individual steps through end-to-end training +between an input and the desired output [55, 29]. The main strength of deep neu- +ral networks lies in their ability to break down a highly complex task, in this case, +the estimation of kinematics from videos, into a sequence of simpler tasks, with- +out the need for intermediate ‘hand-crafted’ representations [30, 2]. Due to the fully +differentiable nature of neural networks, it means that an error in estimation dur- +ing training can influence all stages of the network and adjust them accordingly [2]. +This allows deep neural networks to directly estimate kinematics. +In this work, we challenge the notion of the classical multi-step approach of pose +estimation, fitting of a musculoskeletal model and kinematic estimation. To this +end, we propose a novel end-to-end method that allows for direct estimation of hu- +man kinematics, which is directly optimized for kinematic estimation while treating +pose estimation only as an auxiliary task to constrain the estimations of the net- +work. Figure 1 shows a general overview of our method. + +4 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +Figure 1. +Overview of the proposed direct 3D human kinemat- +ics estimation (D3KE). Instead of using the common ’multi-step’ +approach of predicting pose, fitting it to a model, and estimating +kinematics, our D3KE directly estimates the kinematics. Errors in +earlier steps of the multi-step approach propagate to later steps; +in contrast, our method can correct for errors occurring anywhere +between input and output. +Contributions. To the best of our knowledge, we are the first to present an end- +to-end trainable network that directly generates joint angles, joint positions, scale +factors and marker positions of a biomechanical model from a monocular video. +We propose a method that directly regresses from a video +to joint angles and +scales using deep neural networks. We investigate the influence of various temporal +smoothing methods to increase the accuracy of our algorithm. We introduce a novel +type of network layer that allows for the calculation of the 3D pose from estimated +kinematics during the training process to train the network simultaneously on the +pose and kinematic labels. +2. Materials and Methods +Our method takes videos from a single camera as input and directly estimates +joint angles, which we call direct 3D kinematic estimation (D3KE). The proposed +method first coarsely estimates kinematics per frame by using a convolutional neural +network, and then it uses a sequence network with temporal relations across frames +to re-fine kinematic estimations at each frame. +An overview of our method is shown +in Figure 2. Both networks estimate the scale of body segments, joint angles, and a +rotation matrix from the pelvis to the ground, those serve as input for a skeletal- +model layer in both networks that allows for additional supervision on the pose of +a subject. + +Multi-step approach +Step: Musculoskeletal Modelling +Step: Pose Estimation +Error +Error +Error +Correction +propagation +propagation +70° +50° +OpenSim +D3KE method +Direct Estimation +Deep +Error Correction +Error Correction +70° +Neural Network +50°TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +5 +Figure 2. Taking a single view video as input, D3KE consists of +one convolutional neural network and one sequential network. Per +frame, D3KE outputs joint angles and scales of individual bones in +a skeletal model(scale factors) with a convolutional network. Addi- +tionally, joint angle and scale factor are converted to a pose through +the skeletal-model kinematics (SM) layer. A series of frame esti- +mations in time are then fed into a sequential network to smooth +the estimations and reduce artifacts if one limb occludes another +in the view of the camera (self-occlusion). +In this section, we first describe the deep learning architecture, including a de- +tailed description of the skeletal-model layer. We then describe how the ground +truth data was generated and which pre-processing and hyperparameters were +used for training. +Lastly, we describe the dataset used for training and testing +our method. +2.1. Network Structure. Convolutional neural networks(CNNs) have shown good +accuracy for 2D and 3D pose estimations [10, 51, 39, 42, 48] from single input im- +ages. Conventionally 2D CNNs are used for pose estimation tasks, that takes a +single image as an input and predict the pose of one or multiple people in the +image. For our method, we choose a per-frame convolutional network to coarsely +predict the joint angle and scaling parameters. Inspired by [51], we choose a stan- +dard pre-trained ResNeXt-50 [62] as our convolutional backbone. +To fine-tune the per-frame predicted joint angles and scaling parameters we add +a sequential network. Sequential networks are used in pose estimation to ‘lift’ an +estimated 2D pose to 3D [11, 12]. Recent research combines temporal information +with lifting to improve accuracy during frames where one limb occludes another +in the view of the camera (self-occlusion) or where not all key points were de- +tected [31, 33, 49]. In contrast to CNNs, these sequential networks do not take a +single frame as input but exploit temporal dependencies in the data for their pre- +diction. As the convolutional network outputs per-frame estimates, it cannot take +temporal information into account. We add a sequential network to our architecture +to refine a sequence of estimations made by the convolutional model. Inspired by +works on temporal lifting we experimentally evaluate three sequential networks; an + +Temporal smoothing +Per-frame Estimation +-ngle ['g. +¥0.00 +-0.10 +0.05 +-0.10 +0.00 +Convolutional +to +0 +Neural Network +Forward Kinematics +25 +25 +Layer +50 +50 +Sequential +Convolutional +75 +Network +75 +Neural Network +Forward Kinematics +Forward Kinematics +100 +Layer +100 +Layer +125 +125 +150 +150 +Convolutional +Neural Network +175 +175 +Forward Kinematics +Layer6 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +LSTM [23], a Temporal Convolutional Network (TCN) [49] and a Transformer [31] +to refine the predicted joint angles and scale factors. +Both the convolutional and the sequential network contain a specialized layer +that allows each network to perform the kinematic transformations of a musculo- +skeletal model. Therefore, at train time both networks can be supervised not only +on the estimated joint angles but also on a resulting pose. +Both convolutional and sequential networks are supervised by losses of joint +positions, marker positions, body scales and joint angles. +The overall objective +function L can be expressed in the equation +(1) +L = λ1Ljoint + λ2Lmarker + λ3Lbody + λ4Langle, +where λ1, λ2, λ3, λ4 are weights of losses. We use the root-relative L1 loss in Equa- +tion (2) to define the loss of marker position Lmarker and the loss of joint position +Ljoint. The estimations ˆy and the labels y are first subtracted with each root po- +sition ˆyroot, yroot. For the loss of body scales Lbody and joint angles Langle, we +calculate the L1 norm. +(2) +l = ∥(ˆy − ˆyroot) − (y − yroot)∥1 +The objective of a neural network during training is to minimize the loss function; +in our case, the difference between estimated and ground truth joint angles. How- +ever, this joint angle loss cannot capture the underlying relations and constraints +of individual angles, dictated by the human musculoskeletal system. Intuitively, +small changes in the angles of spine, shoulder, and elbow can accumulate and lead +to large differences in the position of the hand, as illustrated in Figure 3. To ad- +dress this issue, we propose to use a skeletal-model layer to perform the kinematic +transform of a musculoskeletal model. + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +7 +Figure 3. Our skeletal-model layer uses an internal representa- +tion of a skeletal model to convert the predicted joint angles and +scale factors to the positions of individual markers on segments of +the skeletal model. This allows our method to be supervised during +training not only on errors(losses) in the estimation of joint angles +but also on errors in the resulting pose. On the right, we show the +additional error that is created between estimations (gray) and +ground truth (blue). This auxiliary estimation of the pose as 3D +marker positions helps to constrain the estimation of joint angles +as small changes in proximal joints can have a large effect on a +marker at more distal positions. +Skeletal-Model Layer. The skeletal-model layer allows us to convert predicted joint +angles into marker positions on a skeletal model and add them as an additional +loss term. +This loss term represents the cumulative effect of small joint angle +changes on the final pose, indirectly imposing the constraints of a skeletal-model +on the predictions of the network. As the skeletal-model layer does not contain any +learnable parameters, i.e., it cannot change during network training. The accuracy +of the predicted pose is completely determined by the input to the skeletal-model +layer; thus, the pose prediction is only an auxiliary task. +A skeletal model consists of body segments, motions between different body +segments (joints) and points with a vector from a center of its anchor body segment +(markers). Given body scales β, joint angles θ and rotation matrix Rground←pelvis, +we use the skeletal-model layer to calculate marker positions and joint positions. +In the following variables with a hat (ˆx) denote estimated values, variables without +the hat (x) denote the predefined variable from the musculoskeletal model. + +Skeletal Model layer +Direct Estimation +Marker Position Loss +Joint Angles +Segment Rotation +ranslatior +segm +Scale Factors +Marker Distances +Scale Factor Loss +X1.2 +Joint Angle Loss +Backpropagation8 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +First, the translation part T in the transformation from the joint to the body +depends on the subject’s body scale. For example if the subject has longer legs, +the center of the femur will be farther from the hip joint. +We can update the +translation part ˆT by comparing the ratio between predicted body scales ˆβ and +default body scales β in Equation (3), where ⊙ is elementwise multiplication, and ⊘ +is elementwise division. +(3) +ˆT = T ⊙ (ˆβ ⊘ β) +Then, we create a matrix to represent spatial transformation of motions Rmotion +using Equation (4) A1, A2, A3 are the predefined axes ˆθ1, ˆθ2, ˆθ3 and predicted angels +per degree of freedom per joint in axis-angle notation. +G(A, θ) is the standard +function converting an axis-angle representation to a 3x3 transformation matrix. +(4) +Rmotion = +� +��� +R3R2R1 +0 +0 +1 +� +��� +4×4 +(5) +R1 = G(A1, θ1) +(6) +R2 = G(R1A2, θ2) +(7) +R3 = G(R2R1A3, θ3) +Then, we can calculate the estimated transformation from the body to its parent +body ˆRparent←child in Equation (9) using Equation (8) with Oparent, Ochild denoting +predefined orientations from and ˆTparent, ˆTchild the predicted translations from the +joint to the parent/child. +F(Oa) the conversion from euler angles to a 3 × 3 +Rotation matrix. +(8) +Rparent/child←joint(Oparent/child, Tparent/child) = +� +��� +F(Oparent/child) +Tparent/child +0 +1 +� +��� +4×4 +(9) +Rparent←child = Rparent←joint Rmotion R−1 +child←joint +We measure the spatial transform by traversing from the root (pelvis) to leaf +nodes (hands and feet) in the level order. In D3KE, we directly infer the rotation +matrix from the pelvis to the ground Rground←pelvis. Rground←pelvis can initially +be expressed in Equation (10), where I denotes the identity matrix. The rotation +part of Rchild←joint is also a 3 × 3 identity matrix in our musculoskeletal model. +Our method aims to predict the root-relative position so the translation part can +be ignored during the prediction. Moreover, in our musculoskeletal model, only +joint angles of the pelvis are unbounded in [−∞,∞]. Predicting three unbounded +angles to form the rotation matrix in Equation (4) will have the problem of discon- +tinuity [65]. Thus, we directly predict the rotation matrix Rground←pelvis. +(10) +Rground←pelvis = I4×4 Rmotion R−1 +pelvis←joint + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +9 +Last, a marker with a vector of ⃗d from the center of the body is also dependent on +the body scales. The predicted vector of ˆd is updated in Equation (11). The position +of the predicted point is calculated in Equation (12) with ˆRparent←child and ˆd. +(11) +ˆd = ⃗d ⊙ (ˆβ ⊘ β) +(12) +P = +� +parent,child∈path +Rparent←child +� +��� +⃗d +1 +� +��� +4x1 +2.2. Network Training. +2.2.1. Ground Truth Generation. For training our method, we need to create cus- +tom ground truth data that contains all outcomes that our network is predicting +since they are not available in publicly available datasets. Most pose estimation +datasets, provide only video and marker positions from optical motion capture +(OMC) system. For training our method, we need the joint angle and the scales +of individual bones, a rotation matrix of the pelvis to the ground as well as the +marker positions corresponding to them. To generate these, we model the OMC +data, represented as a 3D human mesh model in the OpenSim software [54] and +use inverse kinematics [36, 1] to generate joint angles. The following describes each +step in more detail. +First, we create a general (musculo)skeletal model to fit the data using the +OpenSim software [54, 16]. As we are interested in capturing the complete motion +of the human subject, we model the full body. With the OpenSim software [16, 54] +we create a full-body musculoskeletal model (MSM) by merging existing models of +upper limbs and lower limbs [3, 4, 15, 24, 63] and thoracolumbar spine [8, 7, 9]. +We add wrist and hand [20, 44] models to the MSM, which are not used for ground +truth generation, for the sake of aesthetics. The full-body model contains all bones +in a skeletal system from the head to feet and from the upper arms to the hands. +We do not model every degree of freedom between vertebrae to avoid expensive +computation and the requirement of at least three markers to measure the motions +of one vertebra. Instead, we separate the spine from the fifth lumbar to the first +cervical vertebra into nine segments. +Then, we fit our data to the musculoskeletal model. Instead of using the OMC +marker data directly, we use OMC marker converted to 3D human mesh represen- +tations using the MoSh++ [34] method, to make scaling the model to individual +participants more time efficient and allow us to define an arbitrary number of vir- +tual markers. We fit our data to the musculoskeletal model, by first defining virtual +markers on the vertices of the 3D mesh representation. We then used these virtual +markers as input for the OpenSim software. Then, we used the OpenSim internal +scaling tool to scale the proportion of individual body segments according to the +distances of virtual markers on the 3D mesh. As the sizes of individual body parts +vary across individuals, this step must be conducted individually for each subject +in the dataset. We define the ratio in dimensions between the default and scaled +body segments as scaling factors. Finally, we used the inverse kinematics solver for +the calculation of joint angles. During this process, the MSM is moved for each +time step to a position that minimizes the sum of weighted squared errors between +the virtual markers on the 3D mesh and markers defined on the musculoskeletal + +10 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +model. All joint angles where segments had a higher squared error than 2 cm were +disregarded in the analysis. +The final ground truth values were the calculated joint angles, the scaling factors +per segment as well as the virtual marker positions. Additionally, a pelvis rotation +matrix was generated for each frame, since the pelvis functions as the relative +position of the model to the ground that is free to move in all directions. +2.2.2. Data Preparation and Hyperparameters. To generate the input for our net- +work, each video frame was cropped and augmented. We use the pre-trained Faster +R-CNN [50] with ResNet-50 [22] backbone to extract a square bounding box of the +person in videos and resize it to 256 × 256 pixels as the input image size. Dur- +ing training, we apply data augmentation with scaling, rotation, translation and +noise to simulate occlusions similar to [51]. +Our model was trained using the following hyperparameters and loss. For the +ResNeXt model, we use an Adam optimizer with weight decay [35] of 0.001 and a +batch size of 64. The learning rate exponentially decays in two steps from 5 × 10−4 +to 3.33×10−5 over 28 epochs and from 3.33×10−6 to 10−6 over 2 epochs. For both +sequential and convolutional networks, we set the hyperparameters with λ1 = 1.0, +λ2 = 2.0, λ3 = 0.1 and λ4 = 0.06 experimentally. Due to memory constraints, we +do not train convolutional and sequential models simultaneously, but in succession, +by first training the convolutional model and then refining predictions using the +sequential model. +2.3. Software Tools. All training was conducted in python using the PyTorch li- +brary [47]. The pre-trained ResNext and FasterRCNN networks were obtained from +the torchvision library [59]. All code for training and generation of ground truth +will be made available in a Github repository: https://github.com/bittnerma/ +Direct3DKinematicEstimation. . +2.4. Data. We trained and tested D3KE on the BML-MoVi Database [19]. BML- +Movi is an extensive motion capture and video dataset, it contains recordings of +90 actors that each perform 20 kinds of everyday movements as well as a random +one. Motions were captured using inertial measurement units as well as a Qualisys +optical motion capture system and videos were recorded using two computer-vision +cameras. For this study, we used recordings from the calibrated Point Gray cameras +(PG1, PG2) during recording session F as the full set of optical markers was used +during this session. In accordance with the anatomical plane that each camera is +viewing during the initial T-Pose of the participants, we will refer to the camera +view captured by PG1 as the frontal- and PG2 as the sagittal camera view. For the +generation of ground truth virtual markers, we use the 3D mesh representations of +the Qualisys data that is provided in the larger AMASS dataset [37]. For analysis +of the data, we divided the BML-Movi database into 63 participants for training, +16 participants for the testing, and three participants for validation. This is common +practice in the supervised training of deep neural networks [64]. At training time, +the validation set is used to evaluate the accuracy of kinematic estimation after +each training iteration on a portion of the data the network does not have access +to, to prevent overfitting on the training set. + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +11 +3. Experiments +3.1. Experiment 1: Direct vs. Multi-Step Estimation. To evaluate the ac- +curacy of our direct 3D kinematic estimation approach (D3KE) for joint angle +estimation, we compare its performance against multiple versions of the multi- +step approach. +3.1.1. Experiment 1-A: 3D Pose Based Kinematic Estimation. We first compare +our direct estimation of kinematics and a 3D pose estimation multi-step baseline. +To create a fair comparison between direct and multi-step estimations, we imple- +ment a custom multi-step approach (CMS) that is trained on the same data as our +direct approach. For the CMS, we combine a 3D human pose estimation method +with subsequent musculoskeletal modeling in OpenSim. We modify the metric-scale +heatmaps [51] of the convolutional network to predict marker positions and SMPL +keypoint positions in the metric scale. As for D3KE, we exploit a sequence network +to re-fine marker positions at the target frame. +More specifically, the convolu- +tional network initially infers marker positions under a calibration pose (T-pose), +and OpenSim utilizes the predicted marker data for body scaling, where the gen- +eral musculoskeletal model is scaled to the participant’s body size. Re-fined marker +positions are then used to run inverse kinematics with the scaled musculoskeletal +model to obtain joint angles. The main difference between the CMS approach and +D3KE is that CMS uses multiple steps to estimate the kinematics and is only su- +pervised on the marker/pose estimation task, while D3KE is directly trained on the +kinematic estimation task; this way, we can compare direct vs. multi-step estima- +tion of kinematics. We use multiple metrics for the comparison of D3KE and the +CMS. The mean per bony landmarks position error (MPBLPE) is used to evaluate +bony landmark positions. Bony landmarks are markers placed where bones are +close to the surface, such as the elbow. This metric is inspired by the mean per +joint position error (MPJPE) which is often used in 3D pose estimation. MPBLPE +first aligns estimations and ground truth at the root position and calculates the +average Euclidean distance. We directly evaluate the body scale factors by the root +mean square error RMSEbody on the scalars predicted by the network. However, +to present the results in a more intuitive format, we choose the axis along the longest +dimension in each body scale and convert the scale of the axis into millimeters and +calculate the mean absolute error (MAEbody). +3.1.2. Experiment 1-B: 2D-Pose Based Kinematic Estimation. In the previous ex- +periment, we evaluate the multi-step baseline with the 3D body pose estimation +method. However, the use of fully trained 2D pose estimation algorithms is com- +mon in kinematic estimation works [40, 45, 46]. Therefore, we conduct experiments +to compare our method and these 2D-based kinematic estimation methods. In con- +trast to our CMS method which estimates 3D pose from a single camera estimation, +2D pose estimation methods require at least 2 calibrated cameras for the estima- +tion of 3D keypoints. The use of an additional camera to generate the pose could +be an advantage, which the CMS method does not have. We chose a naive imple- +mentation of the OpenPose algorithm [10, 25], which has extensively been used in +related work [40, 45, 46]. Additionally, we test the MediaPipe implementation of +the blazepose algorithm [6], as a more modern 2D algorithm. MediaPipe is easy +to use since it is available as a python library, however, in contrast to OpenPose + +12 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +it runs faster, allows for additional smoothing of its predictions, provides more key +points, and is labeled on different keypoint labels. +For the OpenPose and MediaPipe, we project the key points to 3D using the +BML-Movi camera parameters https://github.com/saeed1262/MoVi-Toolbox (ac- +cessed on 2 August 2022 ). For OpenPose, we connected missing points (due to +self-occlusion) using linear interpolation. +For MediaPipe, we chose the highest +model complexity (2) and set enabled smooth landmarks for continuous frames of +a video. For modeling and inverse kinematics, we follow the same steps as for the +CMS method only redefining the positions of markers on the OpenSim model to fit +the provided key points. +We compare against an average across both camera views for CMS and D3KE, +as MediaPipe and OpenPose need at least two cameras to work. +We evaluate +performance based on mean absolute error MAEangle(◦), the standard deviation +of errors SDangle(◦) and smoothness of the predictions as the mean velocity of the +angle MVangle (◦/s). The mean velocity error is calculated by the derivative of the +landmark position and joint angle data with respect to time. +(13) +MVangle = +n +� +t=0 +|st−st+1| +∆t +n +with st an individual marker position at time t, ∆t is the amount of time between +time steps and n is the total number of timesteps. +3.2. Experiment 2: Sequential Network Variants. Since we have multiple +options for the sequential networks, we evaluate three to determine whether the +additional modeling of temporal dependencies in the data improves the accuracy +of our method or not. For subsequent smoothing and reduction of self-occlusion +artifacts of the estimations, we test three different networks including LSTM [23], +temporal convolutional networks (TCNs) [49], and a lifting Transformer [31] as +the sequential network. As smoothing is known to improve the accuracy of multi- +step approaches [40], we also evaluate combinations of our CMS model with these +sequential networks. +For the LSTM, we implement a bidirectional architecture with a hidden size +of 128, three recurrent layers and a dropout probability of 0.1. +For TCNs, we +follow [49] to exploit 243 frames as the receptive field and make the momentum +of batch normalization decay from 0.1 to 0.001. For the lifting Transformer, we +use a hidden size of 256 and 8 parallel attention heads in the self-attention layer +and a channel size of 512 in the convolutional layer. Each sequential network is +trained with a sequence length of 243 frames and a batch size of 128 over 50 epochs +with Adam optimizer [28]. The learning rate exponentially decays from 10−3 to +5 × 10−6. +We use the same metrics for the comparison of individual network variants as +we used for the comparison of D3KE and CMS. In addition, we investigate the +smoothness of the predicted sequences, we estimate the mean velocity (MV) on +bony landmark positions and joint angles, denoted as MVBL and MVangle. +3.3. Experiment 3: Processing Speed. One important property of our pro- +posed method for clinical applications is its processing speed. As applications for + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +13 +camera-based kinematic estimation should form an alternative to visual examina- +tions in the future, it should ideally be able to run fast enough to estimate kine- +matics from video frames as fast as they are collected by a camera, mostly between +15 and 30 frames per second. +We compare the running time on Windows 10 with four core CPU, 52 GB RAM +and NVIDIA T4 GPU. We compare the CMS and D3KE method as they both +use the same type of convolutional network. We choose the lifting Transformer +as the sequential architecture in both the CMS and D3KE. For the CMS method, +OpenSim is executed in parallel with four cores. Our report results in frames per +second. +3.4. Experiment 4: Generalization Performance. The goal of camera-based +kinematic estimation is ultimately to create tools for researchers and clinicians to +analyze and diagnose human movement, these tools should not discriminate between +different subjects and movements. We analyze whether our method generalizes to +different subjects, movements, and joints. +To assess how well D3KE generalizes, we compare the estimates of the proposed +method to the ground truth on the time series of each of the 16 participants in the +test set with respect to the performed movement, the joint, the camera view and +the individual participant. For this test, we use the best performing model from +experiment 1 with the lifting transformer. +For all time series of joint angles, mean absolute error (MAE) and Pearson’s +correlation coefficient (ρ) were calculated between the estimation from D3KE and +the ground truth. +Central tendencies in the data are reported as a median and interquartile range +of MAE, RMSE and ρ, as the data are not normally distributed, as assessed through +visual inspection and confirmed by the Shapiro-Wilk test. For completion, mean +and standard deviation are also reported. The absolute values of ρ were categorized +as weak, moderate, strong and excellent for ρ ≤ 0.35, 0.35 < ρ ≤ 0.67, 0.67 < ρ ≤ +0.90 and 0.90 < ρ, respectively [56]. +3.5. Software and Tools. All data analysis was conducted in python 3 [58] using +the pandas library to generate descriptive statistics, SciPy library for the calculation +of MAE, RMSE and Pingouin library for the calculation of ρ. +4. Results +4.1. Direct vs. Multi-Step Estimation. +4.1.1. A: 3D Pose Based Kinematic Estimation. As shown in Table 1, D3KE has +better performance than CMS methods in terms of joint angles and body scales, +and these two factors are the key to kinematic estimation. Significantly, D3KE +reduces 37.4% of errors on MAEangle when comparing the CMS method with the +Transformer architecture. Although the proposed method has a slightly larger MP- +BLPE than the CMS, this metric is not related to the kinematic estimation and is +only used as one of the losses during training in the proposed method (e.g., MP- +BLBE is not minimized). This indicates a gain in accuracy when directly estimating +kinematics from the video instead of the multi-step approach of first estimating pose +and then estimating kinematics. + +14 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +Table 1. Comparison of bony landmarks position (MPBLPE), +body scales (MAEbody) and joint angles (MAEangle) between the +estimation of and the ground truth across all participants, move- +ments, joints and camera views. For the custom multi-step ap- +proach (CMS) as well as our proposed method, we compare convo- +lutional networks with different temporal networks. All versions of +the proposed method show superior performance for the prediction +of body scales and joint angle estimation. All CMSs show supe- +rior performance in estimating marker positions. +Each method +group shows better performance for the task it was optimized for, +highlighting the importance of direct optimization. Bold numbers +indicate the best performance. +MPBLPE +(mm) +MAEbody (mm) +MAEangle (◦) +D3KE +Convolutional +37.78 +6.07 +3.58 +Conv.+ LSTM +37.61 +5.97 +3.57 +Conv.+ TCNs +38.06 +5.93 +3.54 +Conv.+ +Transformer +36.98 +5.90 +3.54 +CMS +Convolutional +35.04 +6.25 +5.89 +Conv.+ LSTM +33.74 +- +5.79 +Conv.+ TCNs +34.52 +- +5.82 +Conv.+ +Transformer +34.00 +- +5.66 +In Table 2, we list RMSE and MAE for body scales of selected segments. The re- +sults show that the CMS performs better than D3KE on lower limbs, and D3KE +performs better than the CMS on upper limbs. The CMS and the proposed method +have comparable performance in scale estimation of the pelvis and lower limbs. +Table 2. Errors in scaling factors of the proposed method and +the baseline compared against the ground truth. +D3KE shows +better performance for the upper extremities and slightly worse +performance for the lower extremities. +RMSEbody (MAEbody (mm)) +CMS +D3KE +pelvis +0.090 (9.58) +0.091 (9.82) +femur +0.073 (10.55) +0.091 (22.21) +tibia +0.060 (9.82) +0.102 (35.00) +humerus +0.102 (14.55) +0.068 (9.41) +ulna +0.395 (24.91) +0.075 (11.59) +radius +0.395 (23.60) +0.075 (10.98) +4.1.2. B: 2D Pose Based Kinematic Estimation. The results of the comparison of +our proposed method, CMS, OpenPose, and MediaPipe are shown in Table 3. We + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +15 +find that algorithms trained on noisy labels, that use fewer key points perform worse +than ours. We see a clear difference between the unsmoothed OpenPose estimations +and the smoothed MediaPipe estimations in the mean velocity of the estimations. +Our proposed method D3KE still performs better showing that even in an ideal sce- +nario (CMS, i.e., no noise in the labels, enough markers, same distribution training +data) direct estimation is preferable. +Table 3. Comparison of popular pose estimation algorithms to +D3KE. As OpenPose and MediaPipe require multiple cameras to +create 3D keypoints, we compare against the average of both cam- +era views for CMS and D3KE. CMS shows better performance than +OpenPose and MediaPipe and D3KE shows the overall best per- +formance. Indicating that direct estimation is preferable to (naive) +implementations of multi-step methods. +MAEangle (◦) +SDangle (◦) +MVangle (◦/s) +OpenPose +16.98 +25.91 +75.15 +MediaPipe +10.60 +18.80 +37.15 +CMS +5.11 +10.27 +15.74 +D3KE +3.41 +6.05 +13.57 +4.2. Sequential Network Variants. Table 1 also shows the results of different +sequential networks for smoothing of the predictions. Although the convolutional +model by itself already has good performance in joint angle and scale factor esti- +mation, using temporal smoothing can additionally reduce the estimation error. +The results of our investigation to reduce the noise in the estimations using +temporal smoothing are shown in Table 4. +The result shows that all temporal +models can improve the smoothness of the sequence. The LSTM achieves the best +performance on MVBL and MVangle among all temporal models, this is contrary to +the results in Table 1, in which using a Transformer as the sequential model yielded +the best results. +Table 4. The mean velocity errors for bony landmarks MVBL and +joint angles MVangle, lower values indicate smoother estimations. +Adding a sequential model most probably improves the continuity +of estimations. +MVBL (mm/s) +MVangle (◦/s) +Convolutional model +378.01 +21.7 +Conv. + LSTM +243.23 +12.19 +Conv. + TCNs +245.35 +12.29 +Conv. + Transformer +262.82 +13.57 +4.3. Processing Speed. Our proposed method achieves 31.96 fps with a batch +size of 256, as shown in Table 5. Since the skeletal-model layer must traverse body +segments in the level order, our proposed method is slower than the CMS for a + +16 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +batch size of 1. However, the support of mini-batch computation in the skeletal- +model layer allows D3KE to run faster than the CMS. Showing that our method +can reach video framerate speeds on a competent GPU. +Table 5. Comparison of processing speed in FPS of D3KE and +the baseline for multiple images or ’batches’ in parallel. OpenSim +does show little change in processing speed for increasing batch +sizes. The proposed method achieves framerate speeds for batches +of 256 images, allowing it to analyze images as fast as a common +webcam or mobile phone camera collects them. +Bold numbers +indicate best performance. +Batch +Size +1 +16 +64 +128 +256 +D3KE +0.92 +8.78 +20.94 +28.25 +31.96 +CMS +7.51 +8.36 +8.43 +8.44 +8.35 +4.4. Generalization Performance. Figure 4 shows that both CMS and D3KE +have relatively little variation across different movements and different participants, +yet larger variations across individual joints. This is also reflected in median MAEs +and ρ per joint (Table 6), with median MAEs for joints varying within a range 3.8° +for joints, while movements and participants vary under 1°. It shows that D3KE +generalizes well to different participants and movements. +Figure 4. Mean absolute error for predicted joint angles per joint, +movement and subset. Across these groups, D3KE shows less varia- +tion compared to the CMS. The low variations indicate that D3KE +is suitable for use on different participants and movements. + +Mean Absolute Error per Group +Action +Joint +Subject +Error +10 +Mean Absolute +0 +Baseline +Baseline +Baseline +Our Method +Our Method +Our MethodTOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +17 +Table 6. Median and inter-quartile ranges (IQR) of joint angles +per joint, movement and participant. MAE, RMSE and correla- +tion are calculated over individual frames, Medians and IQR are +reported due to the skewed distribution of results. Within each +group, both camera views show similar errors. Joint angles show +the highest error and highest spread of values of all groupings. +D3KE generalizes well to different movements, participants and +camera views. +Group +Camera View +MAE (◦) +RMSE (◦) +ρ +Median IQR +Median IQR +Median IQR +Joint +Frontal +2.13 +3.80 +2.54 +3.96 +0.77 +0.16 +Sagittal +2.14 +3.03 +2.55 +3.49 +0.73 +0.21 +Movement +Frontal +1.85 +0.63 +2.19 +0.84 +0.76 +0.11 +Sagittal +1.91 +0.46 +2.30 +0.65 +0.74 +0.11 +Participant +Frontal +1.76 +0.53 +2.03 +0.66 +0.77 +0.04 +Sagittal +1.84 +0.28 +2.14 +0.35 +0.74 +0.04 +4.5. Qualitative Results. We visualize the estimation of musculoskeletal models +from our proposed method with the Transformer architecture in Figure 5. We also +show the comparison between estimation and ground truth of the left knee angle as +an example of joint angle estimation quality. We took the average body scales of the +predicted sequence of scaling factors to scale the model and visualize it in OpenSim +using the predicted joint angles as inputs. From the figure, we can see that the +proposed method can achieve results that are in agreement with the single-view +input video. + +18 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +Figure 5. Qualitative results of D3KE from a ’sitting-down’ +movement in the BML-Movi dataset. The top row shows selected +frames throughout the movement. The middle row shows different +poses of the ground truth skeletal model throughout the movement +(cyan) and the skeletal model (white) based on D3KE’s estimation. +The bottom row shows the changes in flexion/extension of the left +knee throughout the movement with blue being the predicted and +orange being the ground truth angle. +5. Discussion +In summary, we compared a direct approach of estimating joint angles from +video images to the more traditional multi-step approach found in most recent +works. The traditional method first estimates key points from a video of a subject, +then calculates joint angles using a (musculo)skeletal model through an inverse- +kinematics process. We developed a method consisting of a convolutional neural +network and a sequential network both including a specialized layer that performs +kinematic transforms of a (musculo)skeletal model and allows for direct optimiza- +tion of the predicted joint angles (D3KE) and treats the prediction of key points +only as an auxiliary task. We compared our direct estimation approach against +naive implementations of often used algorithms in the related literature, as well as +a self-implemented custom multi-step approach (CMS) that is trained on the same +data as our direct approach. We show that direct estimation of kinematics yields +higher accuracy in predicted joint angles compared to the traditional multi-step +approach. Our results indicate that direct estimation can help the future develop- +ment of algorithms for fast and accessible kinematic analysis for researchers and +clinicians. +5.1. Direct vs. Multi-Step Estimation. +5.1.1. 3D-Pose Kinematic Estimation. To compare direct estimation vs. multi-step +estimation, we compared our D3KE method against a 3D-pose based multi-step +approach (CMS) with comparable network architecture and trained on the same + +Left Knee Angle +0 +Angle +-50 +Ground Truth +Proposed Method +-100 +0 +50 +100 +150 +200 +250 +Frame NumberTOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +19 +training data. Compared to the CMS, our proposed method improves the accuracy +of joint angle estimation. For all model combinations, we can see an improvement +of about 35% in accuracy for the estimation of joint angles. Our results support +the feasibility of our proposed method. It delivers improvements due to directly +optimizing the predicted joint angles and scaling factors while using the pose esti- +mates only as an auxiliary task. The auxiliary task effectively imposes a constraint +on the network estimation. We show that direct optimization is preferable to the +multi-step approach when using videos from a single-camera view. We expect that +using additional specialized layers, a network might be able to directly optimize for +individual muscle forces with comparable accuracy from a monocular video. +5.1.2. 2D-Pose Kinematic Estimation. We compared our direct approach (D3KE) +and the self-trained multi-step approach (CMS) against two multi-step approaches +commonly used in the literature. Compared to the more traditional implementa- +tions of the OpenPose and MediaPipe algorithms, both our proposed and our CMS +method show superior performance. From Tables 4 and 6, we see that estimations +from D3KE are far smoother compared to the traditional methods. This is likely +due to multiple reasons. The predicted key points of OpenPose suffer from system- +atic errors due to inaccuracies in their training data [13], which can explain the drop +in performance, for MediaPipe the accuracy of labels in their training data is not +known as their paper only states that their annotators were human [6], not whether +they had expertise in labeling anatomical key points. The lack of smoothing for the +predicted OpenPose key points can also contribute to its overall worse performance. +MediaPipe, which uses internal smoothing, shows better results in comparison. In +general, we expected worse performance from OpenPose and MediaPipe as they +only predict 18 or 33 key points respectively, while we supervise a total of 77. Both +traditional methods predict keypoints representing joint centers and not markers +on body segments, this makes it hard to distinguish rotations between different +body segments during the musculoskeletal modeling step. +Figure 6. Qualitative comparison of predicted angles of the left +knee, from a ’sitting-down’ movement in the BML-Movi dataset. +While OpenPose shows by far the noisiest estimation, the smooth- +ing of the MediaPipe estimation is clear, our proposed method +and implemented CMS work best, probably due to the restriction +of additional markers. + +Left Knee Flexion/Extension angle during Sitting Down movement +0 +OpenPose +MediaPipe +Angle +-50 +D3KE +CMS +Ground Truth +-100 +0 +50 +100 +150 +200 +Frame #20 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +5.2. Network Variants. We compared three commonly used sequential networks +to improve our estimation. +We show that all sequential networks improve our +estimation accuracy. One possible explanation for this is that the network is better +able to handle ‘self-occlusion’ artifacts. The estimation of a 3D-pose from a single +camera is an ill-posed problem as a 3D point projected to a 2D image can originate +from any position along the ray(s) that fall on the image sensor and form the +corresponding pixel. This can lead to self-occlusion, such as the torso and left arm +occluding the right arm during a right arm swing in frames recorded from the left +sagittal view. During self-occlusion, it is difficult for frame-based networks to make +a good estimate as they lack temporal information of previous angles of the arm to +extrapolate from. Sequential networks on the other hand have access to temporal +information, which can allow for more accurate estimations. Although we only see +a slight increase 0.001° in MAE of the joint angle estimation, we can see a clear +improvement of the sequential models in the smoothness of the predicted angles +Table 4. This might be due to the network learning to interpolate motions during +occurrences of self-occlusion. +5.3. Processing Speed. Compared to the multi-step baseline, CMS, our D3KE +approach shows increased calculation speeds for larger batch sizes. +Both CMS +and D3KE make use of the same ResNeXt50 architecture, which should show ap- +proximately the same performance increase with increasing batch sizes for both +methods. D3KE could be expected to be slower, as it also has the additional time +cost of calculating the pose from the estimated kinematics in the skeletal model +layer. However, due to its multi-step nature, CMS has to perform an additional +inverse kinematics calculation. This calculation seems to form a bottleneck in the +processing speed of the CMS approach restricting it to a framerate of 8 fps. Other +multi-step algorithms will most likely encounter the same problem. In the case of +OpenPose, which runs at about 4 fps [6], even lower frame rates can be expected +for a complete pipeline. This shows the advantage in the processing time of our +direct approach. +For a method to be usable in everyday life, it should be reasonably fast in +running. Processing speeds allowing a method to run between 15 and 30 frames +per second are favorable, as they show that a method can process a video as fast as +its frames are collected. However, our results might not directly translate to every +real-world scenario. To process multiple images simultaneously as batches, D3KE +currently requires GPUs that are not available in mobile devices, which prevents +it from beingportable. +In addition, we use the Faster R-CNN object detection +network to crop our images. This step was not included in the processing speed +evaluation, as it is highly dependent on the chosen object-detection algorithm. +However, with inference speeds of 12fps, the Faster R-CNN object detection would +form a bottleneck in applying our method in real-time applications. +Given the +speed of development in the field of object detection, Faster R-CNN can by now +be regarded as an old algorithm, and newer and faster object detectors should be +used instead. The YoloV7 algorithm [61], which performs object detection at up to +286 fps could be considered. In general, the current architecture is not optimized +for speed or a specific technology and we are using off-the-shelf, fairly standard +convolutional and sequential architectures. +For these architectures, smaller and +faster alternatives might be found in the future. When optimizing for all these +points, we predict the proposed method could run on mobile devices within a few + +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +21 +years, effectively enabling a 3D kinematic analysis instrument to become available +for everyone with a mobile phone or tablet. +5.4. Generalization Performance. Our method generalizes well on the tested +data. As shown in Figure 4 and Table 6, the estimation variations across partic- +ipants and movements are small. +We can conclude that our method shows the +ability to generalize to different camera views, participants, and performed move- +ments, within the tested dataset. Our results indicate that D3KE could be generally +applied to a variety of people and movements, including clinical and sports appli- +cations, e.g., physiotherapists and athletes, when trained on sufficient additional +data. +Although we show good generalization performance on the BMLMovi database, +it is difficult to estimate how well our method will generalize in a real-world scenario. +In machine learning settings, training data is often not representative of the task +of the network in the real world [5] and can introduce biases if applied to scenarios +that are very different from the one represented in the training data. Unfortunately, +there is currently a lack of deep learning datasets for kinematic analysis [52, 40, +13, 60]. In addition, while the BML-Movi database is excellent for training neural +networks due to the large number of participants performing movements and the +diversity of execution styles, it might be not extensive enough to train a network for +biomedical applications in the real world. However, to evaluate the current method +fully, such an extensive dataset would be necessary. In general, we expect a drop +in accuracy when our method is applied to a scenario different from the BMLMovi +database. As we train on just two calibrated cameras, we expect our method to be +most vulnerable to alternative camera positions, that do not show people in either +frontal or sagittal view. Future research should investigate the stability of direct +estimation methods when applied to data that differs significantly from the training +data. +5.5. Future Work. To improve the accuracy of the algorithm and provide further +insight into the strengths and weaknesses of monocular joint angle estimation, a +new dataset with dedicated annotation is needed. A dataset specifically designed +for the estimation of joint angles and/or kinetics could improve the accuracy of the +algorithm. This dataset could be established with a large number of camera views, +and top-down views for better estimation of movements in the transverse plane, +where participants perform movements that exercise the full ROMs of individual +joints including upper extremities, as well as movements that are relevant for health +care professionals such as physiotherapy exercises and other clinical tests. In addi- +tion, the inclusion of abnormal movement patterns could give better insights into +the clinical relevance of newly developed methods. +Transfer learning could be explored to apply 3DKE in settings where little train- +ing data are available. Vdeos that are very different from the BMLMovi training +data, such as people wearing more clothes, are in different surroundings, or are +filmed from a different camera view, will, most probably, yield worse accuracy than +shown in this paper. Transfer learning of a pre-trained D3KE on a minimal portion +of a dataset could be investigated as an alternative to the time-consuming collection +of a novel dataset. +The capabilities of D3KE as an adapter for kinetic analysis of a movement in +OpenSim could be explored. Given data similar to BMLMovi or successful transfer + +22 +TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS +learning on relevant data beforehand, our method provides an easy way to skip the +tedious steps of scaling and running inverse kinematics on an MSM. This enables the +quick generation of MSMs for kinetic analysis from just a single video. Even if this +kinematic estimation comes at the cost of reduced accuracy, it could provide coarse +insights into collected data, which can later be confirmed through finer analysis +with the manually scaled MSMs. +D3KE could be made more generally applicable if the underlying model of the +Skeletal-model layer would not be fixed. Currently, the underlying model is fixed +in the Skeletal-model layer. Future iterations could explore combinations of the +Pytorch and OpenSim python libraries to allow training a network on a self-defined +model or allowing a pre-trained model to be refined through transfer learning for, +e.g., only estimation of joint angles around the shoulder. +Existing Explainable AI tools should be applied to better understand the inner +workings of D3KE. Deep neural networks are capable of high accuracy estimation, +because of their ability to break down highly complex tasks into simpler tasks [2], +but understanding what these simpler tasks are is non-trivial. Research in Explain- +able AI has generated tools and frameworks that allow one to better understand +the basis of the final predictions of a network [53]. Applying these tools could help +users and researchers alike to better understand the biases and limitations of our +method. D3KE can still predict the joint angles even if these joints are occluded; +this means it must make assumptions. What these assumptions are and how they +came to be are important to estimate the trustworthiness of this algorithm in a +real-world scenario. +6. Conclusions +In this paper, we present a novel end-to-end neural network for the estimation +of segment joint angles of the human body. Compared to the previous method, +we directly regress to the joint angle and scale for individual segments from the +input video. We trained our method from scratch on the BML-Movie database and +compared it against a 3D pose estimation method on which we used the inverse +kinematics tool of OpenSim to obtain the kinematics. +We conclude that using direct estimation of joint angles is preferable in a single +camera setting, as it is more accurate compared to the common approach of fitting +an estimated pose to a musculoskeletal model and performing inverse kinematics. +By allowing the network to directly optimize for the joint angles and scaling factors, +our method is less prone to errors in the key point labels used to predict key point +location for pose estimation. In addition, the use of a sequential model is important +when designing a neural network architecture for kinematic estimation, as it allows +to smooth predictions over time to create better estimates of limb position and joint +angles during self-occlusion. +While using deep learning for biomedical solutions is still in its infancy, the pre- +sented method shows that training networks from scratch for specialized tasks is +a viable way to estimate joint angles from a single camera video. With further +advancements in the underlying algorithms as well computational performance, we +predict that the methodology we have presented will assist biomedical and clinic +practitioners to measure and monitor human movement in the near future. +Funding: This work was supported by the Dutch Research Council (NWO) under +the Citius Altius Sanius Perspective Program P16-28 Project 4. + +REFERENCES +23 +Aknowledgements: The authors would like to thank Lisa Noteboom for her +feedback as well as Marco Hoozemans and Dirkjan Veeger for guidance and insight +during our bi-weekly meetings. +Conflict of Interest: The authors declare no conflict of interest. +Abbreviations: The following abbreviations are used in this manuscript: +CMS . . . . . . . . . . . . Custom multi-stage approach +D3KE . . . . . . . . . . . Direct 3D kinematic estimation +IQR . . . . . . . . . . . . . Interquartile Range +MAE . . . . . . . . . . . . Mean absolute error +MMC . . . . . . . . . . . Markerless motion capture +MPBLPE . . . . . . . Mean per bony landmark error +MSM . . . . . . . . . . . . Musculoskeletal model +MVE . . . . . . . . . . . . Mean velocity error +OMC . . . . . . . . . . . . Optical Motion Capture +PCC . . . . . . . . . . . . . Pearson correlation coefficient +RMS . . . . . . . . . . . . Root mean square error +ROM . . . . . . . . . . . . Range of motion +SD . . . . . . . . . . . . . . . Standard Deviation +TCN . . . . . . . . . . . . Temporal Convolutional Network +References +[1] +Mazen Al Borno, Johanna O’Day, Vanessa Ibarra, James Dunne, Ajay Seth, +Ayman Habib, Carmichael Ong, Jennifer Hicks, Scott Uhlrich, and Scott +Delp. “OpenSense: An open-source toolbox for inertial-measurement-unit- +based measurement of lower extremity kinematics over long durations”. 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On the Con- +tinuity of Rotation Representations in Neural Networks. 2020. arXiv: 1812. +07035 [cs.LG]. + diff --git a/k9E5T4oBgHgl3EQfHA4j/content/tmp_files/load_file.txt b/k9E5T4oBgHgl3EQfHA4j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7eeab635fa63db551ab850cb4211404708f3b82d --- /dev/null +++ b/k9E5T4oBgHgl3EQfHA4j/content/tmp_files/load_file.txt @@ -0,0 +1,1622 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf,len=1621 +page_content='TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS MARIAN BITTNER 1,2,3,*,†, WEI-TSE YANG 2,†, XUCONG ZHANG 2, AJAY SETH 3, JAN VAN GEMERT 2, AND FRANS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' VAN DER HELM 3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Markerless estimation of 3D Kinematics has the great potential to clinically diagnose and monitor movement disorders without referrals to expensive motion capture labs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' however, current approaches are limited by performing multiple de-coupled steps to estimate the kinematics of a person from videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Most current techniques work in a multi-step approach by first detecting the pose of the body and then fitting a musculoskeletal model to the data for accurate kinematic estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Errors in training data of the pose detection algorithms, model scaling, as well the requirement of multiple cameras limit the use of these techniques in a clinical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our goal is to pave the way toward fast, easily applicable and accurate 3D kinematic estimation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To this end, we propose a novel approach for direct 3D human kinematic estimation D3KE from videos using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our experiments demonstrate that the proposed end-to-end training is robust and outperforms 2D and 3D markerless motion capture based kinematic estimation pipelines in terms of joint angles error by a large margin (35% from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='44 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='54 degrees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We show that D3KE is superior to the multi-step approach and can run at video framerate speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This technology shows the potential for clinical analysis from mobile devices in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Introduction 3D Human kinematics involves measuring joint angles between body segments, which is essential in the day-to-day practice of experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Skilled physicians could judge, just by looking at a specific motion of their patient, whether it is healthy or abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Skilled sports coaches can help their coachees achieve better perfor- mance and lower injury risk by evaluating their movements through observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, these visual examinations of human kinematics remain inherently subjec- tive, leading to variation between and within human observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Modern systems and sensors could reduce these variations through more objective observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Yet, these systems make the measurement of human motion more costly and more time- consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' A system with the availability and ease of use of visual estimation would help physicians and coaches make more objective observations more often, ultimately raising their own and their subjects quality of life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Digital cameras have made the estimation of human kinematics more accessible but come at the 1 Vicarious Perception Technologies (VicarVision), 1015 AH Amsterdam, The Netherlands 2 Computer Vision Lab, Delft University of Technology, 2628 XE Delft, The Netherlands 3 Biomechanical Engineering, Delft University of Technology, 2628 CN Delft, The Netherlands Corresponding author: mbittner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='work@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='com †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='05435v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='CV] 13 Jan 2023 2 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS cost of reduced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Compared to the more traditional Optical Motion cap- ture (OMC) systems, markerless motion capture (MMC) systems do not require specialized cameras and markers attached to the subject being monitored, but use normal RGB cameras in combination with image-based automatic pose estimation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Instead of specific markers, pose estimation algorithms detect the cen- ters of major joints of the human body, such as the shoulders, hips, and knees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' These detected centers are usually referred to as key points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Multiple commonly used markerless motion capture methods rely on 2D pose estimation methods [27, 45, 46, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Often these methods still need more than one camera to generate a good estimation of the keypoints in 3D, which again requires additional cameras to be set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' On the other hand, an increasing number of meth- ods are using single-view (monocular) 3D pose estimation methods [21, 32, 43], which allow to estimate a 3D pose just by using a single camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This makes MMC systems faster and more accessible as they do not require the additional time and expertise to place markers on the subject or calibrate multiple cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, MMC systems assume that current pose estimation algorithms can accurately re- place markerless motion capture systems for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', biomechanical applications [52, 13, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Commonly used pose estimation algorithms introduce mistakes in kinematic es- timation pipelines due to systematic errors in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To detect key points, most pose estimation methods are trained on a combination of images of a person and ground truth annotations which map pixels in the image to their corre- sponding joint center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='These ground truth annotations are often manually conducted by non-expert annotators, leading to errors caused by personal biases for training and inaccuracies in the pose estimations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For example, Needham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [41] compared three often used pose estimation algorithms OpenPose [10], DeepLab- Cut [38] and AlphaPose [17] algorithm against an OMC system and showed errors in the estimation of joint centers of 30 mm to 50 mm with variations in 12 mm to 25 mm in marker placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Cronin [13] provides an overview of additional problems with 2D pose estimation for kinematic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' These differences are most likely due to a difference between the application that pose estimation algo- rithms are often developed for and their application to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', the biomedical do- main, which has different accuracy requirements [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Wade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [60] proposed to solve this problem by re-annotating existing large-scale datasets, this, however, is a time-consuming process, when for example considering the COCO-keypoint dataset https://cocodataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='org/#keypoints-2020 (accessed on 2 December 2022 ) consists of more than 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='000 labeled poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the evaluation of pose es- timation algorithms, these labeling errors will just appear as a baseline error that all algorithms training on the same data will have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, for applications in the biomedical domain and in situations such as kinematic estimation, where the pose is just an intermediate step errors can propagate to subsequent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Errors in the estimated pose cannot not be corrected by most kinematic es- timation pipelines because they all roughly follow a ‘multi-step’ approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The ‘multi-step’ approach consists of Detection of the 3D pose (in one or more steps);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (Optional) modeling of the pose with a (musculo)skeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Calculation of kinematics and/or downstream tasks such as gait parameters or dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 3 For example, Kidzinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [27] used OpenPose to first predict key points from a video and then trained a convolutional neural network (CNN) to predict the walk- ing parameters of patients with cerebral palsy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [32] first model the 2D pose in OpenPose then create a 3D pose using data-driven matching and finally estimate 3D gait parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Noteboom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [43] first used VideoPose3D [49] to estimate a 3D pose, followed by modeling in OpenSim [54] for the estimation of dy- namics from a single camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Pagnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [45, 46] developed the handy Pose2Sim tool, which first combines 2D OpenPose pose estimations from multiple cameras into a 3D pose then models it in OpenSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Because the pose estimation step is de-coupled from kinematic estimation, errors in pose estimation propagate through to the estimation of kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Uchida and Seth [57] showed that 20 mm of marker uncertainty leads to a variation of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='9◦ in peak ankle plantarflexion angle and im- pacts downstream tasks such as joint moment estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Della Croce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [14] showed precision variation 13 mm to 25 mm, which leads to differences in estimated joint angles up to 10°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Fonseca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [18] showed that misplacement of markers up to 10 mm can lead to errors of 7◦ depending on the marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' With estimation errors of 30 mm to 50 mm in keypoint estimation [41], it is to be expected that these errors will substantially influence kinematic estimation from markerless motion capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Low-pass filter [40, 45] or bi-directional Kalman-filter [40] has been applied to com- pensate for noisy key point estimations, but cannot correct for faults in keypoint detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Subsequent modeling and kinematic calculation steps can only compen- sate for these inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This ‘multi-step’ approach is probably inspired by the steps of a traditional OMC method, as in the traditional OMC systems the pose detection step is done using a different system and is thus isolated from the other steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In camera-based kinematic estimation pipelines, however, the de-coupling of individual steps is no longer necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Deep neural networks have often demonstrated their ability to outperform multi- step systems, by implicitly learning individual steps through end-to-end training between an input and the desired output [55, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The main strength of deep neu- ral networks lies in their ability to break down a highly complex task, in this case, the estimation of kinematics from videos, into a sequence of simpler tasks, with- out the need for intermediate ‘hand-crafted’ representations [30, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Due to the fully differentiable nature of neural networks, it means that an error in estimation dur- ing training can influence all stages of the network and adjust them accordingly [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This allows deep neural networks to directly estimate kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In this work, we challenge the notion of the classical multi-step approach of pose estimation, fitting of a musculoskeletal model and kinematic estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To this end, we propose a novel end-to-end method that allows for direct estimation of hu- man kinematics, which is directly optimized for kinematic estimation while treating pose estimation only as an auxiliary task to constrain the estimations of the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Figure 1 shows a general overview of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 4 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Overview of the proposed direct 3D human kinemat- ics estimation (D3KE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Instead of using the common ’multi-step’ approach of predicting pose, fitting it to a model, and estimating kinematics, our D3KE directly estimates the kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Errors in earlier steps of the multi-step approach propagate to later steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' in contrast, our method can correct for errors occurring anywhere between input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To the best of our knowledge, we are the first to present an end- to-end trainable network that directly generates joint angles, joint positions, scale factors and marker positions of a biomechanical model from a monocular video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We propose a method that directly regresses from a video to joint angles and scales using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We investigate the influence of various temporal smoothing methods to increase the accuracy of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We introduce a novel type of network layer that allows for the calculation of the 3D pose from estimated kinematics during the training process to train the network simultaneously on the pose and kinematic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Materials and Methods Our method takes videos from a single camera as input and directly estimates joint angles, which we call direct 3D kinematic estimation (D3KE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The proposed method first coarsely estimates kinematics per frame by using a convolutional neural network, and then it uses a sequence network with temporal relations across frames to re-fine kinematic estimations at each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' An overview of our method is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Both networks estimate the scale of body segments, joint angles, and a rotation matrix from the pelvis to the ground, those serve as input for a skeletal- model layer in both networks that allows for additional supervision on the pose of a subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Multi-step approach Step: Musculoskeletal Modelling Step: Pose Estimation Error Error Error Correction propagation propagation 70° 50° OpenSim D3KE method Direct Estimation Deep Error Correction Error Correction 70° Neural Network 50°TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Taking a single view video as input, D3KE consists of one convolutional neural network and one sequential network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Per frame, D3KE outputs joint angles and scales of individual bones in a skeletal model(scale factors) with a convolutional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Addi- tionally, joint angle and scale factor are converted to a pose through the skeletal-model kinematics (SM) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' A series of frame esti- mations in time are then fed into a sequential network to smooth the estimations and reduce artifacts if one limb occludes another in the view of the camera (self-occlusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In this section, we first describe the deep learning architecture, including a de- tailed description of the skeletal-model layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We then describe how the ground truth data was generated and which pre-processing and hyperparameters were used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Lastly, we describe the dataset used for training and testing our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Network Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Convolutional neural networks(CNNs) have shown good accuracy for 2D and 3D pose estimations [10, 51, 39, 42, 48] from single input im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Conventionally 2D CNNs are used for pose estimation tasks, that takes a single image as an input and predict the pose of one or multiple people in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For our method, we choose a per-frame convolutional network to coarsely predict the joint angle and scaling parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Inspired by [51], we choose a stan- dard pre-trained ResNeXt-50 [62] as our convolutional backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To fine-tune the per-frame predicted joint angles and scaling parameters we add a sequential network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Sequential networks are used in pose estimation to ‘lift’ an estimated 2D pose to 3D [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Recent research combines temporal information with lifting to improve accuracy during frames where one limb occludes another in the view of the camera (self-occlusion) or where not all key points were de- tected [31, 33, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In contrast to CNNs, these sequential networks do not take a single frame as input but exploit temporal dependencies in the data for their pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As the convolutional network outputs per-frame estimates, it cannot take temporal information into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We add a sequential network to our architecture to refine a sequence of estimations made by the convolutional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Inspired by works on temporal lifting we experimentally evaluate three sequential networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=" an Temporal smoothing Per-frame Estimation ngle ['g." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' ¥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='00 Convolutional to 0 Neural Network Forward Kinematics 25 25 Layer 50 50 Sequential Convolutional 75 Network 75 Neural Network Forward Kinematics Forward Kinematics 100 Layer 100 Layer 125 125 150 150 Convolutional Neural Network 175 175 Forward Kinematics Layer6 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS LSTM [23], a Temporal Convolutional Network (TCN) [49] and a Transformer [31] to refine the predicted joint angles and scale factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Both the convolutional and the sequential network contain a specialized layer that allows each network to perform the kinematic transformations of a musculo- skeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Therefore, at train time both networks can be supervised not only on the estimated joint angles but also on a resulting pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Both convolutional and sequential networks are supervised by losses of joint positions, marker positions, body scales and joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The overall objective function L can be expressed in the equation (1) L = λ1Ljoint + λ2Lmarker + λ3Lbody + λ4Langle, where λ1, λ2, λ3, λ4 are weights of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We use the root-relative L1 loss in Equa- tion (2) to define the loss of marker position Lmarker and the loss of joint position Ljoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The estimations ˆy and the labels y are first subtracted with each root po- sition ˆyroot, yroot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the loss of body scales Lbody and joint angles Langle, we calculate the L1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (2) l = ∥(ˆy − ˆyroot) − (y − yroot)∥1 The objective of a neural network during training is to minimize the loss function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' in our case, the difference between estimated and ground truth joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' How- ever, this joint angle loss cannot capture the underlying relations and constraints of individual angles, dictated by the human musculoskeletal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Intuitively, small changes in the angles of spine, shoulder, and elbow can accumulate and lead to large differences in the position of the hand, as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To ad- dress this issue, we propose to use a skeletal-model layer to perform the kinematic transform of a musculoskeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our skeletal-model layer uses an internal representa- tion of a skeletal model to convert the predicted joint angles and scale factors to the positions of individual markers on segments of the skeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This allows our method to be supervised during training not only on errors(losses) in the estimation of joint angles but also on errors in the resulting pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' On the right, we show the additional error that is created between estimations (gray) and ground truth (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This auxiliary estimation of the pose as 3D marker positions helps to constrain the estimation of joint angles as small changes in proximal joints can have a large effect on a marker at more distal positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Skeletal-Model Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The skeletal-model layer allows us to convert predicted joint angles into marker positions on a skeletal model and add them as an additional loss term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This loss term represents the cumulative effect of small joint angle changes on the final pose, indirectly imposing the constraints of a skeletal-model on the predictions of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As the skeletal-model layer does not contain any learnable parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', it cannot change during network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The accuracy of the predicted pose is completely determined by the input to the skeletal-model layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' thus, the pose prediction is only an auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' A skeletal model consists of body segments, motions between different body segments (joints) and points with a vector from a center of its anchor body segment (markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Given body scales β, joint angles θ and rotation matrix Rground←pelvis, we use the skeletal-model layer to calculate marker positions and joint positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In the following variables with a hat (ˆx) denote estimated values, variables without the hat (x) denote the predefined variable from the musculoskeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Skeletal Model layer Direct Estimation Marker Position Loss Joint Angles Segment Rotation ranslatior segm Scale Factors Marker Distances Scale Factor Loss X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2 Joint Angle Loss Backpropagation8 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS First, the translation part T in the transformation from the joint to the body depends on the subject’s body scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For example if the subject has longer legs, the center of the femur will be farther from the hip joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We can update the translation part ˆT by comparing the ratio between predicted body scales ˆβ and default body scales β in Equation (3), where ⊙ is elementwise multiplication, and ⊘ is elementwise division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (3) ˆT = T ⊙ (ˆβ ⊘ β) Then, we create a matrix to represent spatial transformation of motions Rmotion using Equation (4) A1, A2, A3 are the predefined axes ˆθ1, ˆθ2, ˆθ3 and predicted angels per degree of freedom per joint in axis-angle notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' G(A, θ) is the standard function converting an axis-angle representation to a 3x3 transformation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (4) Rmotion = � ��� R3R2R1 0 0 1 � ��� 4×4 (5) R1 = G(A1, θ1) (6) R2 = G(R1A2, θ2) (7) R3 = G(R2R1A3, θ3) Then, we can calculate the estimated transformation from the body to its parent body ˆRparent←child in Equation (9) using Equation (8) with Oparent, Ochild denoting predefined orientations from and ˆTparent, ˆTchild the predicted translations from the joint to the parent/child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' F(Oa) the conversion from euler angles to a 3 × 3 Rotation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (8) Rparent/child←joint(Oparent/child, Tparent/child) = � ��� F(Oparent/child) Tparent/child 0 1 � ��� 4×4 (9) Rparent←child = Rparent←joint Rmotion R−1 child←joint We measure the spatial transform by traversing from the root (pelvis) to leaf nodes (hands and feet) in the level order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In D3KE, we directly infer the rotation matrix from the pelvis to the ground Rground←pelvis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Rground←pelvis can initially be expressed in Equation (10), where I denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The rotation part of Rchild←joint is also a 3 × 3 identity matrix in our musculoskeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our method aims to predict the root-relative position so the translation part can be ignored during the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Moreover, in our musculoskeletal model, only joint angles of the pelvis are unbounded in [−∞,∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Predicting three unbounded angles to form the rotation matrix in Equation (4) will have the problem of discon- tinuity [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Thus, we directly predict the rotation matrix Rground←pelvis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (10) Rground←pelvis = I4×4 Rmotion R−1 pelvis←joint TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 9 Last, a marker with a vector of ⃗d from the center of the body is also dependent on the body scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The predicted vector of ˆd is updated in Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The position of the predicted point is calculated in Equation (12) with ˆRparent←child and ˆd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (11) ˆd = ⃗d ⊙ (ˆβ ⊘ β) (12) P = � parent,child∈path Rparent←child � ��� ⃗d 1 � ��� 4x1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Network Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Ground Truth Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For training our method, we need to create cus- tom ground truth data that contains all outcomes that our network is predicting since they are not available in publicly available datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Most pose estimation datasets, provide only video and marker positions from optical motion capture (OMC) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For training our method, we need the joint angle and the scales of individual bones, a rotation matrix of the pelvis to the ground as well as the marker positions corresponding to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To generate these, we model the OMC data, represented as a 3D human mesh model in the OpenSim software [54] and use inverse kinematics [36, 1] to generate joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The following describes each step in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' First, we create a general (musculo)skeletal model to fit the data using the OpenSim software [54, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As we are interested in capturing the complete motion of the human subject, we model the full body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' With the OpenSim software [16, 54] we create a full-body musculoskeletal model (MSM) by merging existing models of upper limbs and lower limbs [3, 4, 15, 24, 63] and thoracolumbar spine [8, 7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We add wrist and hand [20, 44] models to the MSM, which are not used for ground truth generation, for the sake of aesthetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The full-body model contains all bones in a skeletal system from the head to feet and from the upper arms to the hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We do not model every degree of freedom between vertebrae to avoid expensive computation and the requirement of at least three markers to measure the motions of one vertebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Instead, we separate the spine from the fifth lumbar to the first cervical vertebra into nine segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Then, we fit our data to the musculoskeletal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Instead of using the OMC marker data directly, we use OMC marker converted to 3D human mesh represen- tations using the MoSh++ [34] method, to make scaling the model to individual participants more time efficient and allow us to define an arbitrary number of vir- tual markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We fit our data to the musculoskeletal model, by first defining virtual markers on the vertices of the 3D mesh representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We then used these virtual markers as input for the OpenSim software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Then, we used the OpenSim internal scaling tool to scale the proportion of individual body segments according to the distances of virtual markers on the 3D mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As the sizes of individual body parts vary across individuals, this step must be conducted individually for each subject in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We define the ratio in dimensions between the default and scaled body segments as scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Finally, we used the inverse kinematics solver for the calculation of joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' During this process, the MSM is moved for each time step to a position that minimizes the sum of weighted squared errors between the virtual markers on the 3D mesh and markers defined on the musculoskeletal 10 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' All joint angles where segments had a higher squared error than 2 cm were disregarded in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The final ground truth values were the calculated joint angles, the scaling factors per segment as well as the virtual marker positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Additionally, a pelvis rotation matrix was generated for each frame, since the pelvis functions as the relative position of the model to the ground that is free to move in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Data Preparation and Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To generate the input for our net- work, each video frame was cropped and augmented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We use the pre-trained Faster R-CNN [50] with ResNet-50 [22] backbone to extract a square bounding box of the person in videos and resize it to 256 × 256 pixels as the input image size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Dur- ing training, we apply data augmentation with scaling, rotation, translation and noise to simulate occlusions similar to [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our model was trained using the following hyperparameters and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the ResNeXt model, we use an Adam optimizer with weight decay [35] of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='001 and a batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The learning rate exponentially decays in two steps from 5 × 10−4 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='33×10−5 over 28 epochs and from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='33×10−6 to 10−6 over 2 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For both sequential and convolutional networks, we set the hyperparameters with λ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='0, λ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='0, λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1 and λ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='06 experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Due to memory constraints, we do not train convolutional and sequential models simultaneously, but in succession, by first training the convolutional model and then refining predictions using the sequential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Software Tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' All training was conducted in python using the PyTorch li- brary [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The pre-trained ResNext and FasterRCNN networks were obtained from the torchvision library [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' All code for training and generation of ground truth will be made available in a Github repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='com/bittnerma/ Direct3DKinematicEstimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We trained and tested D3KE on the BML-MoVi Database [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' BML- Movi is an extensive motion capture and video dataset, it contains recordings of 90 actors that each perform 20 kinds of everyday movements as well as a random one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Motions were captured using inertial measurement units as well as a Qualisys optical motion capture system and videos were recorded using two computer-vision cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For this study, we used recordings from the calibrated Point Gray cameras (PG1, PG2) during recording session F as the full set of optical markers was used during this session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In accordance with the anatomical plane that each camera is viewing during the initial T-Pose of the participants, we will refer to the camera view captured by PG1 as the frontal- and PG2 as the sagittal camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the generation of ground truth virtual markers, we use the 3D mesh representations of the Qualisys data that is provided in the larger AMASS dataset [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For analysis of the data, we divided the BML-Movi database into 63 participants for training, 16 participants for the testing, and three participants for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This is common practice in the supervised training of deep neural networks [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' At training time, the validation set is used to evaluate the accuracy of kinematic estimation after each training iteration on a portion of the data the network does not have access to, to prevent overfitting on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiments 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiment 1: Direct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Multi-Step Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To evaluate the ac- curacy of our direct 3D kinematic estimation approach (D3KE) for joint angle estimation, we compare its performance against multiple versions of the multi- step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiment 1-A: 3D Pose Based Kinematic Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We first compare our direct estimation of kinematics and a 3D pose estimation multi-step baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To create a fair comparison between direct and multi-step estimations, we imple- ment a custom multi-step approach (CMS) that is trained on the same data as our direct approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the CMS, we combine a 3D human pose estimation method with subsequent musculoskeletal modeling in OpenSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We modify the metric-scale heatmaps [51] of the convolutional network to predict marker positions and SMPL keypoint positions in the metric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As for D3KE, we exploit a sequence network to re-fine marker positions at the target frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' More specifically, the convolu- tional network initially infers marker positions under a calibration pose (T-pose), and OpenSim utilizes the predicted marker data for body scaling, where the gen- eral musculoskeletal model is scaled to the participant’s body size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Re-fined marker positions are then used to run inverse kinematics with the scaled musculoskeletal model to obtain joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The main difference between the CMS approach and D3KE is that CMS uses multiple steps to estimate the kinematics and is only su- pervised on the marker/pose estimation task, while D3KE is directly trained on the kinematic estimation task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' this way, we can compare direct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' multi-step estima- tion of kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We use multiple metrics for the comparison of D3KE and the CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The mean per bony landmarks position error (MPBLPE) is used to evaluate bony landmark positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Bony landmarks are markers placed where bones are close to the surface, such as the elbow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This metric is inspired by the mean per joint position error (MPJPE) which is often used in 3D pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MPBLPE first aligns estimations and ground truth at the root position and calculates the average Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We directly evaluate the body scale factors by the root mean square error RMSEbody on the scalars predicted by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, to present the results in a more intuitive format, we choose the axis along the longest dimension in each body scale and convert the scale of the axis into millimeters and calculate the mean absolute error (MAEbody).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiment 1-B: 2D-Pose Based Kinematic Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In the previous ex- periment, we evaluate the multi-step baseline with the 3D body pose estimation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, the use of fully trained 2D pose estimation algorithms is com- mon in kinematic estimation works [40, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Therefore, we conduct experiments to compare our method and these 2D-based kinematic estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In con- trast to our CMS method which estimates 3D pose from a single camera estimation, 2D pose estimation methods require at least 2 calibrated cameras for the estima- tion of 3D keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The use of an additional camera to generate the pose could be an advantage, which the CMS method does not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We chose a naive imple- mentation of the OpenPose algorithm [10, 25], which has extensively been used in related work [40, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Additionally, we test the MediaPipe implementation of the blazepose algorithm [6], as a more modern 2D algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MediaPipe is easy to use since it is available as a python library, however, in contrast to OpenPose 12 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS it runs faster, allows for additional smoothing of its predictions, provides more key points, and is labeled on different keypoint labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the OpenPose and MediaPipe, we project the key points to 3D using the BML-Movi camera parameters https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='com/saeed1262/MoVi-Toolbox (ac- cessed on 2 August 2022 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For OpenPose, we connected missing points (due to self-occlusion) using linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For MediaPipe, we chose the highest model complexity (2) and set enabled smooth landmarks for continuous frames of a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For modeling and inverse kinematics, we follow the same steps as for the CMS method only redefining the positions of markers on the OpenSim model to fit the provided key points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We compare against an average across both camera views for CMS and D3KE, as MediaPipe and OpenPose need at least two cameras to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We evaluate performance based on mean absolute error MAEangle(◦), the standard deviation of errors SDangle(◦) and smoothness of the predictions as the mean velocity of the angle MVangle (◦/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The mean velocity error is calculated by the derivative of the landmark position and joint angle data with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' (13) MVangle = n � t=0 |st−st+1| ∆t n with st an individual marker position at time t, ∆t is the amount of time between time steps and n is the total number of timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiment 2: Sequential Network Variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Since we have multiple options for the sequential networks, we evaluate three to determine whether the additional modeling of temporal dependencies in the data improves the accuracy of our method or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For subsequent smoothing and reduction of self-occlusion artifacts of the estimations, we test three different networks including LSTM [23], temporal convolutional networks (TCNs) [49], and a lifting Transformer [31] as the sequential network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As smoothing is known to improve the accuracy of multi- step approaches [40], we also evaluate combinations of our CMS model with these sequential networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the LSTM, we implement a bidirectional architecture with a hidden size of 128, three recurrent layers and a dropout probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For TCNs, we follow [49] to exploit 243 frames as the receptive field and make the momentum of batch normalization decay from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the lifting Transformer, we use a hidden size of 256 and 8 parallel attention heads in the self-attention layer and a channel size of 512 in the convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Each sequential network is trained with a sequence length of 243 frames and a batch size of 128 over 50 epochs with Adam optimizer [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The learning rate exponentially decays from 10−3 to 5 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We use the same metrics for the comparison of individual network variants as we used for the comparison of D3KE and CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In addition, we investigate the smoothness of the predicted sequences, we estimate the mean velocity (MV) on bony landmark positions and joint angles, denoted as MVBL and MVangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiment 3: Processing Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' One important property of our pro- posed method for clinical applications is its processing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As applications for TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 13 camera-based kinematic estimation should form an alternative to visual examina- tions in the future, it should ideally be able to run fast enough to estimate kine- matics from video frames as fast as they are collected by a camera, mostly between 15 and 30 frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We compare the running time on Windows 10 with four core CPU, 52 GB RAM and NVIDIA T4 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We compare the CMS and D3KE method as they both use the same type of convolutional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We choose the lifting Transformer as the sequential architecture in both the CMS and D3KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the CMS method, OpenSim is executed in parallel with four cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our report results in frames per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Experiment 4: Generalization Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The goal of camera-based kinematic estimation is ultimately to create tools for researchers and clinicians to analyze and diagnose human movement, these tools should not discriminate between different subjects and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We analyze whether our method generalizes to different subjects, movements, and joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To assess how well D3KE generalizes, we compare the estimates of the proposed method to the ground truth on the time series of each of the 16 participants in the test set with respect to the performed movement, the joint, the camera view and the individual participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For this test, we use the best performing model from experiment 1 with the lifting transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For all time series of joint angles, mean absolute error (MAE) and Pearson’s correlation coefficient (ρ) were calculated between the estimation from D3KE and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Central tendencies in the data are reported as a median and interquartile range of MAE, RMSE and ρ, as the data are not normally distributed, as assessed through visual inspection and confirmed by the Shapiro-Wilk test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For completion, mean and standard deviation are also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The absolute values of ρ were categorized as weak, moderate, strong and excellent for ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='35 < ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='67, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='67 < ρ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='90 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='90 < ρ, respectively [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Software and Tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' All data analysis was conducted in python 3 [58] using the pandas library to generate descriptive statistics, SciPy library for the calculation of MAE, RMSE and Pingouin library for the calculation of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Direct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Multi-Step Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' A: 3D Pose Based Kinematic Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As shown in Table 1, D3KE has better performance than CMS methods in terms of joint angles and body scales, and these two factors are the key to kinematic estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Significantly, D3KE reduces 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='4% of errors on MAEangle when comparing the CMS method with the Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Although the proposed method has a slightly larger MP- BLPE than the CMS, this metric is not related to the kinematic estimation and is only used as one of the losses during training in the proposed method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', MP- BLBE is not minimized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This indicates a gain in accuracy when directly estimating kinematics from the video instead of the multi-step approach of first estimating pose and then estimating kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 14 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Comparison of bony landmarks position (MPBLPE), body scales (MAEbody) and joint angles (MAEangle) between the estimation of and the ground truth across all participants, move- ments, joints and camera views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For the custom multi-step ap- proach (CMS) as well as our proposed method, we compare convo- lutional networks with different temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' All versions of the proposed method show superior performance for the prediction of body scales and joint angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' All CMSs show supe- rior performance in estimating marker positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Each method group shows better performance for the task it was optimized for, highlighting the importance of direct optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Bold numbers indicate the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MPBLPE (mm) MAEbody (mm) MAEangle (◦) D3KE Convolutional 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='58 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='+ LSTM 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='57 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='+ TCNs 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='54 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='+ Transformer 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='54 CMS Convolutional 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='04 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='89 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='+ LSTM 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='79 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='+ TCNs 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='82 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='+ Transformer 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='66 In Table 2, we list RMSE and MAE for body scales of selected segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The re- sults show that the CMS performs better than D3KE on lower limbs, and D3KE performs better than the CMS on upper limbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The CMS and the proposed method have comparable performance in scale estimation of the pelvis and lower limbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Errors in scaling factors of the proposed method and the baseline compared against the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' D3KE shows better performance for the upper extremities and slightly worse performance for the lower extremities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' RMSEbody (MAEbody (mm)) CMS D3KE pelvis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='090 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='58) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='091 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='82) femur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='073 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='091 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='21) tibia 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='060 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='82) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='102 (35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='00) humerus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='102 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='068 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='41) ulna 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='395 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='91) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='075 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='59) radius 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='395 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='60) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='075 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='98) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' B: 2D Pose Based Kinematic Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The results of the comparison of our proposed method, CMS, OpenPose, and MediaPipe are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 15 find that algorithms trained on noisy labels, that use fewer key points perform worse than ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We see a clear difference between the unsmoothed OpenPose estimations and the smoothed MediaPipe estimations in the mean velocity of the estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our proposed method D3KE still performs better showing that even in an ideal sce- nario (CMS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', no noise in the labels, enough markers, same distribution training data) direct estimation is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Comparison of popular pose estimation algorithms to D3KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As OpenPose and MediaPipe require multiple cameras to create 3D keypoints, we compare against the average of both cam- era views for CMS and D3KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' CMS shows better performance than OpenPose and MediaPipe and D3KE shows the overall best per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Indicating that direct estimation is preferable to (naive) implementations of multi-step methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MAEangle (◦) SDangle (◦) MVangle (◦/s) OpenPose 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='98 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='91 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='15 MediaPipe 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='60 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='80 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='15 CMS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='27 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='74 D3KE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='05 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Sequential Network Variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Table 1 also shows the results of different sequential networks for smoothing of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Although the convolutional model by itself already has good performance in joint angle and scale factor esti- mation, using temporal smoothing can additionally reduce the estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The results of our investigation to reduce the noise in the estimations using temporal smoothing are shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The result shows that all temporal models can improve the smoothness of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The LSTM achieves the best performance on MVBL and MVangle among all temporal models, this is contrary to the results in Table 1, in which using a Transformer as the sequential model yielded the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The mean velocity errors for bony landmarks MVBL and joint angles MVangle, lower values indicate smoother estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Adding a sequential model most probably improves the continuity of estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MVBL (mm/s) MVangle (◦/s) Convolutional model 378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='01 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='7 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' + LSTM 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='23 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='19 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' + TCNs 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='35 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='29 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' + Transformer 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='82 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Processing Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our proposed method achieves 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='96 fps with a batch size of 256, as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Since the skeletal-model layer must traverse body segments in the level order, our proposed method is slower than the CMS for a 16 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS batch size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, the support of mini-batch computation in the skeletal- model layer allows D3KE to run faster than the CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Showing that our method can reach video framerate speeds on a competent GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Comparison of processing speed in FPS of D3KE and the baseline for multiple images or ’batches’ in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' OpenSim does show little change in processing speed for increasing batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The proposed method achieves framerate speeds for batches of 256 images, allowing it to analyze images as fast as a common webcam or mobile phone camera collects them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Bold numbers indicate best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Batch Size 1 16 64 128 256 D3KE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='92 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='78 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='94 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='25 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='96 CMS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='51 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='36 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='43 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='44 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Generalization Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Figure 4 shows that both CMS and D3KE have relatively little variation across different movements and different participants, yet larger variations across individual joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This is also reflected in median MAEs and ρ per joint (Table 6), with median MAEs for joints varying within a range 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='8° for joints, while movements and participants vary under 1°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' It shows that D3KE generalizes well to different participants and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Mean absolute error for predicted joint angles per joint, movement and subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Across these groups, D3KE shows less varia- tion compared to the CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The low variations indicate that D3KE is suitable for use on different participants and movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Mean Absolute Error per Group Action Joint Subject Error 10 Mean Absolute 0 Baseline Baseline Baseline Our Method Our Method Our MethodTOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 17 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Median and inter-quartile ranges (IQR) of joint angles per joint, movement and participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MAE, RMSE and correla- tion are calculated over individual frames, Medians and IQR are reported due to the skewed distribution of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Within each group, both camera views show similar errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Joint angles show the highest error and highest spread of values of all groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' D3KE generalizes well to different movements, participants and camera views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Group Camera View MAE (◦) RMSE (◦) ρ Median IQR Median IQR Median IQR Joint Frontal 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='54 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='16 Sagittal 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='21 Movement Frontal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='11 Sagittal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='11 Participant Frontal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='04 Sagittal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We visualize the estimation of musculoskeletal models from our proposed method with the Transformer architecture in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We also show the comparison between estimation and ground truth of the left knee angle as an example of joint angle estimation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We took the average body scales of the predicted sequence of scaling factors to scale the model and visualize it in OpenSim using the predicted joint angles as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' From the figure, we can see that the proposed method can achieve results that are in agreement with the single-view input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 18 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Qualitative results of D3KE from a ’sitting-down’ movement in the BML-Movi dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The top row shows selected frames throughout the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The middle row shows different poses of the ground truth skeletal model throughout the movement (cyan) and the skeletal model (white) based on D3KE’s estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The bottom row shows the changes in flexion/extension of the left knee throughout the movement with blue being the predicted and orange being the ground truth angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Discussion In summary, we compared a direct approach of estimating joint angles from video images to the more traditional multi-step approach found in most recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The traditional method first estimates key points from a video of a subject, then calculates joint angles using a (musculo)skeletal model through an inverse- kinematics process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We developed a method consisting of a convolutional neural network and a sequential network both including a specialized layer that performs kinematic transforms of a (musculo)skeletal model and allows for direct optimiza- tion of the predicted joint angles (D3KE) and treats the prediction of key points only as an auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We compared our direct estimation approach against naive implementations of often used algorithms in the related literature, as well as a self-implemented custom multi-step approach (CMS) that is trained on the same data as our direct approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We show that direct estimation of kinematics yields higher accuracy in predicted joint angles compared to the traditional multi-step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our results indicate that direct estimation can help the future develop- ment of algorithms for fast and accessible kinematic analysis for researchers and clinicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Direct vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Multi-Step Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 3D-Pose Kinematic Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To compare direct estimation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' multi-step estimation, we compared our D3KE method against a 3D-pose based multi-step approach (CMS) with comparable network architecture and trained on the same Left Knee Angle 0 Angle 50 Ground Truth Proposed Method 100 0 50 100 150 200 250 Frame NumberTOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 19 training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Compared to the CMS, our proposed method improves the accuracy of joint angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For all model combinations, we can see an improvement of about 35% in accuracy for the estimation of joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our results support the feasibility of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' It delivers improvements due to directly optimizing the predicted joint angles and scaling factors while using the pose esti- mates only as an auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The auxiliary task effectively imposes a constraint on the network estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We show that direct optimization is preferable to the multi-step approach when using videos from a single-camera view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We expect that using additional specialized layers, a network might be able to directly optimize for individual muscle forces with comparable accuracy from a monocular video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 2D-Pose Kinematic Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We compared our direct approach (D3KE) and the self-trained multi-step approach (CMS) against two multi-step approaches commonly used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Compared to the more traditional implementa- tions of the OpenPose and MediaPipe algorithms, both our proposed and our CMS method show superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' From Tables 4 and 6, we see that estimations from D3KE are far smoother compared to the traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This is likely due to multiple reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The predicted key points of OpenPose suffer from system- atic errors due to inaccuracies in their training data [13], which can explain the drop in performance, for MediaPipe the accuracy of labels in their training data is not known as their paper only states that their annotators were human [6], not whether they had expertise in labeling anatomical key points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The lack of smoothing for the predicted OpenPose key points can also contribute to its overall worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' MediaPipe, which uses internal smoothing, shows better results in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In general, we expected worse performance from OpenPose and MediaPipe as they only predict 18 or 33 key points respectively, while we supervise a total of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Both traditional methods predict keypoints representing joint centers and not markers on body segments, this makes it hard to distinguish rotations between different body segments during the musculoskeletal modeling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Qualitative comparison of predicted angles of the left knee, from a ’sitting-down’ movement in the BML-Movi dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' While OpenPose shows by far the noisiest estimation, the smooth- ing of the MediaPipe estimation is clear, our proposed method and implemented CMS work best, probably due to the restriction of additional markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Left Knee Flexion/Extension angle during Sitting Down movement 0 OpenPose MediaPipe Angle 50 D3KE CMS Ground Truth 100 0 50 100 150 200 Frame #20 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Network Variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We compared three commonly used sequential networks to improve our estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We show that all sequential networks improve our estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' One possible explanation for this is that the network is better able to handle ‘self-occlusion’ artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The estimation of a 3D-pose from a single camera is an ill-posed problem as a 3D point projected to a 2D image can originate from any position along the ray(s) that fall on the image sensor and form the corresponding pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This can lead to self-occlusion, such as the torso and left arm occluding the right arm during a right arm swing in frames recorded from the left sagittal view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' During self-occlusion, it is difficult for frame-based networks to make a good estimate as they lack temporal information of previous angles of the arm to extrapolate from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Sequential networks on the other hand have access to temporal information, which can allow for more accurate estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Although we only see a slight increase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='001° in MAE of the joint angle estimation, we can see a clear improvement of the sequential models in the smoothness of the predicted angles Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This might be due to the network learning to interpolate motions during occurrences of self-occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Processing Speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Compared to the multi-step baseline, CMS, our D3KE approach shows increased calculation speeds for larger batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Both CMS and D3KE make use of the same ResNeXt50 architecture, which should show ap- proximately the same performance increase with increasing batch sizes for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' D3KE could be expected to be slower, as it also has the additional time cost of calculating the pose from the estimated kinematics in the skeletal model layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, due to its multi-step nature, CMS has to perform an additional inverse kinematics calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This calculation seems to form a bottleneck in the processing speed of the CMS approach restricting it to a framerate of 8 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Other multi-step algorithms will most likely encounter the same problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In the case of OpenPose, which runs at about 4 fps [6], even lower frame rates can be expected for a complete pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This shows the advantage in the processing time of our direct approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For a method to be usable in everyday life, it should be reasonably fast in running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Processing speeds allowing a method to run between 15 and 30 frames per second are favorable, as they show that a method can process a video as fast as its frames are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, our results might not directly translate to every real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To process multiple images simultaneously as batches, D3KE currently requires GPUs that are not available in mobile devices, which prevents it from beingportable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In addition, we use the Faster R-CNN object detection network to crop our images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This step was not included in the processing speed evaluation, as it is highly dependent on the chosen object-detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, with inference speeds of 12fps, the Faster R-CNN object detection would form a bottleneck in applying our method in real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Given the speed of development in the field of object detection, Faster R-CNN can by now be regarded as an old algorithm, and newer and faster object detectors should be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The YoloV7 algorithm [61], which performs object detection at up to 286 fps could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In general, the current architecture is not optimized for speed or a specific technology and we are using off-the-shelf, fairly standard convolutional and sequential architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' For these architectures, smaller and faster alternatives might be found in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' When optimizing for all these points, we predict the proposed method could run on mobile devices within a few TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS 21 years, effectively enabling a 3D kinematic analysis instrument to become available for everyone with a mobile phone or tablet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Generalization Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our method generalizes well on the tested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As shown in Figure 4 and Table 6, the estimation variations across partic- ipants and movements are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We can conclude that our method shows the ability to generalize to different camera views, participants, and performed move- ments, within the tested dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Our results indicate that D3KE could be generally applied to a variety of people and movements, including clinical and sports appli- cations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', physiotherapists and athletes, when trained on sufficient additional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Although we show good generalization performance on the BMLMovi database, it is difficult to estimate how well our method will generalize in a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In machine learning settings, training data is often not representative of the task of the network in the real world [5] and can introduce biases if applied to scenarios that are very different from the one represented in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Unfortunately, there is currently a lack of deep learning datasets for kinematic analysis [52, 40, 13, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In addition, while the BML-Movi database is excellent for training neural networks due to the large number of participants performing movements and the diversity of execution styles, it might be not extensive enough to train a network for biomedical applications in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' However, to evaluate the current method fully, such an extensive dataset would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In general, we expect a drop in accuracy when our method is applied to a scenario different from the BMLMovi database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' As we train on just two calibrated cameras, we expect our method to be most vulnerable to alternative camera positions, that do not show people in either frontal or sagittal view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Future research should investigate the stability of direct estimation methods when applied to data that differs significantly from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Future Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' To improve the accuracy of the algorithm and provide further insight into the strengths and weaknesses of monocular joint angle estimation, a new dataset with dedicated annotation is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' A dataset specifically designed for the estimation of joint angles and/or kinetics could improve the accuracy of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This dataset could be established with a large number of camera views, and top-down views for better estimation of movements in the transverse plane, where participants perform movements that exercise the full ROMs of individual joints including upper extremities, as well as movements that are relevant for health care professionals such as physiotherapy exercises and other clinical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In addi- tion, the inclusion of abnormal movement patterns could give better insights into the clinical relevance of newly developed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Transfer learning could be explored to apply 3DKE in settings where little train- ing data are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Vdeos that are very different from the BMLMovi training data, such as people wearing more clothes, are in different surroundings, or are filmed from a different camera view, will, most probably, yield worse accuracy than shown in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Transfer learning of a pre-trained D3KE on a minimal portion of a dataset could be investigated as an alternative to the time-consuming collection of a novel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' The capabilities of D3KE as an adapter for kinetic analysis of a movement in OpenSim could be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Given data similar to BMLMovi or successful transfer 22 TOWARDS SINGLE CAMERA HUMAN 3D-KINEMATICS learning on relevant data beforehand, our method provides an easy way to skip the tedious steps of scaling and running inverse kinematics on an MSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' This enables the quick generation of MSMs for kinetic analysis from just a single video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Even if this kinematic estimation comes at the cost of reduced accuracy, it could provide coarse insights into collected data, which can later be confirmed through finer analysis with the manually scaled MSMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' D3KE could be made more generally applicable if the underlying model of the Skeletal-model layer would not be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Currently, the underlying model is fixed in the Skeletal-model layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Future iterations could explore combinations of the Pytorch and OpenSim python libraries to allow training a network on a self-defined model or allowing a pre-trained model to be refined through transfer learning for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=', only estimation of joint angles around the shoulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Existing Explainable AI tools should be applied to better understand the inner workings of D3KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Deep neural networks are capable of high accuracy estimation, because of their ability to break down highly complex tasks into simpler tasks [2], but understanding what these simpler tasks are is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Research in Explain- able AI has generated tools and frameworks that allow one to better understand the basis of the final predictions of a network [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Applying these tools could help users and researchers alike to better understand the biases and limitations of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' D3KE can still predict the joint angles even if these joints are occluded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' this means it must make assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' What these assumptions are and how they came to be are important to estimate the trustworthiness of this algorithm in a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Conclusions In this paper, we present a novel end-to-end neural network for the estimation of segment joint angles of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Compared to the previous method, we directly regress to the joint angle and scale for individual segments from the input video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We trained our method from scratch on the BML-Movie database and compared it against a 3D pose estimation method on which we used the inverse kinematics tool of OpenSim to obtain the kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' We conclude that using direct estimation of joint angles is preferable in a single camera setting, as it is more accurate compared to the common approach of fitting an estimated pose to a musculoskeletal model and performing inverse kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' By allowing the network to directly optimize for the joint angles and scaling factors, our method is less prone to errors in the key point labels used to predict key point location for pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In addition, the use of a sequential model is important when designing a neural network architecture for kinematic estimation, as it allows to smooth predictions over time to create better estimates of limb position and joint angles during self-occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' While using deep learning for biomedical solutions is still in its infancy, the pre- sented method shows that training networks from scratch for specialized tasks is a viable way to estimate joint angles from a single camera video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' With further advancements in the underlying algorithms as well computational performance, we predict that the methodology we have presented will assist biomedical and clinic practitioners to measure and monitor human movement in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Funding: This work was supported by the Dutch Research Council (NWO) under the Citius Altius Sanius Perspective Program P16-28 Project 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' REFERENCES 23 Aknowledgements: The authors would like to thank Lisa Noteboom for her feedback as well as Marco Hoozemans and Dirkjan Veeger for guidance and insight during our bi-weekly meetings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Conflict of Interest: The authors declare no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Abbreviations: The following abbreviations are used in this manuscript: CMS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Custom multi-stage approach D3KE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Direct 3D kinematic estimation IQR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Interquartile Range MAE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Mean absolute error MMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Markerless motion capture MPBLPE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Mean per bony landmark error MSM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Musculoskeletal model MVE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Mean velocity error OMC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Optical Motion Capture PCC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Pearson correlation coefficient RMS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Root mean square error ROM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Range of motion SD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Standard Deviation TCN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Temporal Convolutional Network References [1] Mazen Al Borno, Johanna O’Day, Vanessa Ibarra, James Dunne, Ajay Seth, Ayman Habib, Carmichael Ong, Jennifer Hicks, Scott Uhlrich, and Scott Delp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' “OpenSense: An open-source toolbox for inertial-measurement-unit- based measurement of lower extremity kinematics over long durations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' In: Journal of NeuroEngineering and Rehabilitation 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='1 (Feb.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='02696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' url: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='02696 (visited on 12/12/2022).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' sciencedirect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' com / science / article/pii/0021929089901796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' [64] Aston Zhang, Zachary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Lipton, Mu Li, and Alexander J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' Smola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' “Dive into deep learning”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content=' 07035 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} +page_content='LG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E5T4oBgHgl3EQfHA4j/content/2301.05435v1.pdf'} diff --git a/k9FLT4oBgHgl3EQfdS-S/vector_store/index.faiss b/k9FLT4oBgHgl3EQfdS-S/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0b6ebed0b5ba5bd7e818a0431e108ccb4984b54f --- /dev/null +++ b/k9FLT4oBgHgl3EQfdS-S/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:024abe9b193c2bb965664595904ad4a456566126cd77400d657df83c57ff3de9 +size 1769517 diff --git a/kNE2T4oBgHgl3EQfIgYI/content/tmp_files/2301.03680v1.pdf.txt b/kNE2T4oBgHgl3EQfIgYI/content/tmp_files/2301.03680v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b3531d98a26ce806c123d3636d03a96a86c4772 --- /dev/null +++ b/kNE2T4oBgHgl3EQfIgYI/content/tmp_files/2301.03680v1.pdf.txt @@ -0,0 +1,607 @@ +arXiv:2301.03680v1 [math.GR] 9 Jan 2023 +CONGRUENCE SOLVABILITY IN FINITE MOUFANG LOOPS +OF ORDER COPRIME TO THREE +ALEˇS DR´APAL AND PETR VOJTˇECHOVSK´Y +Abstract. We prove that a normal subloop X of a Moufang loop Q induces an abelian +congruence of Q if and only if each inner mapping of Q restricts to an automorphism of X +and u(xy) = (uy)x for all x, y ∈ X and u ∈ Q. The former condition can be omitted when +X is 3-divisible. This characterization is then used to show that classically solvable finite +3-divisible Moufang loops are congruence solvable. +1. Introduction +There are two notions of solvability in loop theory. Classical solvability generalizes the +standard definition of group theory, so a loop Q is classically solvable if there exists a series +Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1 such that Qi+1 ⊴ Qi and Qi/Qi+1 is a commutative group. +Equivalently, a loop Q is classically solvable it there exists a series Q = Q0 ≥ Q1 ≥ · · · ≥ +Qn = 1 such that Qi+1 ⊴ Q and Qi/Qi+1 is a commutative group. Congruence solvability +specializes a definition from congruence modular varieties, so a loop Q is congruence solvable +if there exists a series Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1 such that Qi+1 ⊴ Q and the normal +subloop Qi/Qi+1 of Q/Qi+1 induces an abelian congruence of Q/Qi+1. +Every congruence solvable loop is classically solvable but the converse does not hold in +general. It is an open problem whether every classically solvable Moufang loop is congruence +solvable. We proved in [5] that the two notions of solvability coincide in Moufang loops of +odd order and in 6-divisible Moufang loops. +A classically solvable nontrivial Moufang loop Q contains a nontrivial commutative sub- +group X that is normal in Q. If Q is also congruence solvable, X can be assumed to satisfy +additional properties which are easy to express (cf. Theorem 2.2) and which have structural +implications for Q. Whether the two notions of solvability coincide in Moufang loops or not, +congruence solvability has the potential to significantly influence and deepen the theory of +Moufang loops. +In this paper we characterize normal subloops of 3-divisible Moufang loops that induce an +abelian congruence, and then we prove that the two notions of solvability coincide in finite +3-divisible Moufang loops, improving upon the 6-divisibility result of [5] in the finite case. +2020 Mathematics Subject Classification. 20N05. +Key words and phrases. Congruence solvability, solvability, abelian congruence, abelian extension, Mo- +ufang loop, 3-divisible Moufang loop. +A. Dr´apal supported by the INTER-EXCELLENCE project LTAUSA19070 of MˇSMT Czech Republic. +P. Vojtˇechovsk´y supported by the Simons Foundation Mathematics and Physical Sciences Collaboration +Grant for Mathematicians no. 855097 and by the PROF grant of the University of Denver. +1 + +Here is a summary of the paper. The congruence commutator theory for loops was de- +scribed in [15]. +In the follow-up paper [16], Stanovsk´y and Vojtˇechovsk´y listed several +equivalent conditions that characterize a normal subloop X of Q that induces an abelian +congruence of Q, cf. [16, Theorem 4.1]. +We record the relevant parts of their result as +Theorem 2.1. Building upon Theorem 2.1, we show that in the case of Moufang loops, a +normal subloop X induces an abelian congruence of Q if and only if (i) each inner map- +ping of Q restricts to an automorphism of X, and (ii) u(xy) = (uy)x for all x, y ∈ X and +u ∈ Q, cf. Theorem 2.2. We can further improve upon Theorem 2.2 in the case of 3-divisible +Moufang loops when condition (i) is automatically satisfied, cf. Theorem 2.5. +Given a 3-divisible Moufang loop Q and its normal subloop S, we then construct a cer- +tain subnormal series (3.6) in the multiplication group Mlt(Q) that starts with the relative +multiplication group MltQ(S). One of the intermediate subloops of the series is constructed +upon considering pseudoautomorphisms of Q. +In Section 5 we prove that every classically solvable finite 3-divisible Moufang loop is +congruence solvable, cf. Theorem 5.7. The proof is an induction on the order of a smallest +counterexample Q. As the three Isomorphism Theorems and the Correspondence Theorem +hold in loops, the standard proof from group theory goes through to establish the following +result: If X is a normal subloop of a loop Q, then Q is classically solvable if and only if both +X and Q/X are classically solvable. Similarly, if X is a normal subloop of Q that induces +an abelian congruence of Q, then Q is congruence solvable if and only if both X and Q/X +are congruence solvable. +It therefore suffices to find a nontrivial normal subloop X of Q that induces an abelian +congruence of Q. If the nucleus Nuc(Q) of Q is nontrivial, it contains a nontrivial normal +commutative subgroup X, and every such subgroup induces an abelian congruence of Q by +Proposition 5.5. +If Nuc(Q) is trivial then Mlt(Q) admits certain triality automorphisms by [9, Theorem 6]. +We study Moufang loops that admit triality automorphisms in the short Section 4. Con- +tinuing the inductive proof, we then show that Q contains a nontrivial normal p-subgroup +S. The series (3.6) yields a subnormal series that starts with the p-group MltQ(S) (cf. The- +orem 5.4) and terminates with Mlt(Q) ⋊ S3, the multiplication group of Q extended by +the triality automorphisms. This gives rise to a nontrivial normal triality p-subgroup U of +Mlt(Q) (cf. Proposition 5.6). Finally, X = U(1) is then the sought-after normal subloop +of Q that induces an abelian congruence of Q (cf. Proposition 4.2, whose proof uses the +characterization of Theorem 2.5). +The main results of this paper are Theorems 2.2, 2.5 and 5.7. However, many of the +auxiliary results are likely to be useful in future investigations of congruence solvability in +Moufang loops and in our understanding of the structure of Moufang loops. +2. A characterization of abelian congruences in Moufang loops +See [1, 13] for an introduction to loop theory and [6] for an introduction to commutator +theory in congruence modular varieties. +Let Q = (Q, ·, \, /, 1) be a loop, a universal algebra satisfying the identities x\(x · y) = +x · (x\y) = y = (y · x)/x = (y/x) · x and x · 1 = x = 1 · x for all x, y ∈ Q. As usual, we also +denote the multiplication operation · by juxtaposition and we assume that juxtaposition is +more binding than the operations ·, \ and /. For instance, xy · (u\v) = (x · y) · (u\v). +2 + +A loop Q is an inverse property loop if for every x ∈ Q there is x−1 ∈ Q such that +x−1 · xy = y = yx · x−1. This implies 1 = xx−1 = x−1x, (x−1)−1 = x, x\y = x−1y and +x/y = xy−1. Inverse property loops can therefore we treated as universal algebras in the +signature (·, −1, 1) familiar from groups. +An associative subloop of a loop Q will be refereed to as a subgroup of Q. A loop Q is +power associative if any element of Q generates a subgroup, and diassociative if any two +elements of Q generate a subgroup. We will often not specify unnecessary parentheses in +diassociative loops, e.g., we write xyx instead of x · yx or xy · x. +The loop identities +x(y · xz) = (xy · x)z, (zx · y)x = z(x · yx), x(yz · x) = xy · zx, (x · yz)x = xy · zx +(M) +are pairwise equivalent and a loop satisfying any one (and hence all) of the identities is called +a Moufang loop. Every Moufang loop is diassociative [12]. +A power associative loop Q is d-divisible if the mapping x �→ xd is onto Q. We claim that +a finite Moufang loop Q is 3-divisible if and only if the order of Q is coprime to 3. Certainly +if Q is not 3-divisible then there are x ̸= y such that x3 = y3, so the group ⟨x, y⟩ is not +3-divisible, it contains an element of order 3, and hence 3 divides |Q| by the elementwise +Lagrange theorem for Moufang loops. Conversely, if 3 divides |Q| then Q contains an element +of order 3 by [2, Lemma 4] and hence Q is not 3-divisible. +If u is an element of a loop Q, then the left translation and the right translations by u are +defined by Lu(v) = uv and Ru(v) = vu, respectively. The group Mlt(Q) = ⟨Lu, Ru : u ∈ Q⟩ +is the multiplication group of Q. +An element of Mlt(Q) that fixes 1 is called an inner +mapping. The inner mappings of Q form a subgroup Inn(Q) of Mlt(Q). It is well known +that Inn(Q) = ⟨Tu, Lu,v, Ru,v : u, v ∈ Q⟩, where +Tu = R−1 +u Lu, Lu,v = L−1 +uv LuLv and Ru,v = R−1 +uv RvRu, +(2.1) +are the standard generators of Inn(Q). +Let A be a universal algebra on an underlying set A. A congruence of A is an equivalence +relation on A that commutes with all operations of A. As in [6], a congruence α centralizes +congruence β over congruence γ if for any (n+1)ary term operation t in the signature of A +the following implication holds for all a, b, u1, v1, . . . , un, vn ∈ A satisfying a α b and u1 β v1, +. . . , un β vn: +t(a, u1, . . . , un) δ t(a, v1, . . . , vn) ⇒ t(b, u1, . . . , un) δ t(b, v1, . . . , vn) +(2.2) +A congruence α is abelian if the condition (2.2) holds with β = α and δ = {(a, a) : a ∈ A}. +There is a one-to-one correspondence between normal subloops of Q and congruences of +Q. For X ⊴ Q, the induced congruence modX is defined by u modX v iff uX = vX. For a +congruence α of Q, the induced normal subloop is the equivalence class of α containing 1. +Let X ⊴ Q. If modX is an abelian congruence of Q then X is a commutative group, but +the converse is not true in general loops, not even in Moufang loops. +The following result is excerpted from [16, Theorem 4.1]. It uses the somewhat unusual +commutators and associators employed in [16], namely, [x, y] = ((xy)/x)/y and [x, y, z] = +((xy · z)/(yz))/x. Note that we have [x, y] = 1 iff xy = yx and [x, y, z] = 1 iff xy · z = x · yz. +Theorem 2.1 ([16]). Let X be a normal subloop of a loop Q. Then the following conditions +are equivalent: +(i) X induces an abelian congruence of Q, +3 + +(ii) every inner mapping of Q restricts to an automorphism of X and for every x, y ∈ X +and u, v, w ∈ Q with v modX w we have [x, y] = [x, y, u] = [x, u, y] = [u, x, y] = 1 and +[x, u, v] = [x, u, w], +(iii) Q is an abelian extension of X by Q/X, that is, X is a commutative group, there is +a transversal T to X in Q containing 1, for every r, s ∈ T there are automorphisms +ϕr,s, ψr,s ∈ Aut(X) and θr,s ∈ X such that ϕr,1 = ψ1,s = idX, θ1,s = θr,1 = 1, and +rx · sy = t · ϕr,s(x)ψr,s(y)θr,s, where t is the unique element of T ∩ (rs)X. +We can substantially simplify condition (ii) in the case of Moufang loops: +Theorem 2.2. Let X be a normal subloop of a Moufang loop Q. Then X induces an abelian +congruence of Q if and only if every inner mapping of Q restricts to an automorphism of X +and u · xy = uy · x for all u ∈ Q and x, y ∈ X. +Proof. The direct implication follows from Theorem 2.1(ii) as follows. Certainly every inner +mapping of Q restrict to an automorphism of X. Since [x, y] = 1 for all x, y ∈ X, the normal +subloop X is commutative. Using commutativity and [u, x, y] = 1 for u ∈ Q, x, y ∈ X, we +deduce u · yx = u · xy = ux · y. +For the converse implication, suppose that every inner mapping of Q restricts to an auto- +morphism of X and u · xy = uy · x for all u ∈ Q and x, y ∈ X. The latter condition implies +that X is a commutative group. We will show that Q is an abelian extension of X by Q/X, +as described in item (iii) of Theorem 2.1. +Let T be a transversal to X in Q with 1 ∈ T. Let r, s ∈ T and x, y ∈ X. In the following +calculation we will use without reference the fact that every inner mapping of Q restricts +to an automorphism of X. We will also write [Q, X, X] = 1 as a justification comment +whenever we use u · xy = ux · y with u ∈ Q and x, y ∈ X. Finally, we omit parentheses in +subterms involving only factors from the commutative group X. Now: +rx · sy = rx · Ts(y)s = s(s−1r · Ls−1,r(x)) · Ts(y)s += s · (s−1r · Ls−1,r(x))Ts(y) · s +by (M) += s · (s−1r · Ls−1,r(x)Ts(y)) · s +[Q, X, X] = 1 += s(s−1r) · Ls−1,r(x)Ts(y)s = r · Ls−1,r(x)Ts(y)s +by (M) += r · sT −1 +s (Ls−1,r(x)Ts(y)) = rs · Lr,sT −1 +s (Ls−1,r(x)Ts(y)) += tz · Lr,sT −1 +s (Ls−1,r(x)Ts(y)) +t ∈ T ∩ (rs)X, z ∈ X += t · zLr,sT −1 +s (Ls−1,r(x)Ts(y)) +[Q, X, X] = 1 += t · Lr,sT −1 +s (Ls−1,r(x)Ts(y))z +X is commutative += t · (Lr,sT −1 +s Ls−1,r(x) · Lr,sT −1 +s Ts(y) · z). +Hence rx · sy = t · ϕr,s(x)ψr,s(y)θr,s, where ϕr,s is the restriction of Lr,sT −1 +s Ls−1,r to X, +ψr,s is the restriction of Lr,sT −1 +s Ts = Lr,s to X and θr,s = z ∈ X. If r = 1 then t is the +unique element of T ∩ sX, so t = s and θ1,s = z = 1. Similarly, θr,1 = 1. We also have +ϕr,1 = Lr,1T −1 +1 L1,r = idX and ψ1,s = L1,s = idX. +□ +We proceed to improve upon Theorem 2.2 in the case of 3-divisible Moufang loops. We +start by recalling two results of Gagola. +4 + +Proposition 2.3 (dual of [7, Theorem 1]). Let Q be a Moufang loop. Then +u3ix · u3jy = u3(i+j)T −i−2j +u +(T i−j +u +(x)T i−j +u +(y)) +for all u, x, y ∈ Q and all i, j ∈ Z. +We will only need Proposition 2.3 for the case i = 1 and j = 0. See [5] for a quick proof +of that case. +Proposition 2.4 ([8, Theorem 1]). Let Q be a Moufang loop generated by a set of elements, +each of which is a cube of an element of Q. Then Inn(Q) = ⟨Tu : u ∈ Q⟩. +Theorem 2.5. Let X be a normal subloop of a 3-divisible Moufang loop Q. Then X induces +an abelian congruence of Q if and only if u · xy = uy · x for all u ∈ Q and x, y ∈ X. +Proof. The direct implication follows from Theorem 2.2. For the converse implication, sup- +pose that u · xy = uy · x for all u ∈ Q and x, y ∈ X so that X is a commutative group. +Since Q is 3-divisible, every element of Q is a cube of some element of Q and Propo- +sition 2.4 therefore applies. +In view of Proposition 2.4 and Theorem 2.2, it suffices to +show that Tu restricts to an automorphism of X for every u ∈ Q. Let x, y ∈ X. With +i = 1 and j = 0, the formula of Proposition 2.3 becomes u3x · y = u3T −1 +u (Tu(x)Tu(y)). +Since u3x · y = u3 · yx = u3 · xy is assumed, we have u3 · xy = u3T −1 +u (Tu(x)Tu(y)), +xy = T −1 +u (Tu(x)Tu(y)) and Tu(xy) = Tu(x)Tu(y). +□ +3. A subnormal series in the multiplication group +If S ≤ Q and Q is a loop then MltQ(S) = ⟨Ls, Rs : s ∈ S⟩ ≤ Mlt(Q) is the relative +multiplication group of S in Q. +If S ⊴ Q then Mlt(Q/S) is an image of Mlt(Q) under the homomorphism determined by +Lx �→ LxS and Rx �→ RxS. Let C(Q, S) be the kernel of this homomorphism. It is not hard +to show that C(Q, S) consists of all ϕ ∈ Mlt(Q) that centralize the cosets of S in Q, that is, +ϕ(uS) = uS for all u ∈ Q. +Lemma 3.1. Let S be a normal subloop of Q. Then +C(Q, S) = {Lsσ, Rsσ : s ∈ S, σ ∈ Inn(Q) ∩ C(Q, S)}. +Proof. Given ϕ ∈ Mlt(Q), there are uniquely determined v ∈ Q and σ ∈ Inn(Q) such that +ϕ = Lvσ. If σ ∈ Inn(Q) ∩ C(Q, S) and s ∈ S then Lsσ(uS) = Ls(uS) = Ls(Su) = s(Su) = +(sS)u = Su = uS for all u ∈ Q, and therefore σ ∈ Inn(Q) ∩ C(Q, S) and Lsσ ∈ C(Q, S). +Conversely, let ϕ = Lvσ ∈ C(Q, S). Since S = ϕ(S) = Lvσ(S) = Lv(S) = vS, it follows that +v ∈ S. Moreover, σ(uS) = L−1 +v Lvσ(uS) = L−1 +v (uS) = uS since v ∈ S. The argument for +Rsσ is similar. +□ +In particular, MltQ(S) = ⟨Ls, Rs : s ∈ S⟩ ≤ C(Q, S) ⊴ Mlt(Q). In this section we will +refine this series into a subnormal series in the case of 3-divisible Moufang loops. +For a loop Q, define the nucleus Nuc(Q) and the left nucleus Nucℓ(Q) by +Nuc(Q) = {a ∈ Q : a(uv) = (au)v, u(av) = (ua)v, u(va) = (uv)a for all u, v ∈ Q}, +Nucℓ(Q) = {a ∈ Q : a(uv) = (au)v for all u, v ∈ Q}. +Bruck proved in [1] that the nuclei are subgroups of Q, and if Q is Moufang then Nuc(Q) = +Nucℓ(Q) is a normal subloop of Q. +5 + +A (left) pseudoautomorphism ϕ of Q is a permutation of Q for which there exists some +c ∈ Q such that +cϕ(x) · ϕ(y) = cϕ(xy) +for all x, y ∈ Q. The element c is called a (left) companion of ϕ. Then d ∈ Q is another +companion of ϕ if and only if d/c ∈ Nucℓ(Q). +Denote by Psaℓ(Q) the set of all pairs (c, ϕ) such that c is a left companion of a pseudoau- +tomorphism ϕ of Q. Then Psaℓ(Q) is a group with operations +(c, ϕ)(d, ψ) = (cϕ(d), ϕψ) and (c, ϕ)−1 = (ϕ−1(c−1), ϕ−1). +(3.1) +Pseudoautomorphisms may be interpreted by means of autotopisms. A triple (α, β, γ) of +permutations of Q is an autotopism of Q if +α(x)β(y) = γ(xy) +(3.2) +for all x, y ∈ Q. Clearly, +(c, ϕ) ∈ Psaℓ(Q) +⇔ +(Lcϕ, ϕ, Lcϕ) ∈ Atp(Q). +(3.3) +The set of all autotopisms of Q forms the autotopism group Atp(Q) under componentwise +composition. If (α, β, γ) ∈ Atp(Q) and β(1) = 1, then α = γ and (α(1), β) = (γ(1), β) ∈ +Psaℓ(Q). This can be easily seen by setting first y = 1 and then x = 1 in (3.2). +Let us now summarize basic facts pertaining to autotopisms and pseudoautomorphisms +in Moufang loops, cf. [1]. +Let Q be a Moufang loop. Every inner mapping of Q is a pseudoautomorphism. This can +be seen by verifying the well-known identities +(x−3, Tx) ∈ Psaℓ(Q), ([x−1, y], [Lx, Ry]) ∈ Psaℓ(Q), +Lx,y = [Lx, R−1 +y ] = [R−1 +x , Ly] and Rx,y = [L−1 +y , Rx] = [Ry, L−1 +x ]. +(3.4) +Let Mx = LxRx. Since Q is Moufang, Mx is also equal to RxLx. If x ∈ Q then Atp(Q) +contains each of the triples +(Lx, Rx, Mx), (Mx, L−1 +x , Lx) and (R−1 +x , Mx, Rx). +(3.5) +A semiautomorphism of a Moufang loop Q is a bijection ϕ of Q satisfying ϕ(1) = 1 and +ϕ(xyx) = ϕ(x)ϕ(y)ϕ(x) for all x, y ∈ Q. Every semiautomorphism satisfies ϕ(xi) = ϕ(x)i for +all x ∈ Q and i ∈ Z. Every pseudoautomorphism of a Moufang loop is a semiautomorphism. +In particular, ϕ(x−1) = x−1 if ϕ ∈ Inn(Q). +Lemma 3.2. Let Q be a Moufang loop. Suppose that x ∈ Q and (c, ϕ) ∈ Psaℓ(Q). Then +(cϕ(x−1), L−1 +ϕ(x)ϕLx) ∈ Psaℓ(Q), +where cy = y−1cy. +Proof. Let ψ = L−1 +ϕ(x)ϕLx and note that ψ(1) = 1. By (3.3) and (3.5) the composition +(Mϕ(x), L−1 +ϕ(x), Lϕ(x))(Lcϕ, ϕ, Lcϕ)(M−1 +x , Lx, L−1 +x ) +is an autotopisms of Q. Hence a companion of ψ = L−1 +ϕ(x)ϕLx is equal to Lϕ(x)LcϕL−1 +x (1) = +ϕ(x)cϕ(x−1) = ϕ(x−1)−1cϕ(x−1). +□ +The following result will be useful in the inductive proof of Theorem 5.7. +6 + +Lemma 3.3. Let A be a finite commutative normal subgroup of a Moufang loop Q. Let p be +a prime dividing |A|. Then there is a nontrivial normal p-subgroup of A that is normal in +Q. +Proof. Let S be the p-primary component of the commutative group A. Let ϕ be an inner +mapping of Q. Since A ⊴ Q, ϕ(A) = A. As ϕ is a semiautomorphism of Q, it restricts to a +semiautomorphism of A. Recall that ϕ(xi) = ϕ(x)i for all x ∈ Q and i ∈ Z. In particular, if +x ∈ S then |ϕ(x)| divides |x| and therefore ϕ(x) ∈ S. +□ +Let S be a normal subloop of a Moufang loop Q. Recall the normal subgroup C(Q, S) +of Mlt(Q) and define C0(Q, S) as the set of all ϕ ∈ C(Q, S) such that when ϕ is written as +ϕ = Lsσ with s ∈ S and σ ∈ Inn(Q) (cf. Lemma 3.1) then the pseudoautomorphism σ has +a companion in S. +Proposition 3.4. Let S be a normal subloop of a Moufang loop Q. Then C0(Q, S) is a +normal subgroup of C(Q, S). +Proof. Let ϕ = Lsσ ∈ C(Q, S) with s ∈ S and σ ∈ Inn(Q), and let c ∈ Q be a companion of +σ. We show that the mapping +f : C(Q, S) → Q/(Nuc(Q)S), +Lsσ �→ c(Nuc(Q)S) +is a well-defined homomorphism with kernel C0(Q, S). +Since since both S and Nuc(Q) are normal in Q, they generate the normal subloop +Nuc(Q)S ⊴ Q. +If c and d are companions of σ then d ∈ c Nuc(Q) ⊆ c Nuc(Q)S, so +f is well-defined. +For the homomorphic property, consider Lsσ, Ltτ ∈ C(Q, S) (with +s, t ∈ S and σ, τ ∈ Inn(Q)) such that (c, σ), (d, τ) ∈ Psaℓ(Q) for some c, d ∈ Q. +We +have f(Lsσ)f(Ltτ) = c Nuc(Q)S · d Nuc(Q)S = (cd) Nuc(Q)S. Let +ψ = L−1 +sσ(t)LsσLtτ = L−1 +sσ(t)LsLσ(t)L−1 +σ(t)σLtτ +and observe that ψ(1) = 1, so ψ ∈ Inn(Q). +To compute a companion of ψ, note that +[s−1, σ(t−1)] is a companion of L−1 +sσ(t)LsLσ(t) = Ls,σ(t) = [Ls, R−1 +σ(t)], by (3.4), and that cσ(t−1) +is a companion of L−1 +σ(t)σLt, by Lemma 3.2. Using (3.1), L−1 +σ(t)σLtτ has a companion cσ(t−1) · +L−1 +σ(t)σLt(d) = cσ(t−1) · σ(t−1)σ(td). Using (3.1) again shows that ψ possesses a companion +e = [s−1, σ(t−1)] [Ls, R−1 +σ(t)] +� +cσ(t−1) · σ(t−1)σ(td) +� +. +By Lemma 3.1, each of σ, τ, Lsσ(t), Lσ(t) and Ls belong to C(Q, S). Hence ψ ∈ Inn(Q) ∩ +C(Q, S). Since sσ(t) ∈ S, we see that LsσLtτ decomposes as Lsσ(t)ψ and f(ψ) = e Nuc(Q)S. +For the homomorphic property, it remains to show that e ≡ cd modulo Nuc(Q)S. Since σ +centralizes cosets of S and we work modulo Nuc(Q)S ≥ S, e is equivalent to +[s−1, t−1][Ls, R−1 +t ](c(t−1) · t−1(td)). +Since s, t ∈ S, we have further e ≡ [Ls, R−1 +t ](cd). The left translation Ls is identical modulo +S, and e ≡ cd follows. +The kernel of f consist of all Lsσ ∈ C(Q, S) such that σ has a companion in Nuc(Q)S. +Since the companions are determined up to Nuc(Q), the kernel coincides with C0(Q, S). +□ +Proposition 3.5. Let S be a normal subloop of a Moufang loop Q. Is S is 3-divisible then +MltQ(S) ⊴ C0(Q, S). +7 + +Proof. Let t ∈ S and Lsσ ∈ C0(Q, S), where s ∈ S and σ ∈ Inn(Q) ∩ C(Q, S) has a +companion c ∈ S. We need to show that LLsσ +t +, RLsσ +t +∈ MltQ(S). Since LLsσ +t += (L−1 +s LtLs)σ = +(L−1 +s )σLσ +t Lσ +s, (L−1 +s )σ = (Lσ +s )−1 and similarly for RLsσ +t +, we only need to show that Lσ +s , Rσ +s ∈ +MltQ(S). We will prove Lσ +s ∈ MltQ(S), the other case following dually. +Since S is 3-divisible, there is d ∈ S such that d3 = c. Note that Td(c) = c. By (3.1), +Psaℓ(Q) contains (c−1, Td)(c, σ) = (c−1Td(c), Tdσ) = (1, Tdσ), which means that α = Tdσ +is an automorphism of Q. Recall that for all x ∈ Q we have Lα +x = α−1Lxα = Lα−1(x), +Rα +x = Rα−1(x) and thus T α +x = Tα−1(x). Also, T −1 +x += Tx−1 in a Moufang loop. Therefore +Lσ +s = (L +T −1 +d +s +)α = (TdLsTd−1)α = Tα−1(d)Lα−1(s)Tα−1(d−1) +is in MltQ(S), since α(S) = Tdσ(S) = Td(S) = S. +□ +Corollary 3.6. Let S be a normal 3-divisible subloop of a Moufang loop Q. Then +MltQ(S) ⊴ C0(Q, S) ⊴ C(Q, S) ⊴ Mlt(Q). +(3.6) +4. Moufang loops admitting triality automorphisms +Glauberman observed in [9, Theorem 6] that if Q is a Moufang loop with trivial nucleus, +then the mappings +σ : Lx �→ R−1 +x , Rx �→ L−1 +x +ρ : Lx �→ Rx, Rx �→ M−1 +x += L−1 +x R−1 +x +(and Mx �→ L−1 +x ) +extend uniquely to automorphisms of Mlt(Q). See [2, 14] and [11, Chapter 13] for more +information on groups with triality and Moufang loops. +We say that a Moufang loop Q admits triality automorphisms if the above maps σ and ρ +extend into automorphisms of Mlt(Q). (There exist Moufang loops with nontrivial nucleus +that admit triality automorphisms.) +Suppose that a Moufang loop Q admits triality automorphisms. Since the subgroup ⟨σ, ρ⟩ +satisfies the standard presenting relations of the symmetric group S3, it is isomorphic to a +homomorphic image of S3. Hence Q induces a semidirect product Mlt(Q) ⋊ S3 (with an +action of S3 that might not be faithful). Let α = σρ and β = σρ2, so that α(Lx) = L−1 +x , +α(Rx) = Mx, β(Lx) = Mx, β(Rx) = R−1 +x +and σ = αβα. Note that σ centralizes Inn(Q). +Indeed, σ(Tx) = σ(R−1 +x Lx) = LxR−1 +x += Tx, σ(Lx,y) = σ([Lx, R−1 +y ]) = [R−1 +x , Ly] = Lx,y and +σ(Rx,y) = σ([L−1 +y , Rx]) = [Ry, L−1 +x ] = Rx,y, where we have used (3.4). +Let Q be a Moufang loop that admits triality automorphisms. A subgroup U ≤ Mlt(Q) is +a triality subgroup if U is invariant under the triality automorphisms σ and ρ. If U ⊴Mlt(Q) +then U is a triality subgroup if and only if U ⊴ Mlt(Q) ⋊ S3 under the induced action of S3. +Lemma 4.1. Suppose that Q is a Moufang loop that admits triality automorphisms and that +U ⊴ Mlt(Q) is a triality subgroup. Let S = U(1) be the orbit of U that contains the element +1. Then S is a normal subloop of Q and MltQ(S) ≤ U. +Proof. Since U is normal, the blocks conjugate to S in Mlt(Q) form equivalence classes of a +congruence of Q. This implies that S is normal in Q. For each s ∈ S there is ϕ ∈ Inn(Q) +such that Lsϕ ∈ U. Then Ms = LsRs = Lsϕ(R−1 +s ϕ)−1 = Lsϕσ(Lsϕ)−1 ∈ U. Hence also +Rs = α(Ms) ∈ U and Ls = β(Ms) ∈ U. +□ +8 + +Proposition 4.2. Let Q be a 3-divisible Moufang loop that admits triality automorphisms. +Let U ⊴Mlt(Q) be a nontrivial commutative triality subgroup. Then Q possesses a nontrivial +subloop X ⊴ Q that induces an abelian congruence of Q. +Proof. By Lemma 4.1, X = U(1) is a normal subloop of Q such Lx, Rx ∈ U for all x ∈ X. +Since U is commutative, [Lx, Ly] = idQ = [Lx, Ry] for all x, y ∈ Q. The first condition implies +that X is commutative. The second condition says that x · uy = xu · y for all x, y ∈ X, +u ∈ Q. Then, by Moufang Theorem [12, 3], u · xy = ux · y for all x, y ∈ X, u ∈ Q. Hence +u · yx = u · xy = ux · y for all x, y ∈ X, u ∈ Q. By Theorem 2.5, X induces an abelian +congruence of Q. +□ +5. Solvability in finite Moufang loops of order coprime to three +In this section we show that the two concepts of solvability coincide for finite 3-divisible +Moufang loops. We start with a well known fact of elementary group theory: +Lemma 5.1. Let H be a subnormal subgroup of a finite group G. If there exists a nontrivial +p-group N ⊴ H, p a prime, then G contains a nontrivial normal p-subgroup. +Proof. Without loss of generality it may be assumed that N is a minimal normal p-subgroup +of H. Thus there exists U ≤ H such that N ≤ U and U is a minimal characteristic subgroup +of H. Since N ⊴ U, the group U has to be a p-group. +If H ⊴ G then U ⊴ G since U is characteristic in H. This proves the case k = 1 of the +general case N ⊴ H = Hk ⊴ · · · ⊴ H1 ⊴ H0 = G. Suppose that k > 1 and proceed by +induction. By the induction assumption there exists a nontrivial normal p-subgroup of H1. +We are done by the case k = 1. +□ +Next, let us recall a result of Glauberman and Wright on Moufang loops of prime power +order. The odd case was established in [9] and the even case in [10]. A new proof that covers +both cases can be found in [4]. +Theorem 5.2 ([9, 10]). Let Q be a Moufang loop of prime power order pk. Then Q is +centrally nilpotent and Mlt(Q) is a p-group. +We will not use Theorem 5.3 below but we offer it as a motivation for Theorem 5.4, which +is taken from [4]. We have included a proof of Theorem 5.4 for the convenience of the reader. +For a set of primes π, a power associative loop Q is a π-loop if for every x ∈ Q the order +of x is a power of a prime from π. +Theorem 5.3 ([9, Theorem 3]). Let Q be a Moufang loop of odd order, π a set of primes +and S a classically solvable π-subloop of Q. Then MltQ(S) is a solvable π-group. +Theorem 5.4 ([4]). Let Q be a finite Moufang loop, p a prime and S a p-subloop of Q. +Then MltQ(S) is a p-group. +Proof. Restricting ϕ ∈ MltQ(S) to Mlt(S) yields an epimorphism with kernel FixQ(S) = +{ϕ ∈ MltQ(S) : ϕ(s) = s for all s ∈ S}. By Theorem 5.2, Mlt(S) is a p-group and therefore +MltQ(S)/ FixQ(S) is a p-group. +Consider the group InnQ(S) = MltQ(S) ∩ Inn(Q) = ⟨Ts, Ls,t, Rs,t : s, t ∈ S⟩. For each +ϕ ∈ InnQ(S), let C(ϕ) be the set of all companions of ϕ, a coset of N = Nuc(Q). Since the +9 + +standard generators of InnQ(S) are pseudoautomorphisms with companions in S and every +ϕ ∈ InnQ(S) satisfies ϕ(S) = S, it follows from (3.1) that C(ϕ) ∩ S ̸= ∅. +For ϕ, ψ ∈ FixQ(S), we therefore certainly have c, d ∈ S such that (c, ϕ), (d, ψ) ∈ Psaℓ(Q). +Since ϕ(d) = d, we also have (cd, ϕψ) = (c, ϕ)(d, ψ) ∈ Psaℓ(Q) by (3.1), proving that +cd ∈ C(ϕψ). The element cd also belongs to C(ϕ)C(ψ) = cNdN = cdN. Hence ϕ �→ C(ϕ) +is a homomorphism from the group FixQ(S) to the loop SN/N ∼= S/(S ∩ N). Let A be the +kernel of this homomorphism. Being associative, the image of the homomorphism is equal +to a subgroup of S/(S ∩ N), some p-group. Hence FixQ(S)/A is a p-group. +The kernel A consists of all automorphisms of Q that are contained in FixQ(S). Now, +αLsα−1 = Lα(s) = Ls for every s ∈ S and α ∈ A. Similarly for Rs. Hence A ≤ Z(MltQ(S)) +is a commutative group. Write A as B × D, where B is the p-primary component of A. +Both B and D are normal subgroups of MltQ(S), being central. All three MltQ(S)/ FixQ(S), +FixQ(S)/A and A/D are p-groups. Hence MltQ(S)/D is a p-group. The subgroup D is +commutative, normal and of order coprime to MltQ(S)/D. Hence it possesses a complement +in MltQ(S), say P, a Sylow p-subgroup of MltQ(S). Since D ≤ Z(MltQ(S)), P is a normal +subgroup of MltQ(S) and hence the unique Sylow p-subgroup of MltQ(S). +For s ∈ S, +we have |Ls| = |Rs| = |s| by diassociativity. +Since S is a p-loop and the elementwise +Lagrange theorem holds in Moufang loops, it follows that both Ls and Rs belong to P. +Hence P = MltQ(S) = ⟨Ls, Rs : s ∈ S⟩. +□ +Proposition 5.5. Let X be a commutative normal subloop of a Moufang loop Q. If X ≤ +Nuc(Q) then X induces an abelian congruence of Q. +Proof. By [1], Nuc(Q) ⊴ Q. Hence if a, b ∈ Nuc(Q) and x ∈ Q then +Tx(ab) = (x · ab)x−1 = (xa · b)x−1 = (Tx(a)x · b)x−1 = (Tx(a) · xb)x−1 = Tx(a)Tx(b), +(5.1) +where we used Tx(a) ∈ Nuc(Q) in the last step. The remaining standard generators of Inn(Q) +act trivially on Nuc(Q). By Theorem 2.2, X induces an abelian congruence of Q. +□ +Proposition 5.6. Let Q be a finite 3-divisible Moufang loop with a nontrivial normal p- +subloop S, p ̸= 3 a prime. Then Mlt(Q) contains a nontrivial normal elementary abelian +p-subgroup. Furthermore, if Q also admits triality automorphisms, then Mlt(Q) contains a +nontrivial normal elementary abelian triality p-subgroup. +Proof. It suffices to prove that Mlt(Q) contains a nontrivial normal p-subgroup since the +center of such a p-group is characteristic, and the socle of an abelian p-group is characteristic +too. By Lemma 5.1, it even suffices to show the existence of a subnormal p-group. +Corollary 3.6 implies the existence of the subnormal series (3.6). By Theorem 5.4, MltQ(S) +is a p-group. +If Q admits triality automorphisms, the subnormal series (3.6) may be extended by +Mlt(Q) ⊴ Mlt(Q) ⋊ S3. +□ +Theorem 5.7. Let Q be a finite 3-divisible Moufang loop. Then Q is classically solvable if +and only if it is congruence solvable. +Proof. The converse implication holds in general. For the direct implication, let Q be a +smallest 3-divisible Moufang loop that is classically solvable but not congruence solvable. +If there is a normal subloop X of Q that induces an abelian congruence of Q, then Q/X +is 3-divisible and classically solvable, hence congruence solvable by minimality of Q, but +10 + +then Q is congruence solvable, being an abelian extension of a commutative group X by a +congruence solvable loop Q/X, a contradiction. +Suppose that 1 < N = Nuc(Q) ⊴ Q. +Since Q is classically solvable, the group N is +solvable. It therefore contains a nontrivial characteristic commutative subgroup X ⊴ N. By +(5.1), Inn(Q) acts upon X as a subgroup of Aut(N). Hence ϕ(X) = X for each ϕ ∈ Inn(Q). +But this means that X is a normal subloop of Q. Then X induces an abelian congruence of +Q by Proposition 5.5. +Now suppose that Nuc(Q) = 1. Then Q admits triality automorphisms, cf. Section 4. +Since Q is classically solvable, it has a series of normal subloops of Q with factors being +commutative groups. In particular, Q contains a nontrivial normal commutative subgroup +A, the first nontrivial term of the series. +Let p be any prime dividing |A|, necessarily +p ̸= 3. By Lemma 3.3, Q contains a nontrivial normal p-subgroup S (contained in A). By +Proposition 5.6, Mlt(Q) contains a nontrivial normal commutative triality subgroup. By +Proposition 4.2, Q contains a nontrivial normal subloop that induces an abelian congruence +of Q. +□ +References +[1] R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1971. +[2] S. Doro, Simple Moufang loops, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 377–392. +[3] A. Dr´apal, A simplified proof of Moufang’s theorem, Proc. Amer. Math. Soc. 139 (2011), no. 1, 93–98. +[4] A. Dr´apal, A short proof for the central nilpotency of Moufang loops of prime power order, submitted. +[5] A. Dr´apal and P. Vojtˇechovsk´y, Abelian congruences and solvability in Moufang loops, submitted. +[6] R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, London Mathematical +Society Lecture Note Series 125, Cambridge University Press, Cambridge, 1987. +[7] S.M. Gagola, III, Cyclic extensions of Moufang loops induced by semi-automorphisms, J. Algebra Appl. +13 (2014), no. 4, 1350128, 7 pp. +[8] S.M. Gagola, III, When are inner mapping groups generated by conjugation maps?, Arch. Math. (Basel) +101 (2013), 207–212. +[9] G. Glauberman, On loops of odd order. II., J. Algebra 8 (1968), 393–414. +[10] G. Glauberman and C.R.B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968), 415–417. +[11] J.I. Hall, Moufang loops and groups with triality are essentially the same thing, Mem. Amer. Math. Soc. +260 (2019), no. 1252. +[12] R. Moufang, Zur Struktur von Alternativk¨orpern, Math. Ann. 110 (1935), no. 1, 416–430. +[13] H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann, Berlin (1990). +[14] J.D. Phillips, Moufang loop multiplication groups with triality, Rocky Mountain J. Math. 29 (1999), +no. 4, 1483–1490. +[15] D. Stanovsk´y and P. Vojtˇechovsk´y, Commutator theory for loops, J. Algebra 399 (2014), 290–322. +[16] D. Stanovsk´y and P. Vojtˇechovsk´y, Abelian extensions and solvable loops, Results Math. 66 (2014), +367–384. +(Dr´apal) Dept. of Mathematics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech +Republic +Email address, Dr´apal: drapal@karlin.mff.cuni.cz +(Vojtˇechovsk´y) Dept. of Mathematics, University of Denver, 2390 S. York St., Denver, CO +80208, USA +Email address, Vojtˇechovsk´y: petr@math.du.edu +11 + diff --git a/kNE2T4oBgHgl3EQfIgYI/content/tmp_files/load_file.txt b/kNE2T4oBgHgl3EQfIgYI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..354438ec6238859a4b520069b9d686f0ce9b8a4d --- /dev/null +++ b/kNE2T4oBgHgl3EQfIgYI/content/tmp_files/load_file.txt @@ -0,0 +1,628 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf,len=627 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='03680v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='GR] 9 Jan 2023 CONGRUENCE SOLVABILITY IN FINITE MOUFANG LOOPS OF ORDER COPRIME TO THREE ALEˇS DR´APAL AND PETR VOJTˇECHOVSK´Y Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We prove that a normal subloop X of a Moufang loop Q induces an abelian congruence of Q if and only if each inner mapping of Q restricts to an automorphism of X and u(xy) = (uy)x for all x, y ∈ X and u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The former condition can be omitted when X is 3-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Introduction There are two notions of solvability in loop theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Classical solvability generalizes the standard definition of group theory, so a loop Q is classically solvable if there exists a series Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1 such that Qi+1 ⊴ Qi and Qi/Qi+1 is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Equivalently, a loop Q is classically solvable it there exists a series Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1 such that Qi+1 ⊴ Q and Qi/Qi+1 is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Congruence solvability specializes a definition from congruence modular varieties, so a loop Q is congruence solvable if there exists a series Q = Q0 ≥ Q1 ≥ · · · ≥ Qn = 1 such that Qi+1 ⊴ Q and the normal subloop Qi/Qi+1 of Q/Qi+1 induces an abelian congruence of Q/Qi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Every congruence solvable loop is classically solvable but the converse does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It is an open problem whether every classically solvable Moufang loop is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We proved in [5] that the two notions of solvability coincide in Moufang loops of odd order and in 6-divisible Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A classically solvable nontrivial Moufang loop Q contains a nontrivial commutative sub- group X that is normal in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If Q is also congruence solvable, X can be assumed to satisfy additional properties which are easy to express (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2) and which have structural implications for Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Whether the two notions of solvability coincide in Moufang loops or not, congruence solvability has the potential to significantly influence and deepen the theory of Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In this paper we characterize normal subloops of 3-divisible Moufang loops that induce an abelian congruence, and then we prove that the two notions of solvability coincide in finite 3-divisible Moufang loops, improving upon the 6-divisibility result of [5] in the finite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 20N05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Congruence solvability, solvability, abelian congruence, abelian extension, Mo- ufang loop, 3-divisible Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Dr´apal supported by the INTER-EXCELLENCE project LTAUSA19070 of MˇSMT Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Vojtˇechovsk´y supported by the Simons Foundation Mathematics and Physical Sciences Collaboration Grant for Mathematicians no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 855097 and by the PROF grant of the University of Denver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 1 Here is a summary of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The congruence commutator theory for loops was de- scribed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In the follow-up paper [16], Stanovsk´y and Vojtˇechovsk´y listed several equivalent conditions that characterize a normal subloop X of Q that induces an abelian congruence of Q, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' [16, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We record the relevant parts of their result as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Building upon Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1, we show that in the case of Moufang loops, a normal subloop X induces an abelian congruence of Q if and only if (i) each inner map- ping of Q restricts to an automorphism of X, and (ii) u(xy) = (uy)x for all x, y ∈ X and u ∈ Q, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We can further improve upon Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2 in the case of 3-divisible Moufang loops when condition (i) is automatically satisfied, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Given a 3-divisible Moufang loop Q and its normal subloop S, we then construct a cer- tain subnormal series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6) in the multiplication group Mlt(Q) that starts with the relative multiplication group MltQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' One of the intermediate subloops of the series is constructed upon considering pseudoautomorphisms of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In Section 5 we prove that every classically solvable finite 3-divisible Moufang loop is congruence solvable, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The proof is an induction on the order of a smallest counterexample Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' As the three Isomorphism Theorems and the Correspondence Theorem hold in loops, the standard proof from group theory goes through to establish the following result: If X is a normal subloop of a loop Q, then Q is classically solvable if and only if both X and Q/X are classically solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Similarly, if X is a normal subloop of Q that induces an abelian congruence of Q, then Q is congruence solvable if and only if both X and Q/X are congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It therefore suffices to find a nontrivial normal subloop X of Q that induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If the nucleus Nuc(Q) of Q is nontrivial, it contains a nontrivial normal commutative subgroup X, and every such subgroup induces an abelian congruence of Q by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If Nuc(Q) is trivial then Mlt(Q) admits certain triality automorphisms by [9, Theorem 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We study Moufang loops that admit triality automorphisms in the short Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Con- tinuing the inductive proof, we then show that Q contains a nontrivial normal p-subgroup S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6) yields a subnormal series that starts with the p-group MltQ(S) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The- orem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4) and terminates with Mlt(Q) ⋊ S3, the multiplication group of Q extended by the triality automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This gives rise to a nontrivial normal triality p-subgroup U of Mlt(Q) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Finally, X = U(1) is then the sought-after normal subloop of Q that induces an abelian congruence of Q (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2, whose proof uses the characterization of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The main results of this paper are Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' However, many of the auxiliary results are likely to be useful in future investigations of congruence solvability in Moufang loops and in our understanding of the structure of Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A characterization of abelian congruences in Moufang loops See [1, 13] for an introduction to loop theory and [6] for an introduction to commutator theory in congruence modular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q = (Q, ·, \\, /, 1) be a loop, a universal algebra satisfying the identities x\\(x · y) = x · (x\\y) = y = (y · x)/x = (y/x) · x and x · 1 = x = 1 · x for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' As usual, we also denote the multiplication operation · by juxtaposition and we assume that juxtaposition is more binding than the operations ·, \\ and /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For instance, xy · (u\\v) = (x · y) · (u\\v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 2 A loop Q is an inverse property loop if for every x ∈ Q there is x−1 ∈ Q such that x−1 · xy = y = yx · x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This implies 1 = xx−1 = x−1x, (x−1)−1 = x, x\\y = x−1y and x/y = xy−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Inverse property loops can therefore we treated as universal algebras in the signature (·, −1, 1) familiar from groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' An associative subloop of a loop Q will be refereed to as a subgroup of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A loop Q is power associative if any element of Q generates a subgroup, and diassociative if any two elements of Q generate a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We will often not specify unnecessary parentheses in diassociative loops, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=', we write xyx instead of x · yx or xy · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The loop identities x(y · xz) = (xy · x)z, (zx · y)x = z(x · yx), x(yz · x) = xy · zx, (x · yz)x = xy · zx (M) are pairwise equivalent and a loop satisfying any one (and hence all) of the identities is called a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Every Moufang loop is diassociative [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A power associative loop Q is d-divisible if the mapping x �→ xd is onto Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We claim that a finite Moufang loop Q is 3-divisible if and only if the order of Q is coprime to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Certainly if Q is not 3-divisible then there are x ̸= y such that x3 = y3, so the group ⟨x, y⟩ is not 3-divisible, it contains an element of order 3, and hence 3 divides |Q| by the elementwise Lagrange theorem for Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Conversely, if 3 divides |Q| then Q contains an element of order 3 by [2, Lemma 4] and hence Q is not 3-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If u is an element of a loop Q, then the left translation and the right translations by u are defined by Lu(v) = uv and Ru(v) = vu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The group Mlt(Q) = ⟨Lu, Ru : u ∈ Q⟩ is the multiplication group of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' An element of Mlt(Q) that fixes 1 is called an inner mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The inner mappings of Q form a subgroup Inn(Q) of Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It is well known that Inn(Q) = ⟨Tu, Lu,v, Ru,v : u, v ∈ Q⟩, where Tu = R−1 u Lu, Lu,v = L−1 uv LuLv and Ru,v = R−1 uv RvRu, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1) are the standard generators of Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let A be a universal algebra on an underlying set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A congruence of A is an equivalence relation on A that commutes with all operations of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' As in [6], a congruence α centralizes congruence β over congruence γ if for any (n+1)ary term operation t in the signature of A the following implication holds for all a, b, u1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' , un, vn ∈ A satisfying a α b and u1 β v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' , un β vn: t(a, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' , un) δ t(a, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' , vn) ⇒ t(b, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' , un) δ t(b, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' , vn) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2) A congruence α is abelian if the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2) holds with β = α and δ = {(a, a) : a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' There is a one-to-one correspondence between normal subloops of Q and congruences of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For X ⊴ Q, the induced congruence modX is defined by u modX v iff uX = vX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For a congruence α of Q, the induced normal subloop is the equivalence class of α containing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let X ⊴ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If modX is an abelian congruence of Q then X is a commutative group, but the converse is not true in general loops, not even in Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The following result is excerpted from [16, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It uses the somewhat unusual commutators and associators employed in [16], namely, [x, y] = ((xy)/x)/y and [x, y, z] = ((xy · z)/(yz))/x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Note that we have [x, y] = 1 iff xy = yx and [x, y, z] = 1 iff xy · z = x · yz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1 ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let X be a normal subloop of a loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then the following conditions are equivalent: (i) X induces an abelian congruence of Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 3 (ii) every inner mapping of Q restricts to an automorphism of X and for every x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' y ∈ X and u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' w ∈ Q with v modX w we have [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' y] = [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' u] = [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' y] = [u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' y] = 1 and [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' v] = [x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' w],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (iii) Q is an abelian extension of X by Q/X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' X is a commutative group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' there is a transversal T to X in Q containing 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' for every r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' s ∈ T there are automorphisms ϕr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' ψr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s ∈ Aut(X) and θr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s ∈ X such that ϕr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1 = ψ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s = idX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s = θr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' and rx · sy = t · ϕr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s(x)ψr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s(y)θr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' where t is the unique element of T ∩ (rs)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We can substantially simplify condition (ii) in the case of Moufang loops: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let X be a normal subloop of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then X induces an abelian congruence of Q if and only if every inner mapping of Q restricts to an automorphism of X and u · xy = uy · x for all u ∈ Q and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The direct implication follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1(ii) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Certainly every inner mapping of Q restrict to an automorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since [x, y] = 1 for all x, y ∈ X, the normal subloop X is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Using commutativity and [u, x, y] = 1 for u ∈ Q, x, y ∈ X, we deduce u · yx = u · xy = ux · y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For the converse implication, suppose that every inner mapping of Q restricts to an auto- morphism of X and u · xy = uy · x for all u ∈ Q and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The latter condition implies that X is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We will show that Q is an abelian extension of X by Q/X, as described in item (iii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let T be a transversal to X in Q with 1 ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let r, s ∈ T and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In the following calculation we will use without reference the fact that every inner mapping of Q restricts to an automorphism of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We will also write [Q, X, X] = 1 as a justification comment whenever we use u · xy = ux · y with u ∈ Q and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Finally, we omit parentheses in subterms involving only factors from the commutative group X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Now: rx · sy = rx · Ts(y)s = s(s−1r · Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)) · Ts(y)s = s · (s−1r · Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x))Ts(y) · s by (M) = s · (s−1r · Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)) · s [Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' X] = 1 = s(s−1r) · Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)s = r · Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)s by (M) = r · sT −1 s (Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)) = rs · Lr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='sT −1 s (Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)) = tz · Lr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='sT −1 s (Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)) t ∈ T ∩ (rs)X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' z ∈ X = t · zLr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='sT −1 s (Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y)) [Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' X] = 1 = t · Lr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='sT −1 s (Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x)Ts(y))z X is commutative = t · (Lr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='sT −1 s Ls−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='r(x) · Lr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='sT −1 s Ts(y) · z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence rx · sy = t · ϕr,s(x)ψr,s(y)θr,s, where ϕr,s is the restriction of Lr,sT −1 s Ls−1,r to X, ψr,s is the restriction of Lr,sT −1 s Ts = Lr,s to X and θr,s = z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If r = 1 then t is the unique element of T ∩ sX, so t = s and θ1,s = z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Similarly, θr,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We also have ϕr,1 = Lr,1T −1 1 L1,r = idX and ψ1,s = L1,s = idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ We proceed to improve upon Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2 in the case of 3-divisible Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We start by recalling two results of Gagola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 4 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3 (dual of [7, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then u3ix · u3jy = u3(i+j)T −i−2j u (T i−j u (x)T i−j u (y)) for all u, x, y ∈ Q and all i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We will only need Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3 for the case i = 1 and j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' See [5] for a quick proof of that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4 ([8, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop generated by a set of elements, each of which is a cube of an element of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Inn(Q) = ⟨Tu : u ∈ Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let X be a normal subloop of a 3-divisible Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then X induces an abelian congruence of Q if and only if u · xy = uy · x for all u ∈ Q and x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The direct implication follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For the converse implication, sup- pose that u · xy = uy · x for all u ∈ Q and x, y ∈ X so that X is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since Q is 3-divisible, every element of Q is a cube of some element of Q and Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4 therefore applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In view of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2, it suffices to show that Tu restricts to an automorphism of X for every u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' With i = 1 and j = 0, the formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3 becomes u3x · y = u3T −1 u (Tu(x)Tu(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since u3x · y = u3 · yx = u3 · xy is assumed, we have u3 · xy = u3T −1 u (Tu(x)Tu(y)), xy = T −1 u (Tu(x)Tu(y)) and Tu(xy) = Tu(x)Tu(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A subnormal series in the multiplication group If S ≤ Q and Q is a loop then MltQ(S) = ⟨Ls, Rs : s ∈ S⟩ ≤ Mlt(Q) is the relative multiplication group of S in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If S ⊴ Q then Mlt(Q/S) is an image of Mlt(Q) under the homomorphism determined by Lx �→ LxS and Rx �→ RxS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let C(Q, S) be the kernel of this homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It is not hard to show that C(Q, S) consists of all ϕ ∈ Mlt(Q) that centralize the cosets of S in Q, that is, ϕ(uS) = uS for all u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let S be a normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then C(Q, S) = {Lsσ, Rsσ : s ∈ S, σ ∈ Inn(Q) ∩ C(Q, S)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Given ϕ ∈ Mlt(Q), there are uniquely determined v ∈ Q and σ ∈ Inn(Q) such that ϕ = Lvσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If σ ∈ Inn(Q) ∩ C(Q, S) and s ∈ S then Lsσ(uS) = Ls(uS) = Ls(Su) = s(Su) = (sS)u = Su = uS for all u ∈ Q, and therefore σ ∈ Inn(Q) ∩ C(Q, S) and Lsσ ∈ C(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Conversely, let ϕ = Lvσ ∈ C(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since S = ϕ(S) = Lvσ(S) = Lv(S) = vS, it follows that v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Moreover, σ(uS) = L−1 v Lvσ(uS) = L−1 v (uS) = uS since v ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The argument for Rsσ is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ In particular, MltQ(S) = ⟨Ls, Rs : s ∈ S⟩ ≤ C(Q, S) ⊴ Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In this section we will refine this series into a subnormal series in the case of 3-divisible Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For a loop Q, define the nucleus Nuc(Q) and the left nucleus Nucℓ(Q) by Nuc(Q) = {a ∈ Q : a(uv) = (au)v, u(av) = (ua)v, u(va) = (uv)a for all u, v ∈ Q}, Nucℓ(Q) = {a ∈ Q : a(uv) = (au)v for all u, v ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Bruck proved in [1] that the nuclei are subgroups of Q, and if Q is Moufang then Nuc(Q) = Nucℓ(Q) is a normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 5 A (left) pseudoautomorphism ϕ of Q is a permutation of Q for which there exists some c ∈ Q such that cϕ(x) · ϕ(y) = cϕ(xy) for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The element c is called a (left) companion of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then d ∈ Q is another companion of ϕ if and only if d/c ∈ Nucℓ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Denote by Psaℓ(Q) the set of all pairs (c, ϕ) such that c is a left companion of a pseudoau- tomorphism ϕ of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Psaℓ(Q) is a group with operations (c, ϕ)(d, ψ) = (cϕ(d), ϕψ) and (c, ϕ)−1 = (ϕ−1(c−1), ϕ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1) Pseudoautomorphisms may be interpreted by means of autotopisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A triple (α, β, γ) of permutations of Q is an autotopism of Q if α(x)β(y) = γ(xy) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2) for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Clearly, (c, ϕ) ∈ Psaℓ(Q) ⇔ (Lcϕ, ϕ, Lcϕ) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3) The set of all autotopisms of Q forms the autotopism group Atp(Q) under componentwise composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If (α, β, γ) ∈ Atp(Q) and β(1) = 1, then α = γ and (α(1), β) = (γ(1), β) ∈ Psaℓ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This can be easily seen by setting first y = 1 and then x = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let us now summarize basic facts pertaining to autotopisms and pseudoautomorphisms in Moufang loops, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Every inner mapping of Q is a pseudoautomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This can be seen by verifying the well-known identities (x−3, Tx) ∈ Psaℓ(Q), ([x−1, y], [Lx, Ry]) ∈ Psaℓ(Q), Lx,y = [Lx, R−1 y ] = [R−1 x , Ly] and Rx,y = [L−1 y , Rx] = [Ry, L−1 x ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4) Let Mx = LxRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since Q is Moufang, Mx is also equal to RxLx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If x ∈ Q then Atp(Q) contains each of the triples (Lx, Rx, Mx), (Mx, L−1 x , Lx) and (R−1 x , Mx, Rx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5) A semiautomorphism of a Moufang loop Q is a bijection ϕ of Q satisfying ϕ(1) = 1 and ϕ(xyx) = ϕ(x)ϕ(y)ϕ(x) for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Every semiautomorphism satisfies ϕ(xi) = ϕ(x)i for all x ∈ Q and i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Every pseudoautomorphism of a Moufang loop is a semiautomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In particular, ϕ(x−1) = x−1 if ϕ ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Suppose that x ∈ Q and (c, ϕ) ∈ Psaℓ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then (cϕ(x−1), L−1 ϕ(x)ϕLx) ∈ Psaℓ(Q), where cy = y−1cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let ψ = L−1 ϕ(x)ϕLx and note that ψ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5) the composition (Mϕ(x), L−1 ϕ(x), Lϕ(x))(Lcϕ, ϕ, Lcϕ)(M−1 x , Lx, L−1 x ) is an autotopisms of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence a companion of ψ = L−1 ϕ(x)ϕLx is equal to Lϕ(x)LcϕL−1 x (1) = ϕ(x)cϕ(x−1) = ϕ(x−1)−1cϕ(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ The following result will be useful in the inductive proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 6 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let A be a finite commutative normal subgroup of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let p be a prime dividing |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then there is a nontrivial normal p-subgroup of A that is normal in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let S be the p-primary component of the commutative group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let ϕ be an inner mapping of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since A ⊴ Q, ϕ(A) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' As ϕ is a semiautomorphism of Q, it restricts to a semiautomorphism of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Recall that ϕ(xi) = ϕ(x)i for all x ∈ Q and i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In particular, if x ∈ S then |ϕ(x)| divides |x| and therefore ϕ(x) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Let S be a normal subloop of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Recall the normal subgroup C(Q, S) of Mlt(Q) and define C0(Q, S) as the set of all ϕ ∈ C(Q, S) such that when ϕ is written as ϕ = Lsσ with s ∈ S and σ ∈ Inn(Q) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1) then the pseudoautomorphism σ has a companion in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let S be a normal subloop of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then C0(Q, S) is a normal subgroup of C(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let ϕ = Lsσ ∈ C(Q, S) with s ∈ S and σ ∈ Inn(Q), and let c ∈ Q be a companion of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We show that the mapping f : C(Q, S) → Q/(Nuc(Q)S), Lsσ �→ c(Nuc(Q)S) is a well-defined homomorphism with kernel C0(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since since both S and Nuc(Q) are normal in Q, they generate the normal subloop Nuc(Q)S ⊴ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If c and d are companions of σ then d ∈ c Nuc(Q) ⊆ c Nuc(Q)S, so f is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For the homomorphic property, consider Lsσ, Ltτ ∈ C(Q, S) (with s, t ∈ S and σ, τ ∈ Inn(Q)) such that (c, σ), (d, τ) ∈ Psaℓ(Q) for some c, d ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We have f(Lsσ)f(Ltτ) = c Nuc(Q)S · d Nuc(Q)S = (cd) Nuc(Q)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let ψ = L−1 sσ(t)LsσLtτ = L−1 sσ(t)LsLσ(t)L−1 σ(t)σLtτ and observe that ψ(1) = 1, so ψ ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' To compute a companion of ψ, note that [s−1, σ(t−1)] is a companion of L−1 sσ(t)LsLσ(t) = Ls,σ(t) = [Ls, R−1 σ(t)], by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4), and that cσ(t−1) is a companion of L−1 σ(t)σLt, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1), L−1 σ(t)σLtτ has a companion cσ(t−1) · L−1 σ(t)σLt(d) = cσ(t−1) · σ(t−1)σ(td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1) again shows that ψ possesses a companion e = [s−1, σ(t−1)] [Ls, R−1 σ(t)] � cσ(t−1) · σ(t−1)σ(td) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1, each of σ, τ, Lsσ(t), Lσ(t) and Ls belong to C(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence ψ ∈ Inn(Q) ∩ C(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since sσ(t) ∈ S, we see that LsσLtτ decomposes as Lsσ(t)ψ and f(ψ) = e Nuc(Q)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For the homomorphic property, it remains to show that e ≡ cd modulo Nuc(Q)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since σ centralizes cosets of S and we work modulo Nuc(Q)S ≥ S, e is equivalent to [s−1, t−1][Ls, R−1 t ](c(t−1) · t−1(td)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since s, t ∈ S, we have further e ≡ [Ls, R−1 t ](cd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The left translation Ls is identical modulo S, and e ≡ cd follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The kernel of f consist of all Lsσ ∈ C(Q, S) such that σ has a companion in Nuc(Q)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since the companions are determined up to Nuc(Q), the kernel coincides with C0(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let S be a normal subloop of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Is S is 3-divisible then MltQ(S) ⊴ C0(Q, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let t ∈ S and Lsσ ∈ C0(Q, S), where s ∈ S and σ ∈ Inn(Q) ∩ C(Q, S) has a companion c ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We need to show that LLsσ t , RLsσ t ∈ MltQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since LLsσ t = (L−1 s LtLs)σ = (L−1 s )σLσ t Lσ s, (L−1 s )σ = (Lσ s )−1 and similarly for RLsσ t , we only need to show that Lσ s , Rσ s ∈ MltQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We will prove Lσ s ∈ MltQ(S), the other case following dually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since S is 3-divisible, there is d ∈ S such that d3 = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Note that Td(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1), Psaℓ(Q) contains (c−1, Td)(c, σ) = (c−1Td(c), Tdσ) = (1, Tdσ), which means that α = Tdσ is an automorphism of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Recall that for all x ∈ Q we have Lα x = α−1Lxα = Lα−1(x), Rα x = Rα−1(x) and thus T α x = Tα−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Also, T −1 x = Tx−1 in a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Therefore Lσ s = (L T −1 d s )α = (TdLsTd−1)α = Tα−1(d)Lα−1(s)Tα−1(d−1) is in MltQ(S), since α(S) = Tdσ(S) = Td(S) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let S be a normal 3-divisible subloop of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then MltQ(S) ⊴ C0(Q, S) ⊴ C(Q, S) ⊴ Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Moufang loops admitting triality automorphisms Glauberman observed in [9, Theorem 6] that if Q is a Moufang loop with trivial nucleus, then the mappings σ : Lx �→ R−1 x , Rx �→ L−1 x ρ : Lx �→ Rx, Rx �→ M−1 x = L−1 x R−1 x (and Mx �→ L−1 x ) extend uniquely to automorphisms of Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' See [2, 14] and [11, Chapter 13] for more information on groups with triality and Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We say that a Moufang loop Q admits triality automorphisms if the above maps σ and ρ extend into automorphisms of Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (There exist Moufang loops with nontrivial nucleus that admit triality automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=') Suppose that a Moufang loop Q admits triality automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since the subgroup ⟨σ, ρ⟩ satisfies the standard presenting relations of the symmetric group S3, it is isomorphic to a homomorphic image of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence Q induces a semidirect product Mlt(Q) ⋊ S3 (with an action of S3 that might not be faithful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let α = σρ and β = σρ2, so that α(Lx) = L−1 x , α(Rx) = Mx, β(Lx) = Mx, β(Rx) = R−1 x and σ = αβα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Note that σ centralizes Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Indeed, σ(Tx) = σ(R−1 x Lx) = LxR−1 x = Tx, σ(Lx,y) = σ([Lx, R−1 y ]) = [R−1 x , Ly] = Lx,y and σ(Rx,y) = σ([L−1 y , Rx]) = [Ry, L−1 x ] = Rx,y, where we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop that admits triality automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A subgroup U ≤ Mlt(Q) is a triality subgroup if U is invariant under the triality automorphisms σ and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If U ⊴Mlt(Q) then U is a triality subgroup if and only if U ⊴ Mlt(Q) ⋊ S3 under the induced action of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Suppose that Q is a Moufang loop that admits triality automorphisms and that U ⊴ Mlt(Q) is a triality subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let S = U(1) be the orbit of U that contains the element 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then S is a normal subloop of Q and MltQ(S) ≤ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since U is normal, the blocks conjugate to S in Mlt(Q) form equivalence classes of a congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This implies that S is normal in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For each s ∈ S there is ϕ ∈ Inn(Q) such that Lsϕ ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Ms = LsRs = Lsϕ(R−1 s ϕ)−1 = Lsϕσ(Lsϕ)−1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence also Rs = α(Ms) ∈ U and Ls = β(Ms) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ 8 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a 3-divisible Moufang loop that admits triality automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let U ⊴Mlt(Q) be a nontrivial commutative triality subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Q possesses a nontrivial subloop X ⊴ Q that induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1, X = U(1) is a normal subloop of Q such Lx, Rx ∈ U for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since U is commutative, [Lx, Ly] = idQ = [Lx, Ry] for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The first condition implies that X is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The second condition says that x · uy = xu · y for all x, y ∈ X, u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then, by Moufang Theorem [12, 3], u · xy = ux · y for all x, y ∈ X, u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence u · yx = u · xy = ux · y for all x, y ∈ X, u ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5, X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Solvability in finite Moufang loops of order coprime to three In this section we show that the two concepts of solvability coincide for finite 3-divisible Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We start with a well known fact of elementary group theory: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let H be a subnormal subgroup of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If there exists a nontrivial p-group N ⊴ H, p a prime, then G contains a nontrivial normal p-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Without loss of generality it may be assumed that N is a minimal normal p-subgroup of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Thus there exists U ≤ H such that N ≤ U and U is a minimal characteristic subgroup of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since N ⊴ U, the group U has to be a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If H ⊴ G then U ⊴ G since U is characteristic in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' This proves the case k = 1 of the general case N ⊴ H = Hk ⊴ · · · ⊴ H1 ⊴ H0 = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Suppose that k > 1 and proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By the induction assumption there exists a nontrivial normal p-subgroup of H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We are done by the case k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Next, let us recall a result of Glauberman and Wright on Moufang loops of prime power order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The odd case was established in [9] and the even case in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' A new proof that covers both cases can be found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2 ([9, 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop of prime power order pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Q is centrally nilpotent and Mlt(Q) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We will not use Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3 below but we offer it as a motivation for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4, which is taken from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' We have included a proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4 for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For a set of primes π, a power associative loop Q is a π-loop if for every x ∈ Q the order of x is a power of a prime from π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3 ([9, Theorem 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a Moufang loop of odd order, π a set of primes and S a classically solvable π-subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then MltQ(S) is a solvable π-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a finite Moufang loop, p a prime and S a p-subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then MltQ(S) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Restricting ϕ ∈ MltQ(S) to Mlt(S) yields an epimorphism with kernel FixQ(S) = {ϕ ∈ MltQ(S) : ϕ(s) = s for all s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2, Mlt(S) is a p-group and therefore MltQ(S)/ FixQ(S) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Consider the group InnQ(S) = MltQ(S) ∩ Inn(Q) = ⟨Ts, Ls,t, Rs,t : s, t ∈ S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For each ϕ ∈ InnQ(S), let C(ϕ) be the set of all companions of ϕ, a coset of N = Nuc(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since the 9 standard generators of InnQ(S) are pseudoautomorphisms with companions in S and every ϕ ∈ InnQ(S) satisfies ϕ(S) = S, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1) that C(ϕ) ∩ S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For ϕ, ψ ∈ FixQ(S), we therefore certainly have c, d ∈ S such that (c, ϕ), (d, ψ) ∈ Psaℓ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since ϕ(d) = d, we also have (cd, ϕψ) = (c, ϕ)(d, ψ) ∈ Psaℓ(Q) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1), proving that cd ∈ C(ϕψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The element cd also belongs to C(ϕ)C(ψ) = cNdN = cdN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence ϕ �→ C(ϕ) is a homomorphism from the group FixQ(S) to the loop SN/N ∼= S/(S ∩ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let A be the kernel of this homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Being associative, the image of the homomorphism is equal to a subgroup of S/(S ∩ N), some p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence FixQ(S)/A is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The kernel A consists of all automorphisms of Q that are contained in FixQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Now, αLsα−1 = Lα(s) = Ls for every s ∈ S and α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Similarly for Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence A ≤ Z(MltQ(S)) is a commutative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Write A as B × D, where B is the p-primary component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Both B and D are normal subgroups of MltQ(S), being central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' All three MltQ(S)/ FixQ(S), FixQ(S)/A and A/D are p-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence MltQ(S)/D is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The subgroup D is commutative, normal and of order coprime to MltQ(S)/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence it possesses a complement in MltQ(S), say P, a Sylow p-subgroup of MltQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since D ≤ Z(MltQ(S)), P is a normal subgroup of MltQ(S) and hence the unique Sylow p-subgroup of MltQ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For s ∈ S, we have |Ls| = |Rs| = |s| by diassociativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since S is a p-loop and the elementwise Lagrange theorem holds in Moufang loops, it follows that both Ls and Rs belong to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence P = MltQ(S) = ⟨Ls, Rs : s ∈ S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let X be a commutative normal subloop of a Moufang loop Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If X ≤ Nuc(Q) then X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By [1], Nuc(Q) ⊴ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence if a, b ∈ Nuc(Q) and x ∈ Q then Tx(ab) = (x · ab)x−1 = (xa · b)x−1 = (Tx(a)x · b)x−1 = (Tx(a) · xb)x−1 = Tx(a)Tx(b), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1) where we used Tx(a) ∈ Nuc(Q) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The remaining standard generators of Inn(Q) act trivially on Nuc(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2, X induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a finite 3-divisible Moufang loop with a nontrivial normal p- subloop S, p ̸= 3 a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Mlt(Q) contains a nontrivial normal elementary abelian p-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Furthermore, if Q also admits triality automorphisms, then Mlt(Q) contains a nontrivial normal elementary abelian triality p-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It suffices to prove that Mlt(Q) contains a nontrivial normal p-subgroup since the center of such a p-group is characteristic, and the socle of an abelian p-group is characteristic too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1, it even suffices to show the existence of a subnormal p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6 implies the existence of the subnormal series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='4, MltQ(S) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If Q admits triality automorphisms, the subnormal series (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6) may be extended by Mlt(Q) ⊴ Mlt(Q) ⋊ S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let Q be a finite 3-divisible Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Q is classically solvable if and only if it is congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' The converse implication holds in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' For the direct implication, let Q be a smallest 3-divisible Moufang loop that is classically solvable but not congruence solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' If there is a normal subloop X of Q that induces an abelian congruence of Q, then Q/X is 3-divisible and classically solvable, hence congruence solvable by minimality of Q, but 10 then Q is congruence solvable, being an abelian extension of a commutative group X by a congruence solvable loop Q/X, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Suppose that 1 < N = Nuc(Q) ⊴ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since Q is classically solvable, the group N is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' It therefore contains a nontrivial characteristic commutative subgroup X ⊴ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='1), Inn(Q) acts upon X as a subgroup of Aut(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Hence ϕ(X) = X for each ϕ ∈ Inn(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' But this means that X is a normal subloop of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then X induces an abelian congruence of Q by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Now suppose that Nuc(Q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Then Q admits triality automorphisms, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Since Q is classically solvable, it has a series of normal subloops of Q with factors being commutative groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' In particular, Q contains a nontrivial normal commutative subgroup A, the first nontrivial term of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Let p be any prime dividing |A|, necessarily p ̸= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='3, Q contains a nontrivial normal p-subgroup S (contained in A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='6, Mlt(Q) contains a nontrivial normal commutative triality subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='2, Q contains a nontrivial normal subloop that induces an abelian congruence of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' □ References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Bruck, A Survey of Binary Systems, Springer-Verlag, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Doro, Simple Moufang loops, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Proc.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Vojtˇechovsk´y, Commutator theory for loops, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Algebra 399 (2014), 290–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Stanovsk´y and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' Vojtˇechovsk´y, Abelian extensions and solvable loops, Results Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' 66 (2014), 367–384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' (Dr´apal) Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' of Mathematics, Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic Email address, Dr´apal: drapal@karlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='cz (Vojtˇechovsk´y) Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' of Mathematics, University of Denver, 2390 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=' York St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content=', Denver, CO 80208, USA Email address, Vojtˇechovsk´y: petr@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} +page_content='edu 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE2T4oBgHgl3EQfIgYI/content/2301.03680v1.pdf'} diff --git a/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf b/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b3c65e72fdd740103dd83d8e4316e595df242529 --- /dev/null +++ b/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d99506561f7ee28ef0ce5087ec0eaf3c1336c655f36c5be58dc0dacd1727bbfe +size 5809261 diff --git a/l9FAT4oBgHgl3EQfcB1A/content/tmp_files/2301.08561v1.pdf.txt b/l9FAT4oBgHgl3EQfcB1A/content/tmp_files/2301.08561v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb1e8287fa23d5721604b862546154eb5a0f3294 --- /dev/null +++ b/l9FAT4oBgHgl3EQfcB1A/content/tmp_files/2301.08561v1.pdf.txt @@ -0,0 +1,2020 @@ +arXiv:2301.08561v1 [math.AP] 8 Jan 2023 +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +FOR A TRIPLY NONLINEAR THERMISTOR PROBLEM +MOULAY RCHID SIDI AMMI, IBRAHIM DAHI, +ABDERRAHMANE EL HACHIMI, AND DELFIM F. M. TORRES +Abstract. We study the existence and uniqueness of a bounded weak solution +for a triply nonlinear thermistor problem in Sobolev spaces. +Furthermore, +we prove the existence of an absorbing set and, consequently, the universal +attractor. +1. Introduction +The thermistor was discovered by Michael Faraday in 1833, who noticed that the +temperature increases when the silver sulfides resistance decreases. A lot of studies +of the thermistor problem can be found in [1, 9, 10, 15, 17]. +A thermistor is a circuit component that may be used as a current limiter or +as a temperature sensor. It is, typically, a tiny cylinder, constructed of a ceramic +substance whose electrical conductivity is highly dependent on temperature. The +thermistor regulates the heat created by an electrical current traveling through a +conductor device. Thermistor problems have received a lot of attention. We refer +the reader to [4, 7, 10, 12, 17, 19] and references therein. +Thermistors are commonly used as temperature control devices in a wide variety +of industrial equipment, ranging from space vehicles to air conditioning controllers. +They are also often used in the medical field, for localized and general body tem- +perature measurement, in meteorology, for weather forecasting, and in chemical +industries as process temperature sensors. A detailed description of thermistors +and their applications in electronics and other industries can be found in [23]. +There are two types of thermistors: NTC and PTC, which have a positive and +negative temperature coefficient, respectively. An NTC thermistor is a tempera- +ture sensor that measures temperature using the resistance qualities of ceramic and +metal composites. NTC sensors provide a number of benefits in terms of tempera- +ture sensing, including small size, great long-term stability, and high accuracy and +precision. The operation of a PTC electric surge device is as follows: when the cir- +cuit’s current is suddenly increased, the device heats up, causing a dramatic decline +in its electrical conductivity, effectively shutting off the circuit. In this paper, we +2010 Mathematics Subject Classification. 35A01, 35A02, 46E35. +Key words and phrases. Existence; uniqueness; thermistor problem; Sobolev spaces; global +attractor; ω−limit; invariant set; absobsing set; semi-group. +This is a 19 pages preprint of a paper whose final and definite form is published in +’Moroccan J. of Pure and Appl. Anal. (MJPAA)’, ISSN: Online 2351-8227 – Print +2605-6364. +1 + +2 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +consider the following general nonlocal thermistor problem: + + + + + + + +∂α(v) +∂s +− ∆mv = κ +f(v) +( +� +Ω f(v)dx)2 , +in +Q, +α(v(x, 0)) = α(v0), +in +Ω, +v = 0, +on +Γ×]0, M[. +(1.1) +Problem (1.1) models the diffusion of the temperature produced when an electric +current flows crossing a material, where f(v) is the electrical resistance of the +conductor and +f(v) +( +� +Ω f(v)dx)2 represents the non-local term of (1.1). +Here, Q = +Ω × [0, M], where Ω is an open bounded subset of RN, N ≥ 1, and M is a positive +constant. +Problem (1.1) is a generalization of the problem appearing in the work of Kaval- +laris and Nadzieja [16]. For α(v) = v and m = 2, one gets the classical model of the +thermistor problem appearing in the work of Lacey [17], which is a transformation +of the following problem: +∂v +∂s = ∇ · (κ(v)∇v) + ρ(v)|∇ψ|2, +∇ · (ρ(v)∇ψ) = 0, +(1.2) +where κ is the thermal conductivity, ψ is the electrical potential, and ρ(v) represents +the electrical conductivity, which is normally a positive function supposed to drop +sharply by several orders of magnitude at some critical temperature, and remains +essentially zero for larger temperatures. This feature is essential for the intended +functioning of thermistors as thermoelectric switches. +In the case α(v) = v and m = 2, existence and uniqueness results of bounded +weak solutions to problem (1.1) were established in [10]. Existence of an optimal +control has been obtained by many authors with different assumptions on f and m. +We refer, for instance, to [14]. On the other hand, numerical computations of (1.1) +and (1.2) have been carried out by other authors, see for example [6, 21, 22, 28], +in which the chosen parameters correspond to actual devices. Moreover, a study of +(1.2) in the case N = 1 can be found in [13]. Here, we extend the existing literature +of the nonlocal thermistor problem to a triply nonlinear case. +Let B be the area of Ω, I the current such that κ = I2/B2, and ∆m be defined +by +∆mv = div(| ∇v |m−2 ∇v) ∀m ≥ 2. +We further specify the terms in (1.1). We assume: +(H1) v0 ∈ L∞(Ω); +(H2) α : R −→ R is a Lipschitz continuous increasing function such that α(0) = 0 +and α′(s) ≥ λ > 0 for all s ∈ R; +(H3) f is a Lipshitz continuous function, with compact support, verifying +σ ≤ f(s), for all s ∈ R, for a positive constant σ. +The rest of the paper is organized as follows. In Section 2, we collect some basic +concepts and a few known results that are useful to our development. Section 3 is +devoted to the existence of a classical solution to the regularized problem of (1.1). +In Section 4, existence of a bounded weak solution to the regularized problem +is proved. Then, in Section 5, we provide sufficient conditions under which the + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +3 +solution is unique. Existence of an absorbing set, as well as the global attractor, +are proved in Section 6. Finally, we present some concluding remarks in Section 7. +2. Preliminaries +In this section we collect a few known results that are useful to us. +Definition 1 (See [5]). Let α be a continuous increasing function with α(0) = 0. +For s ∈ R we define +Ψ (s) = +� s +0 +α(t)dt. +The Legendre transform Ψ ∗ of Ψ is defined by +Ψ ∗ (t) = sup +r∈R +{rt − Ψ(t)}. +(2.1) +In particular, we get +Ψ ∗ (α(t)) = tα(t) − Ψ (t) . +(2.2) +Remark 2. If v ∈ L∞(Q), then α(v) ∈ L∞(Q). It turns out, from equality (2.2), +that Ψ ∗ (α(v)) is also bounded. +Lemma 3 (See [26]). Assume that z is a non-negative, absolutely continuous func- +tion, satisfying the following inequality: +z′(s) ≤ hz(s) + g(s), for s ≥ s0, +where h and g are two non-negative integrable functions on [0, M]. Then, for each +s ∈ [0, M], +z(s) ≤ exp +�� s +0 +h(τ)dτ +� +· +� +z(0) + +� s +0 +g(τ)dτ +� +. +Lemma 4 (Ghidaglia lemma [26]). Let z be a positive and absolutely continuous +function on ]0, ∞[ such that the inequality +z′ + δzq ≤ η +holds, where q > 1, δ > 0, η ≥ 0. Then, +z(s) ≤ +�η +δ +�1/q ++ (δ(q − 1)s)−1/(q−1) +for all s ≥ 0. +Lemma 5 (See [2]). If v ∈ Lm � +0, M; W 1,m(Ω) +� +with +∂α(v) +∂s +∈ Lm′ � +0, M; W −1,m′(Ω)) +� +, +then +�∂α(v) +∂s +, v +� +W −1,m′ (Ω),W 1,m(Ω) += d +ds +� +Ω +Ψ ∗(α(v)). +In order to study the existence of the global (universal) attractor, we introduce +the following definitions. +Definition 6 (See [26]). Let us consider B ⊂ F and U an open bounded set such +that U ⊂ B. Then B is an absorbing set in U if the orbit of each bounded set of U +enters into B after a given period of time (which may depend on the set): +∀B0 ⊂ U, +B0 bounded, +∃s0 (B0) such that S(s)B0 ⊂ B, +∀s ≥ s0 (B0). + +4 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +Definition 7 (See [26]). The set A ⊂ F is said to be an universal attractor for the +semigroup (S(s))s≥0, if the following conditions hold: +(1) A ⊂ F is a nonempty invariant compact set, +(2) the set A ⊂ F attracts any bounded set B ⊂ F, that is, +dist (S(s)B, A) → 0 as s → +∞, such that dist(D, B) = supa∈D infb∈B ∥a − b∥F. +3. Regularized problems +In this section, we first present our approximation scheme. Then we proceed to +prove the existence of a weak solution to our regularized problem. To design our +regularized scheme, we consider +αr is of class C1(R) where 0 < λ < α′ +r, +αr(0) = 0, αr −→ α in Cloc(R) and |αr| ≤ |α|, +fr is of class C∞(R), +fr → f, in L1(Q) and a.e in Q, +fr satisfies (H3) . +(3.1) +The initial condition is regularized as in the proof of [11, Proposition 3, p. 761], +that is, +vr,0 ∈ C∞ +c (Ω) such that vr,0 → v0 in L∞(Ω), ∥vr,0∥L∞(Ω) ≤ ∥v0∥L∞(Ω) + 1. +(3.2) +Our regularized problems are then given by + + + + + + + +∂αr(vr) +∂s +− ∆r +mv = κ +fr(vr) +(� +Ω fr(vr)dx)2 , +in +Q = Ω × [0, M], +αr(vr,x(0)) = αr(vr,0), +in +Ω, +vr = 0, +on +Γ×]0, M[, +(3.3) +where ∆r +mv = div + +� +| ∇v |2 +r +�m − 2 +2 +∇v + +, m ≥ 2. +Theorem 8. Assume that hypotheses (H1)–(H3) hold. Then there exists a solution +to problem (3.3). +The following lemma plays a key role in the proof of Theorem 8. +Lemma 9. For all r > 0, we have +∥ vr ∥L∞(Q)≤ C(M, ∥ v0 ∥L∞(Ω)), +where C(M, ∥ v0 ∥L∞(Ω)) is a positive constant. +Proof. Multiplying the first equation of problem (3.3) by +� +(αr(vr) − αr(s0))+�p+1 +(s0 is a positive constant where | vr |> s0) and integrating over Ω, we get +� +Ω +∂αr(vr) +∂s +� +(αr(vr) − αr(s0))+�p+1 +− +� +Ω +∆r +mvr +� +(αr(vr) − αr(s0))+�p+1 += +� +Ω +κ · fr(vr) +�� +Ω fr(vr)dx +�2 +� +(αr(vr) − αr(s0))+�p+1 +. +So, we have + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +5 +1 +p + 2 +� +Ω +∂ +∂s +� +(αr(vr) − αr(s0))+�p+2 += +� +Ω +∆r +mvr +� +(αr(vr) − αr(s0))+�p+1 ++ +� +Ω +κ · fr(vr) +�� +Ω fr(vr)dx +�2 +� +(αr(vr) − αr(s0))+�p+1 +. +Then, +1 +p + 2 +∂ +∂s +� +Ω +� +(αr(vr) − αr(s0))+�p+2 += +� +Ω +∆r +mvr +� +(αr(vr) − αr(s0))+�p+1 ++ +� +Ω +κ · fr(vr) +�� +Ω fr(vr)dx +�2 +� +(αr(vr) − αr(s0))+�p+1 +. +(3.4) +On the other hand, we have +� +Ω +∆r +mvr +� +(αr(vr) − αr(s0))+�p+1 += +� +Ω +div + +� +| ∇vr |2 +r +�m − 2 +2 +∇vr + + +� +(αr(vr) − αr(s0))+�p+1 += −(p + 1) +� +Ω + +� +| ∇vr |2 +r +�m − 2 +2 +| ∇vr |2 + + α′ +r(vr) +� +(αr(vr) − αr(s0))+�p ++ +� +∂Ω + +� +| ∇vr |2 +r +�m − 2 +2 +∂vr +∂ν + + +� +(αr(vr) − αr(s0))+�p+1 +. +Since +� +| ∇vr |2 +r +�m − 2 +2 +| ∇vr |2≥ 0 and α′ +r > 0, we get +1 +p + 2 +∂ +∂s +� +Ω +� +(αr(vr) − αr(s0))+�p+2 +≤ +� +∂Ω + +(| ∇vr |2 +r) +m − 2 +2 +∂vr +∂ν + + +� +(αr(vr) − αr(s0))+�p+1 ++ +� +Ω +κ · fr(v) +�� +Ω fr(v)dx +�2 +� +(αr(vr) − αr(s0))+�p+1 +. +(3.5) +By using (H3), we have +� +Ω +κ · fr(vr) +(� +Ω fr(vr)dx)2 +� +(αr(vr) − αr(s0))+�p+1 +≤ +κ +(σ · meas(Ω))2 +� +Ω +fr(vr) +� +(αr(vr) − αr(s0))+�p+1 +. +Since fr satisfies (H3), it yields +fr (vr(x, t)) = fr (vr(x, t)) χ{vr(x,t)∈supp(f)} + fr (vr(x, t)) χ{vr(x,t)/∈supp(f)} +≤ fr (vr(x, t)) χ{vr(x,t)∈supp(f)}. + +6 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +If vr(x, t) ∈ supp(f), then it follows that (vr(x, t))r is bounded. Thus, there exists +a positive constant C0 such that +� +Ω +fr(vr) +� +(αr(vr) − αr(s0))+�p+1 +≤ C0 +� +Ω +� +(αr(vr) − αr(s0))+�p+1 +. +Keeping that in mind, we have for a positive constant C1 that +1 +p + 2 +∂ +∂s +� +Ω +� +(αr(vr) − αr(s0))+�p+2 +≤ C1 +� +Ω +� +(α(vr) − αr(s0))+�p+1 +. +(3.6) +From H¨older’s inequality, there exists positive constants Cj, j = 2, 3, 4, such that +� +Ω +� +(αr(vr) − αr(s0))+�p+1 +≤ (meas(Ω)) +1 +p + 1 · +�� +Ω +� +(αr(vr) − αr(s0))+�p+2 +�p + 1 +p + 2 +≤ C2 [zp(s)]p+1, +where zp(s) :=∥ (αr(vr) − αr(s0))+ ∥Lp+2(Ω). In view of (3.6), we have +1 +p + 2 +∂ +∂s +� +Ω +� +(αr(vr) − αr(s0))+�p+2 +≤ C3 [zp(s)]p+1. +Then, +1 +p + 2 +∂ +∂s [zp(s)]p+2 ≤ C3 [zp(s)]p+1, +(3.7) +and hence +∂ +∂s [zp(s)] ≤ C3, +from which it follows that +[zp(s) − zp(0)] ≤ C3M, +which implies +zp(s) ≤ zp(0) + C3M. +Letting p go to infinity, we obtain that +∥ (αr(vr) − αr(s0))+ ∥L∞(Ω)≤ C4. +(3.8) +Now, let ur = −vr, and consider the following problem: + + + + + + + + + + + +∂ ˜αr(ur) +∂s +− ∆r +mur = κ +˜fr(ur) +�� +Ω ˜fr(vr)dx +�2 =: ˜g(ur) +in +Q, +˜αr(ux,r(0)) = ˜αr(u0) +in +Ω, +ur = 0 +on +Γ×]0, M[, +(3.9) +where ˜αr(τ) = −αr(−τ), ˜gr(τ) = −gr(−τ) and ˜fr(τ) = −fr(−τ). Those functions +satisfy the same conditions verified by α, g and f, respectively. The same reasoning +done to get (3.8), shows that +∥ (˜αr(ur) − ˜αr(s0))+ ∥L∞(Ω)≤ C5, +(3.10) +which is equivalent to +∥ (−αr(−vr(s)) + αr(−s0))+ ∥L∞(Ω)≤ C5. +From (3.8) and (3.10), we deduce that there exists a positive constant C such that +∥ vr(s) ∥L∞(Ω)≤ C(M, ∥ v0 ∥L∞(Ω)), +for all s ∈ [0, M]. + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +7 +The lemma is proved. +□ +Proof of Theorem 8. From Lemma 9 and hypotheses (H1)–(H3), we conclude, from +the classical results of Ladyzenskaya (see [18, pp. 457–459]), with the existence of +a classical solution to the regularized problem (3.3). +□ +4. Existence of a weak solution +Definition 10. We say that v ∈ L∞(Q) ∩ Lm � +0, M; W 1,m(Ω) +� +∩ L∞ � +t, M; W 1,m(Ω) +� +, +t > 0, is a bounded weak solution of problem (1.1), if it satisfies the following iden- +tity: +� M +0 +�∂α(v) +∂s +, u +� +− +� +Q +| ∇v |m−2 ∇v∇u = κ +� +Q +f(v) +(� +Ω f(v)dx)2 u, +(4.1) +for all u ∈ +� +Lm � +0, M; W 1,m(Ω) +� +∩ L∞(Q) +� +. Furthermore, if we have +u ∈ +� +W 1,1 � +0, M; L1(Ω) +� +∩ Lm � +0, M; W 1,m(Ω) +�� +with u(·, M) = 0, then +� M +0 +�∂α(v) +∂s +, u +� += − +� M +0 +� +Ω +[α(v) − α(v0)] ∂su, +where the duality product is defined by ⟨·, ·⟩ = ⟨·, ·⟩W −1,m′ (Ω),W 1,m(Ω). +Remark 11. Since αr is an increasing function and | αr |≤| α |, then, by using +Lemma 9, we also have that (αr(vr))r is bounded. +Our plan is to derive now enough a priori estimates needed in the sequel. +Lemma 12. For all r > 0, we have +||vr||Lm(0,M;W 1,m(Ω)) ≤ C6, +(4.2) +where C6 is a positive constant independent of r. +Proof. Multiplying the first equation of (3.3) by vr and integrating, we get +� +Ω +∂αr(vr) +∂s +vr − +� +Ω +∆r +mvrvr = +� +Ω +κ +fr(vr) +( +� +Ω fr(vr)dx)2 vr. +(4.3) +Applying (2.2), we obtain that +� +Ω +∂αr(vr) +∂s +vr = +� +Ω +∂ [Ψ ∗ (αr(vr))] +∂s +. +On another hand, by using Green’s formula, we get +� +Ω +∆r +mvrvr = − +� +Ω +� +| ∇vr |2 +r +�m − 2 +2 +∇vr∇vr + +� +∂Ω +� +| ∇vr |2 +r +� ∂vr +∂ν · vr. +Substituting into (4.3), we get +� +Ω +∂αr(vr) +∂s +vr + +� +Ω +� +| ∇vr |2 +r +�m − 2 +2 +| ∇vr |2 = +� +Ω +κ · fr(vr) +( +� +Ω fr(vr)dx)2 vr +− +� +∂Ω +� +| ∇vr |m−2 +r +� ∂vr +∂ν · vr, + +8 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +� +Ω +� +| ∇vr |2 +r +�m − 2 +2 +| ∇vr |2 = +� +Ω +κ · fr(vr) +(� +Ω fr(vr)dx)2 vr − +� +Ω +∂ [Ψ ∗ (αr(vr))] +∂s +− +� +∂Ω +� +| ∇vr |m−2 +r +� ∂vr +∂ν · vr. +Then, using the boundary conditions, we have +� M +0 +� +Ω +� +| ∇vr |2 +r +�m − 2 +2 +| ∇vr |2 = +� M +0 +� +Ω +κ · fr(vr) +( +� +Ω fr(vr)dx)2 vr +− +� M +0 +� +Ω +∂ [Ψ ∗ (αr(vr))] +∂s +. +(4.4) +From Remark 2, we know that (Ψ ∗ (αr(vr)))r is bounded. With the aid of hypoth- +esis (H3) and Lemma 9, there exists a positive constant C7 such that +� M +0 +� +Ω +κ +fr(vr) +( +� +Ω fr(vr)dx)2 vr − +� M +0 +� +Ω +∂ [Ψ ∗ (αr(vr))] +∂s +≤ +� M +0 +� +Ω +κ +fr(vr) +( +� +Ω fr(vr)dx)2 vr +− +� +Ω +Ψ ∗ (αr(vr(·, M))) + +� +Ω +Ψ ∗ (αr(vr(·, 0))) +≤ +κ +(σ · meas(Ω))2 +� M +0 +� +Ω +fr(vr)· | vr | ++ 2 · max +����� +� +Ω +Ψ ∗ (αr(vr(·, M))) +���� , +���� +� +Ω +Ψ ∗ (αr(vr(·, 0))) +���� +� +≤ C7. +It yields that +� M +0 +� +Ω +|∇vr|m ≤ +� M +0 +� +Ω +� +| ∇vr |2 +r +�m − 2 +2 +| ∇vr |2≤ C7. +We deduce that vr ∈ Lm � +0, M; W 1,m(Ω) +� +. +□ +Remark 13. Inequality (4.2), combined with Young’s inequality, imply that + +� +| ∇vr |2 +r +�m − 2 +2 +∇vr + + +r +is bounded in Lm′ � +0, M; W 1,m′(Ω) +� +. +A further upper bound for vr is established in the following lemma. +Lemma 14. For all r, s > 0, there exist positive constants C(t), C(t, M), and +C1(t, M), such that the following inequalities hold: +||vr (s)||W 1,m(Ω) ≤ C(t), +for all s ≥ t, +(4.5) +� M +t +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 +≤ C(t, M), +(4.6) +� M +t +� +Ω +�∂αr(vr) +∂s +�2 +≤ C1(t, M). +(4.7) + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +9 +Proof. Multiplying the first equation of problem (3.3) by ∂vr +∂s , and integrating, we +obtain that � +Ω +∂αr(vr) +∂s +∂vr +∂s − +� +Ω +∆r +mvr +∂vr +∂s = +� +Ω +κ +fr(vr) +( +� +Ω fr(vr)dx)2 +∂vr +∂s . +(4.8) +Since +� +Ω +∂αr(vr) +∂s +∂vr +∂s = +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 +, +the equality (4.8) becomes +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 +− +� +Ω +∆r +mvr +∂vr +∂s = +� +Ω +κ +fr(vr) +( +� +Ω fr(vr)dx)2 +∂vr +∂s . +By applying Green’s formula, we get +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 ++ 1 +m +∂ +∂s +� +Ω +� +| ∇vr |2 +r +�m +2 = +� +Ω +κ +fr(vr) +( +� +Ω fr(vr)dx)2 +∂vr +∂s . +(4.9) +Let Gr(vr) := +� vr +0 +gr(s)ds and gr(s) := +fr(s) +( +� +Ω fr(s)dx)2 . By using the boundedness +of vr and (3.1), we have ∂Gr(vr) +∂s +≤ C8. Then, it yields that +� +Ω +gr(vr)∂vr +∂s ≤ C8 · meas(Ω). +With this in mind, we derive +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 ++ 1 +m +∂ +∂s +� +Ω +� +| ∇vr |2 +r +�m +2 ≤ C9. +(4.10) +Then, +1 +m +∂ +∂s +� +Ω +� +| ∇vr |2 +r +�m +2 ≤ C9 +(4.11) +and, by using Gronwall’s Lemma 3, we get +� +Ω +|∇vr|m ≤ 1 +m +� +Ω +� +| ∇vr |2 +r +�m +2 ≤ C10. +(4.12) +According to Poincar´e’s inequality, it follows that +||vr (s)||W 1,m(Ω) ≤ C(t), +for all s ≥ t. +This, combined with inequality (4.10), yields +� M +t +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 ++ 1 +m +� +Ω +� +| ∇vr(·, M) |2 +r +�m +2 +≤ 1 +m +� +Ω +� +| ∇vr(·, t) |2 +r +�m +2 + C9 (M − t) . +(4.13) +Now, add (4.12) to (4.13), to obtain +� M +t +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 ++ 1 +m +� +Ω +� +| ∇vr(·, M) |2 +r +�m +2 ≤ C(t, M). + +10 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +As a consequence, we have +� M +t +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 +≤ C(t, M). +Since α is a locally Lipschitzian function, then there exists a positive constant L +such that α′ +r ≤ L. Hence, we get +� M +t +� +Ω +�∂αr(vr) +∂s +�2 +≤ L +� M +t +� +Ω +α′ +r(vr) +�∂vr +∂s +�2 +≤ C1(t, M). +The proof is complete. +□ +Theorem 15. Assume that hypotheses (H1)–(H3) hold. Then there exists a weak +bounded solution to problem (3.3). +Proof. To achieve the proof of Theorem 15, we need to pass to the limit in problem +(3.3). By virtue of Lemma 9, there exists a subsequence, still denoted (vr)r, such +that +vr −→ v weakly star in L∞(Q). +Note from estimate (4.2) that +vr −→ v weakly in Lm � +0, M; W 1,m(Ω) +� +. +Since (vr)r is bounded in L∞ � +t, M; W 1,m(Ω) +� +, then +vr −→ v weakly star in L∞ � +t, M; W 1,m +0 +(Ω) +� +. +Under the hypotheses of fr, we have fr −→ f a.e. This, together with Vitali’s +theorem (see [20]), implies the convergence to f(v) in L1(Q). Applying Green’s +formula, +����� +� M +0 +� +Ω +∆r +mvru +����� ≤ +������ +� +Ω +� +| ∇vr |2 +r +�m − 2 +2 +∇vr∇u +������ +, for u ∈ Lm � +0, M; W 1,m +0 +(Ω) +� +. +By using Remark 13, the right-hand side of this inequality is bounded. Then there +exists ϑ ∈ Lm′ � +0, M; W −1,m′(Ω) +� +such that +∆r +mvr −→ ϑ weakly in Lm′ � +0, M; W −1,m′(Ω) +� +. +A classical argument (see [5]), asserts that ϑ = ∆mv. +Combining (4.5) and the smoothness of function αr, yields the boundedness +of the sequence (αr(vr))r in L∞ � +t, M; W 1,m(Ω) +� +. On the other hand, by using +(4.7), we deduce that +�∂αr(vr) +∂s +� +r +is bounded in L2 � +t, M; L2(Ω) +� +, for all t > 0. +Aubin’s lemma (see [25]) allows us to claim that (αr(vr))r is relatively compact in +C +� +]0, M[; L1(Ω) +� +. Therefore, αr(vr) −→ δ strongly in C +� +]0, M[; L1(Ω) +� +. Hence, in +an entirely similar manner as in [5, p. 1048], it can be handled that δ = α(v). For +the continuous of the solution at point s = 0, we proceed as in [3]. From Lemma 14, +we deduce that αr(vr) −→ α(v) strongly in C +� +[0, M]; L1(Ω) +� +. +Let us consider v0 ∈ L∞(Ω) and take a smooth sequence (vr,0) satisfying (3.2). +Hence, (vr,0) is bounded and convergent to v0 in L1(Ω). +Then, thanks to the +dominate convergence theorem, we have α (vr,0) −→ α (v0) in L1(Ω). Now, we deal +with initial data v0 ∈ C1(¯Ω). Choosing the sequence (vr,0) bounded in the space + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +11 +W 1,m(Ω) and verifying hypothesis (3.2), the corresponding α(vr) are continuous at +s = 0. Furthermore, we have +∥α(v(s)) − α(v(0))∥L1(Ω) ≤ ∥α(v(s)) − α (vr(s))∥L1(Ω) + ∥α (vr(s)) − α (vr,0)∥L1(Ω) ++ ∥α (vr,0) − α (v0)∥L1(Ω). +(4.14) +In view of Lemma 16, we have +∥α(v(s)) − α(v(0))∥L1(Ω)≤ eKs ∥α (v0) − α (vr,0)∥L1(Ω) ++ ∥α (vr(s)) − α (vr,0)∥L1(Ω) + ∥α (vr,0) − α (v0)∥L1(Ω). +(4.15) +As s goes to 0 of (4.15), all terms of the right hand side of (4.15) tend to 0. Then, we +deduce that α (v) ∈ C +� +[0, M]; L1(Ω) +� +. Finally, letting r −→ 0 in (3.3), we obtain +the existence of a weak bounded solution. +□ +5. Uniqueness of solution +To prove the uniqueness of the solution, we need to impose some further hypoth- +esis. We assume that there exists a positive constant L2 such that +| f(u) − f(v) |≤ L2 | α(u) − α(v) | . +(5.1) +Lemma 16. Let v and u be two solutions of problem (1.1) with initial data v0 and +u0, respectively. Then, the following inequality holds: +∥α(v(s)) − α(u(s))∥L1(Ω) ≤ eKs ∥α (v0) − α (u0)∥L1(Ω), +(5.2) +where K is a positive constant. +Proof. The proof is similar to the one in [8]. +□ +For the proof of our next result, we need the following lemma. +Lemma 17 (Tartar’s inequality [24]). If a, b ∈ RN, then +� +|a|m−2a − |b|m−2b +� +· (a − b) ≥ C(m) +� +|a − b|m, +if m ≥ 2, +|a−b|2 +(|a|+|b|)2−m , +if 1 < m < 2, +(5.3) +for all m > 1, where C(m) = 22−m when m ≥ 2 and C(m) = m−1 when 1 < m < 2. +Lemma 18. Let us consider two solutions v and u of problem (1.1) with initial +data v0 and u0, respectively, such that v0 = u0. Then, v = u in Q. +Proof. For a small positive µ, let +Hµ(Y ) = min +� +1, max +�Y +µ , 0 +�� +, for all Y ∈ R. +We use Hµ(v − u) as a test function. Multiplying the first equation of problem +(1.1), corresponding to u and v, by Hµ(v − u) and subtracting the two equations, +we derive that +� s +0 +� +Ω +∂ +∂s (α(v) − α(u)) Hµ(v − u) − +� s +0 +� +Ω +(∆mv − ∆mu) Hµ(v − u) += +� s +0 +� +Ω +κ +f(v) +( +� +Ω f(v)dx)2 Hµ(v − u) − +� s +0 +� +Ω +κ +f(u) +( +� +Ω f(u)dx)2 Hµ(v − u). +(5.4) + +12 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +Using Green’s formula and taking into account the boundary conditions, we obtain +that +� s +0 +� +Ω +(∆mv) Hµ(v − u) = − +� s +0 +� +Ω +| ∇v |m−2 ∇v · ∇(v − u) · H′ +µ(v − u). +(5.5) +We easily check that +� s +0 +� +Ω +(∆mu) Hµ(v − u) = − +� s +0 +� +Ω +| ∇u |m−2 ∇u · ∇(v − u) · H′ +µ(v − u). +(5.6) +From (5.5) and (5.6), it follows that +� s +0 +� +Ω +(∆mv − ∆mu) · Hµ(v − u) += − +� s +0 +� +Ω +� +| ∇v |m−2 ∇v− | ∇u |m−2 ∇u +� +∇(v − u) · H′ +µ(v − u). +By using Lemma 17, it follows that +� s +0 +� +Ω +(∆mv − ∆mu) · Hµ(v − u) ≤ 0. +Hence, +� s +0 +� +Ω +∂ +∂s (α(v) − α(u)) Hµ(v − u) ≤ +� s +0 +� +Ω +∂ +∂s (α(v) − α(u)) Hµ(v − u) +− +� s +0 +� +Ω +(∆mv − ∆mu) Hµ(v − u). +(5.7) +Recalling (5.4) and (5.7), we get +� s +0 +� +Ω +∂ +∂s (α(v) − α(u)) · Hµ(v − u) +≤ +� s +0 +� +Ω +γ(x) · Hµ(v − u), +(5.8) +where +γ(x) := κ +f(v) +( +� +Ω f(v)dx)2 − κ +f(u) +( +� +Ω f(u)dx)2 , +γ(x) · χ{v−u>0} = κf(u) +� +Ω [f(u) − f(v)] dx +� +Ω [f(u) + f(v)] dx +�� +Ω f(u)dx +�2 �� +Ω f(v)dx +�2 +· χ{v−u>0} ++ κ f(v) − f(u) +�� +Ω f(v)dx +�2 · χ{v−u>0}. +Adding this to (5.1), +γ(x) · χ{v−u>0} ≤ κL2 +� +Ω (α(v) − α(u)) dx +�� +Ω(f(v) + f(u)) dx +� +�� +Ω f(u)dx +�2 �� +Ω f(v)dx +�2 +f(u) · χ{v−u>0} ++ κL2 +(α(v) − α(u)) +�� +Ω f(v) dx +�2 · χ{v−u>0}. +(5.9) + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +13 +On the other hand, we have +κ · L2 +� s +0 +� +Ω +�� +Ω (α(v) − α(u)) dx +� �� +Ω(f(v) + f(u)) dx +� +�� +Ω f(u)dx +�2 �� +Ω f(v)dx +�2 +f(u) · χ{v−u>0} +≤ 2κ · L2 · meas(Ω) · sup f(a) +a∈supp(f) +� s +0 +� +Ω +�� +Ω (α(v) − α(u)) dx +� +�� +Ω f(u)dx +�2 �� +Ω f(v)dx +�2 f(u) +≤ 2κ · L2 · meas(Ω) · +� +sup f(a) +a∈supp(f) +�2 � s +0 +� +Ω +�� +Ω (α(v) − α(u)) dx +� +�� +Ω f(u)dx +�2 �� +Ω f(v)dx +�2 . +(5.10) +Since +� +Ω +(α(v) − α(u)) dx = +� +Ω +(α(v) − α(u)) · χ{v−u>0} dx ++ +� +Ω +(α(v) − α(u)) · χ{v−u≤0} dx, +and α is an increasing function, we get that +� +Ω +(α(v) − α(u)) dx ≤ +� +Ω +(α(v) − α(u)) · χ{v−u>0} dx ≤ +� +Ω +(α(v) − α(u))+ dx. +(5.11) +Keeping in mind (5.9)–(5.11) and hypothesis (H3) on f, it follows that +� s +0 +� +Ω +γ(x) · χ{v−u>0} dx dt +≤ +κ · L2 +(meas(Ω)σ)2 +� s +0 +� +Ω +(α(v) − α(u))+ dx dt ++ 2κ · L2 · meas(Ω) +(meas(Ω) · σ)4 +· +� +sup f(a) +a∈supp(f) +�2 � s +0 +� +Ω +�� +Ω +(α(v) − α(u))+ dx +� +≤ + + +κL2 +(meas(Ω)σ)2 + 2κ · L2(meas(Ω))2 +(meas(Ω)σ)4 +� +sup f(a) +a∈supp(f) +�2 + +� s +0 +� +Ω +(α(v) − α(u))+ dxdt. +(5.12) +On the another hand, when we tend µ to zero, we get +� s +0 +� +Ω +∂ +∂s (α(v) − α(u)) Hµ(v − u) −→ +� s +0 +� +Ω +∂ +∂s (α(v) − α(u)) · χ{v−u>0}. +We also have that +� s +0 +� +Ω +γ(x) · Hµ(v − u) −→ +� s +0 +� +Ω +γ(x) · χ{v−u>0}. +This, combined with (5.8) and (5.12), yields the existence of a positive constant +C11 such that +� +Ω +(α(v) − α(u))+ ≤ C11 · +� s +0 +� +Ω +(α(v) − α(u))+. +(5.13) +Applying the usual Gronwall’s lemma, we get α(v) ≤ α(u). +Knowing that α is +an increasing function, it follows, in particular, that α(v) = α(u) in {v − u > 0}. +Keeping this and (5.3) in mind, we obtain that ∇(v − u) = 0 in {µ > v − u > 0}. +Hence, max{0, min{v − u, µ}} = C12, where C12 is a positive constant. We deduce + +14 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +that v ≤ u in Q. +Interchanging the role of v and u, the proof of uniqueness is +finished. +□ +6. Existence of an absorbing set and the universal attractor +In this section we prove the existence of an universal attractor by first proving +the existence of an absorbing set. To this end, let us consider (S(s))s≥0 a continuous +semigroup generated by problem (1.1) such that +S(s) : +L∞(Ω) +→ L∞(Ω) +v0 +→ α(v(s)), +(6.1) +where v is the bounded weak solution of problem (1.1). By using Theorem 8, the +map (6.1) is well defined. Now, let us formulate the second main result in this +paper. +Theorem 19. For m > 2, (S(s))s≥0 possesses an universal attractor, which is +bounded in W 1,m +0 +(Ω). +In order to prove Theorem 19, we first show the following result. +Lemma 20. Under assumptions (H1)–(H3), there exists a positive constant ρ such +that +∥ v(s) ∥L∞(Ω)≤ ρ, +for all s > 0. +Proof. Multiplying the first equation of (1.1) by |α(v)|p α(v), and integrating over +Ω, we obtain that +� +Ω +∂α(v) +∂s +|α(v)|p α(v) − +� +Ω +∆mv · |α(v)|p α(v) = κ +� +Ω +f(v) +(� +Ω f(v)dx)2 |α(v)|p α(v). +Then, +1 +p + 2 +∂ +∂s +� +Ω +|α(v)|p+2 − +� +Ω +∆mv · |α(v)|p α(v) = κ +� +Ω +f(v) +( +� +Ω f(v)dx)2 |α(v)|p α(v). +Applying Green’s formula, and using the boundary conditions, we get +1 +p + 2 +∂ +∂s +� +Ω +|α(v)|p+2 + (p + 1) +� +Ω +|∇v|m α′(v) |α(v)|p += κ +� +Ω +f(v) +(� +Ω f(v)dx)2 |α(v)|p α(v). +(6.2) +On the other hand, since α′(v) ≥ λ, we have +� +Ω +|∇v|m α′(v) |α(v)|p ≥ λ +� +Ω +|∇v|m |α(v)|p , +in [0, M]. +Now, we discuss two cases. +Case 1. If |∇v| ≥ |α(v)|, then +� +Ω +|∇v|m α′(v) |α(v)|p ≥ λ +� +Ω +|α(v)|m+p . +(6.3) +Case 2. If |∇(v)| ≤ |α(v)|, we get +� +Ω +|∇v|m α′(v) |α(v)|p ≥ λ +� +Ω +|∇v|m |α(v)|p ≥ λ +� +Ω +|∇v|m+p . + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +15 +By using Poincar´e’s inequality, we derive that +� +Ω +|∇v|m α′(v) |α(v)|p ≥ λ · C13 +� +Ω +|v|m+p, for a positive constant C13. +The smoothness of the function α implies +� +Ω +|∇v|m α′(v) |α(v)|p ≥ λ · C13 +L1 +� +Ω +|α(v)|m+p , +(6.4) +where L1 is the Lipshitzity constant of function α. Recall from (6.2) − (6.4) that +1 +p + 2 +∂ +∂s +� +Ω +|α(v)|p+2 + min +�λ · C13 +L1 +, λ +� +· +� +Ω +|α(v)|m+p +≤ κ +� +Ω +f(v) +�� +Ω f(v)dx +�2 |α(v)|p α(v). +It is easy to check that +1 +p + 2 +∂ +∂s +� +Ω +|α(v)|p+2 + min +�λ · C13 +L1 +, λ +� +· +� +Ω +|α(v)|m+p ≤ C14 +� +Ω +|α(v)|p+1 , +for a positive constant C14. +Set zp(s) :=∥ α(v) ∥Lp+2(Ω) and C15 := min +�λ · C13 +L1 +, λ +� +. Making use of H¨older’s +inequality and the continuous embedding of Lm+p(Ω) in Lp+2(Ω), we obtain that +∂zp(s) +∂s +(zp(s))p+1 + C15 (zp(s))m+p ≤ C14 (zp(s))p+1. +It follows that +∂zp(s) +∂s ++ C15 (zp(s))m−1 ≤ C14. +(6.5) +This puts us in a position to employ Ghidaglia’s Lemma 4, to get +zp(s) ≤ +�C14 +C15 +� +1 +m − 1 + +1 +(C15 (m − 2) s) +1 +m − 2 +:= ρs. +(6.6) +Letting p going to infinity, we obtain that +∥ α(v) ∥L∞(Ω)≤ C(η) +for all s ≥ η > 0. This implies +∥ v(s) ∥L∞(Ω)≤ max +� +| α−1(C(η)) |, | α−1(−C(η)) | +� +. +(6.7) +Let us consider ρ := max +� +| α−1(C(η)) |, | α−1(−C(η)) | +� +as the radius of the ball +centered at 0. This ball is an absorbing set in L∞(Ω). +□ +Remark 21. Existence of an absorbing set in W 1,m(Ω) is obtained due to inequality +(4.5) together with the lower semi-continuity of the norm. It yields that +||v (s)||W 1,m(Ω) ≤ C(t) := ρt, +for all s ≥ t. +Then the ball B (0, ρt) is an absorbing set in W 1,m(Ω). + +16 +M. R. SIDI AMMI, I. DAHI, A. EL HACHIMI, D. F. M. TORRES +Now, in order to prove Lemma 23 below, we show that the solution of problem +(1.1) is H¨older continuous. To this end, we set α(v) := w and we add the following +assumptions: +(H4) α is a strict increasing function and α−1 ∈ C1(R); +(H5) i) +� +α−1(w) +�′ is degenerate in the neighborhood of zero and there exists +z ∈ [−η0, η0], η0 a positive constant, such that +β0 |z|k0 ≤ +� +α−1(w) +�′ ≤ β1 |z|k1 +(6.8) +for positive constants βj and kj, j = 0, 1; +ii) there exists two positive constants e0 and e1 such that +e0 ≤ +� +α−1(w) +�′ ≤ e1, +(6.9) +∂w +∂s − div +���� +� +α−1(w) +�′��� +m−2 +· +� +α−1(w) +�′ |∇w|m−2∇w +� += κ +f(α−1(w)) +�� +Ω f(α−1(w))dx +�2 , +(6.10) +w = 0, +(6.11) +for all z ∈] − ∞, −η0[ +� +]η0, +∞[. +Identifying (6.10) with (1) in the paper [27], and using hypotheses (H3)–(H5), +we can apply the following theorem. +Theorem 22 (See [27]). Suppose that Theorem 8 holds. Then, under assumptions +(H3)–(H5), the solution of problem (1.1) is H¨older continuous. +In the following Lemma we prove that the operator (S(s))s≥0 is uniformly com- +pact for s large enough. +Lemma 23. If B is a bounded set, then +� +s≥s0 +S(s)B +is relatively compact for any s ≥ s0. +Proof. We can derive from Lemma 9 that the set � +s≥s0 S(s)B is bounded in L∞(Ω). +Furthermore, the approximation solution is uniformly bounded. We are in position +to invoke Theorem 22 and, consequently, we deduce, by Ascoli–Arzel`a theorem, +that the set +� +s≥s0 +S(s)B is relatively compact. +□ +Proof of Theorem 19. We have to prove that (S(s))s≥0 related to problem (1.1) +possesses an universal attractor. We consider the following ω-limit: +ω(B0) := {v ∈ L∞(Ω) : ∃sn → +∞, ∃vn ∈ B0 such that S (sn) vn → v in L∞(Ω)}, +where B0 := S(t)B +L∞(Ω) for some t > 0. We apply Lemma 1.1 in [26] to get that +ω(B0) is a nonempty compact invariant set. Then the first condition of Definition 7 +holds. For the second condition of Definition 7, we proceed by absurd. Assume +that A does not attract each bounded set in L∞(Ω). Then there exists a bounded +set B, not attracted by A, and there exists sn → ∞ and ǫ > 0 such that +dist (S(sn)B, A) ≥ ǫ +2, +(6.12) + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +17 +from whence follows that, for every n, there exists dn ∈ B such that +dist (S(sn)dn, A) ≥ ǫ +2. +(6.13) +Knowing that B0 is an absorbing set for B (a bounded set), there exists s such that +s ≥ s1, where s1 is a positive constant, and we have S(s)B ⊂ B0. Since sn → ∞, +then sn ≥ s1 for large enough n and S(sn)B ⊂ B0. As a consequence, we have +S(sn)dn ∈ B0. +(6.14) +On the other hand, recall from Lemma 23 that +� +s≥s0 +S(s)B0 is relatively compact. +Consequently, the sequence (S(sn)dn)n is also relatively compact. So, there exists +a subsequence such that +S(sn)dn −→ ℓ ∈ L∞(Ω), as sn −→ ∞. +With the semi-group propriety, we have +lim +n−→∞ S(sn)dn = +lim +n−→∞ S(sn − s1)S(s1)dn = +lim +n−→∞ S(s′ +n)d′ +n = ℓ, +(6.15) +where s′ +n := sn − s1 and d′ +n := S(s1)dn. We infer that +ω(B0) := {v : ∃sn, dn such that S(sn)dn −→ v}. +(6.16) +In view of the fact that d′ +n ∈ B0, then s′ +n and d′ +n play the role of sn and dn, respec- +tively, in (6.16). Keeping this and (6.15) in mind, we obtain that ℓ ∈ ω(B0) = A. +Then dist (ℓ, A) = 0 < ǫ +2. This is in contradiction with inequality (6.13). Hence, A +is the universal attractor. +□ +7. Conclusions and perspectives +In this paper, we proved existence and uniqueness of a bounded weak solution in +Sobolev spaces for a non-local thermistor problem in the presence of triply nonlinear +terms. We also proved the existence of the global attractor. As future work, we +plan to study the regularity of the global attractor, the stability of the solution, +and the optimal control for the thermistor problem (1.1). +Acknowledgments +Torres was supported by FCT through CIDMA and project UIDB/04106/2020. +References +[1] P. Agarwal, M. R. Sidi Ammi, and J. Asad. Existence and uniqueness results on time scales +for fractional nonlocal thermistor problem in the conformable sense. Advances in Difference +Equations, 2021(1):1–11, 2021. +[2] H. W. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Mathema- +tische Zeitschrift, 183(3):311–341, 1983. +[3] F. Andreu, J. M. Maz´on, F. Simondon, and J. Toledo. Attractor for a degenerate nonlinear +diffusion problem with nonlinear boundary condition. Journal of Dynamics and Differential +Equations, 10(3):347–377, 1998. +[4] S. N. Antontsev and M. Chipot. The thermistor problem: existence, smoothness uniqueness, +blowup. 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The vitali convergence theorem for the vector-valued McShane +integral. Mathematica Bohemica, 129(2):159–176, 2004. +[21] M. R. Sidi Ammi and D. F. M. Torres. Numerical analysis of a nonlocal parabolic prob- +lem resulting from thermistor problem. Math. Comput. Simulation, 77(2-3):291–300, 2008. +arXiv:0709.0129 +[22] M. R. Sidi Ammi and D. F. M. Torres. Optimal control of nonlocal thermistor equations. +Internat. J. Control, 85(11):1789–1801, 2012. arXiv:1206.2873 +[23] M. R. Sidi Ammi and D. F. M. Torres. Galerkin spectral method for the fractional nonlocal +thermistor problem. Comput. Math. Appl., 73(6):1077–1086, 2017. arXiv:1605.07804 +[24] J. Simon. R´egularit´e de la solution d’un probl`eme aux limites non lin´eaires. Ann. Fac. Sci. +Toulouse Math., 3(3-4):247–274, 1981. +[25] J. Simon. Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. (4), 146:65–96, 1987. +[26] R. Temam. Infinite-dimensional dynamical systems in mechanics and physics. Applied Math- +ematical Sciences, 68, 1988. +[27] V. Vespri. On the local behaviour of solutions of a certain class of doubly nonlinear parabolic +equations. Manuscripta Mathematica, 75(1):65–80, 1992. +[28] S. Zhou and D. R. Westbrook. Numerical solutions of the thermistor equations. Journal of +Computational and Applied Mathematics, 79(1):101–118, 1997. +Moulay Rchid Sidi Ammi (corresponding author) +Department of Mathematics, AMNEA Group, MAIS Laboratory, Faculty of Sciences +and Technics, Moulay Ismail B. P. 509, Errachidia, Morocco. +Email address: rachidsidiammi@yahoo.fr + +EXISTENCE RESULT OF THE GLOBAL ATTRACTOR +19 +Ibrahim DAHI +Department of Mathematics, AMNEA Group, MAIS Laboratory, Faculty of Sciences +and Technics, Moulay Ismail B. P. 509, Errachidia, Morocco. +Email address: i.dahi@edu.umi.ac.ma +Abderrahmane El Hachimi +Department of Mathematics, Faculty of Sciences, Mohammed V University of Rabat, +Morocco. +Email address: aelahacimi@yahoo.fr +Delfim F. M. Torres +R&D Unit CIDMA, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, +Portugal. +Email address: delfim@ua.pt + diff --git a/l9FAT4oBgHgl3EQfcB1A/content/tmp_files/load_file.txt b/l9FAT4oBgHgl3EQfcB1A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..530b5663aed989ab7f4980b632aa943e15630b01 --- /dev/null +++ b/l9FAT4oBgHgl3EQfcB1A/content/tmp_files/load_file.txt @@ -0,0 +1,740 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf,len=739 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='08561v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='AP] 8 Jan 2023 EXISTENCE RESULT OF THE GLOBAL ATTRACTOR FOR A TRIPLY NONLINEAR THERMISTOR PROBLEM MOULAY RCHID SIDI AMMI, IBRAHIM DAHI, ABDERRAHMANE EL HACHIMI, AND DELFIM F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We study the existence and uniqueness of a bounded weak solution for a triply nonlinear thermistor problem in Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Furthermore, we prove the existence of an absorbing set and, consequently, the universal attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Introduction The thermistor was discovered by Michael Faraday in 1833, who noticed that the temperature increases when the silver sulfides resistance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' A lot of studies of the thermistor problem can be found in [1, 9, 10, 15, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' A thermistor is a circuit component that may be used as a current limiter or as a temperature sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' It is, typically, a tiny cylinder, constructed of a ceramic substance whose electrical conductivity is highly dependent on temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The thermistor regulates the heat created by an electrical current traveling through a conductor device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Thermistor problems have received a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We refer the reader to [4, 7, 10, 12, 17, 19] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Thermistors are commonly used as temperature control devices in a wide variety of industrial equipment, ranging from space vehicles to air conditioning controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' They are also often used in the medical field, for localized and general body tem- perature measurement, in meteorology, for weather forecasting, and in chemical industries as process temperature sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' A detailed description of thermistors and their applications in electronics and other industries can be found in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' There are two types of thermistors: NTC and PTC, which have a positive and negative temperature coefficient, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' An NTC thermistor is a tempera- ture sensor that measures temperature using the resistance qualities of ceramic and metal composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' NTC sensors provide a number of benefits in terms of tempera- ture sensing, including small size, great long-term stability, and high accuracy and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The operation of a PTC electric surge device is as follows: when the cir- cuit’s current is suddenly increased, the device heats up, causing a dramatic decline in its electrical conductivity, effectively shutting off the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In this paper, we 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 35A01, 35A02, 46E35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Existence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' uniqueness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' thermistor problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Sobolev spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' global attractor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' ω−limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' invariant set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' absobsing set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' semi-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This is a 19 pages preprint of a paper whose final and definite form is published in ’Moroccan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' of Pure and Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (MJPAA)’, ISSN: Online 2351-8227 – Print 2605-6364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES consider the following general nonlocal thermistor problem: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∂α(v) ∂s − ∆mv = κ f(v) ( � Ω f(v)dx)2 , in Q, α(v(x, 0)) = α(v0), in Ω, v = 0, on Γ×]0, M[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) models the diffusion of the temperature produced when an electric current flows crossing a material, where f(v) is the electrical resistance of the conductor and f(v) ( � Ω f(v)dx)2 represents the non-local term of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Here, Q = Ω × [0, M], where Ω is an open bounded subset of RN, N ≥ 1, and M is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) is a generalization of the problem appearing in the work of Kaval- laris and Nadzieja [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For α(v) = v and m = 2, one gets the classical model of the thermistor problem appearing in the work of Lacey [17], which is a transformation of the following problem: ∂v ∂s = ∇ · (κ(v)∇v) + ρ(v)|∇ψ|2, ∇ · (ρ(v)∇ψ) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) where κ is the thermal conductivity, ψ is the electrical potential, and ρ(v) represents the electrical conductivity, which is normally a positive function supposed to drop sharply by several orders of magnitude at some critical temperature, and remains essentially zero for larger temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This feature is essential for the intended functioning of thermistors as thermoelectric switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In the case α(v) = v and m = 2, existence and uniqueness results of bounded weak solutions to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) were established in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Existence of an optimal control has been obtained by many authors with different assumptions on f and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We refer, for instance, to [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' On the other hand, numerical computations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) have been carried out by other authors, see for example [6, 21, 22, 28], in which the chosen parameters correspond to actual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Moreover, a study of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) in the case N = 1 can be found in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Here, we extend the existing literature of the nonlocal thermistor problem to a triply nonlinear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let B be the area of Ω, I the current such that κ = I2/B2, and ∆m be defined by ∆mv = div(| ∇v |m−2 ∇v) ∀m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We further specify the terms in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We assume: (H1) v0 ∈ L∞(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (H2) α : R −→ R is a Lipschitz continuous increasing function such that α(0) = 0 and α′(s) ≥ λ > 0 for all s ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (H3) f is a Lipshitz continuous function, with compact support, verifying σ ≤ f(s), for all s ∈ R, for a positive constant σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In Section 2, we collect some basic concepts and a few known results that are useful to our development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Section 3 is devoted to the existence of a classical solution to the regularized problem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In Section 4, existence of a bounded weak solution to the regularized problem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, in Section 5, we provide sufficient conditions under which the EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 3 solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Existence of an absorbing set, as well as the global attractor, are proved in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Finally, we present some concluding remarks in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Preliminaries In this section we collect a few known results that are useful to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Definition 1 (See [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let α be a continuous increasing function with α(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For s ∈ R we define Ψ (s) = � s 0 α(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The Legendre transform Ψ ∗ of Ψ is defined by Ψ ∗ (t) = sup r∈R {rt − Ψ(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) In particular, we get Ψ ∗ (α(t)) = tα(t) − Ψ (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' If v ∈ L∞(Q), then α(v) ∈ L∞(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' It turns out, from equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2), that Ψ ∗ (α(v)) is also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 3 (See [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Assume that z is a non-negative, absolutely continuous func- tion, satisfying the following inequality: z′(s) ≤ hz(s) + g(s), for s ≥ s0, where h and g are two non-negative integrable functions on [0, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, for each s ∈ [0, M], z(s) ≤ exp �� s 0 h(τ)dτ � � z(0) + � s 0 g(τ)dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 4 (Ghidaglia lemma [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let z be a positive and absolutely continuous function on ]0, ∞[ such that the inequality z′ + δzq ≤ η holds, where q > 1, δ > 0, η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, z(s) ≤ �η δ �1/q + (δ(q − 1)s)−1/(q−1) for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 5 (See [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' If v ∈ Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � with ∂α(v) ∂s ∈ Lm′ � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W −1,m′(Ω)) � , then �∂α(v) ∂s , v � W −1,m′ (Ω),W 1,m(Ω) = d ds � Ω Ψ ∗(α(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In order to study the existence of the global (universal) attractor, we introduce the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Definition 6 (See [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let us consider B ⊂ F and U an open bounded set such that U ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then B is an absorbing set in U if the orbit of each bounded set of U enters into B after a given period of time (which may depend on the set): ∀B0 ⊂ U, B0 bounded, ∃s0 (B0) such that S(s)B0 ⊂ B, ∀s ≥ s0 (B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES Definition 7 (See [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The set A ⊂ F is said to be an universal attractor for the semigroup (S(s))s≥0, if the following conditions hold: (1) A ⊂ F is a nonempty invariant compact set, (2) the set A ⊂ F attracts any bounded set B ⊂ F, that is, dist (S(s)B, A) → 0 as s → +∞, such that dist(D, B) = supa∈D infb∈B ∥a − b∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Regularized problems In this section, we first present our approximation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then we proceed to prove the existence of a weak solution to our regularized problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' To design our regularized scheme, we consider αr is of class C1(R) where 0 < λ < α′ r, αr(0) = 0, αr −→ α in Cloc(R) and |αr| ≤ |α|, fr is of class C∞(R), fr → f, in L1(Q) and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='e in Q, fr satisfies (H3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) The initial condition is regularized as in the proof of [11, Proposition 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 761], that is, vr,0 ∈ C∞ c (Ω) such that vr,0 → v0 in L∞(Ω), ∥vr,0∥L∞(Ω) ≤ ∥v0∥L∞(Ω) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) Our regularized problems are then given by \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∂αr(vr) ∂s − ∆r mv = κ fr(vr) (� Ω fr(vr)dx)2 , in Q = Ω × [0, M], αr(vr,x(0)) = αr(vr,0), in Ω, vr = 0, on Γ×]0, M[, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) where ∆r mv = div \uf8eb \uf8ed� | ∇v |2 +r �m − 2 2 ∇v \uf8f6 \uf8f8, m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Assume that hypotheses (H1)–(H3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then there exists a solution to problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The following lemma plays a key role in the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For all r > 0, we have ∥ vr ∥L∞(Q)≤ C(M, ∥ v0 ∥L∞(Ω)), where C(M, ∥ v0 ∥L∞(Ω)) is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Multiplying the first equation of problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) by � (αr(vr) − αr(s0))+�p+1 (s0 is a positive constant where | vr |> s0) and integrating over Ω, we get � Ω ∂αr(vr) ∂s � (αr(vr) − αr(s0))+�p+1 − � Ω ∆r mvr � (αr(vr) − αr(s0))+�p+1 = � Ω κ · fr(vr) �� Ω fr(vr)dx �2 � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' So, we have EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 5 1 p + 2 � Ω ∂ ∂s � (αr(vr) − αr(s0))+�p+2 = � Ω ∆r mvr � (αr(vr) − αr(s0))+�p+1 + � Ω κ · fr(vr) �� Ω fr(vr)dx �2 � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, 1 p + 2 ∂ ∂s � Ω � (αr(vr) − αr(s0))+�p+2 = � Ω ∆r mvr � (αr(vr) − αr(s0))+�p+1 + � Ω κ · fr(vr) �� Ω fr(vr)dx �2 � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='4) On the other hand, we have � Ω ∆r mvr � (αr(vr) − αr(s0))+�p+1 = � Ω div \uf8eb \uf8ed� | ∇vr |2 +r �m − 2 2 ∇vr \uf8f6 \uf8f8 � (αr(vr) − αr(s0))+�p+1 = −(p + 1) � Ω \uf8eb \uf8ed� | ∇vr |2 +r �m − 2 2 | ∇vr |2 \uf8f6 \uf8f8 α′ r(vr) � (αr(vr) − αr(s0))+�p + � ∂Ω \uf8eb \uf8ed� | ∇vr |2 +r �m − 2 2 ∂vr ∂ν \uf8f6 \uf8f8 � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Since � | ∇vr |2 +r �m − 2 2 | ∇vr |2≥ 0 and α′ r > 0, we get 1 p + 2 ∂ ∂s � Ω � (αr(vr) − αr(s0))+�p+2 ≤ � ∂Ω \uf8eb \uf8ed(| ∇vr |2 +r) m − 2 2 ∂vr ∂ν \uf8f6 \uf8f8 � (αr(vr) − αr(s0))+�p+1 + � Ω κ · fr(v) �� Ω fr(v)dx �2 � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) By using (H3), we have � Ω κ · fr(vr) (� Ω fr(vr)dx)2 � (αr(vr) − αr(s0))+�p+1 ≤ κ (σ · meas(Ω))2 � Ω fr(vr) � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Since fr satisfies (H3), it yields fr (vr(x, t)) = fr (vr(x, t)) χ{vr(x,t)∈supp(f)} + fr (vr(x, t)) χ{vr(x,t)/∈supp(f)} ≤ fr (vr(x, t)) χ{vr(x,t)∈supp(f)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES If vr(x, t) ∈ supp(f), then it follows that (vr(x, t))r is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Thus, there exists a positive constant C0 such that � Ω fr(vr) � (αr(vr) − αr(s0))+�p+1 ≤ C0 � Ω � (αr(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Keeping that in mind, we have for a positive constant C1 that 1 p + 2 ∂ ∂s � Ω � (αr(vr) − αr(s0))+�p+2 ≤ C1 � Ω � (α(vr) − αr(s0))+�p+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='6) From H¨older’s inequality, there exists positive constants Cj, j = 2, 3, 4, such that � Ω � (αr(vr) − αr(s0))+�p+1 ≤ (meas(Ω)) 1 p + 1 · �� Ω � (αr(vr) − αr(s0))+�p+2 �p + 1 p + 2 ≤ C2 [zp(s)]p+1, where zp(s) :=∥ (αr(vr) − αr(s0))+ ∥Lp+2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='6), we have 1 p + 2 ∂ ∂s � Ω � (αr(vr) − αr(s0))+�p+2 ≤ C3 [zp(s)]p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, 1 p + 2 ∂ ∂s [zp(s)]p+2 ≤ C3 [zp(s)]p+1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='7) and hence ∂ ∂s [zp(s)] ≤ C3, from which it follows that [zp(s) − zp(0)] ≤ C3M, which implies zp(s) ≤ zp(0) + C3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Letting p go to infinity, we obtain that ∥ (αr(vr) − αr(s0))+ ∥L∞(Ω)≤ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) Now, let ur = −vr, and consider the following problem: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ∂ ˜αr(ur) ∂s − ∆r mur = κ ˜fr(ur) �� Ω ˜fr(vr)dx �2 =: ˜g(ur) in Q, ˜αr(ux,r(0)) = ˜αr(u0) in Ω, ur = 0 on Γ×]0, M[, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='9) where ˜αr(τ) = −αr(−τ), ˜gr(τ) = −gr(−τ) and ˜fr(τ) = −fr(−τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Those functions satisfy the same conditions verified by α, g and f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The same reasoning done to get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8), shows that ∥ (˜αr(ur) − ˜αr(s0))+ ∥L∞(Ω)≤ C5, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10) which is equivalent to ∥ (−αr(−vr(s)) + αr(−s0))+ ∥L∞(Ω)≤ C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10), we deduce that there exists a positive constant C such that ∥ vr(s) ∥L∞(Ω)≤ C(M, ∥ v0 ∥L∞(Ω)), for all s ∈ [0, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 7 The lemma is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' From Lemma 9 and hypotheses (H1)–(H3), we conclude, from the classical results of Ladyzenskaya (see [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 457–459]), with the existence of a classical solution to the regularized problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Existence of a weak solution Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We say that v ∈ L∞(Q) ∩ Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � ∩ L∞ � t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � , t > 0, is a bounded weak solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1), if it satisfies the following iden- tity: � M 0 �∂α(v) ∂s , u � − � Q | ∇v |m−2 ∇v∇u = κ � Q f(v) (� Ω f(v)dx)2 u, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) for all u ∈ � Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � ∩ L∞(Q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Furthermore, if we have u ∈ � W 1,1 � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' L1(Ω) � ∩ Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) �� with u(·, M) = 0, then � M 0 �∂α(v) ∂s , u � = − � M 0 � Ω [α(v) − α(v0)] ∂su, where the duality product is defined by ⟨·, ·⟩ = ⟨·, ·⟩W −1,m′ (Ω),W 1,m(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Since αr is an increasing function and | αr |≤| α |, then, by using Lemma 9, we also have that (αr(vr))r is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Our plan is to derive now enough a priori estimates needed in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For all r > 0, we have ||vr||Lm(0,M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='W 1,m(Ω)) ≤ C6, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) where C6 is a positive constant independent of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Multiplying the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) by vr and integrating, we get � Ω ∂αr(vr) ∂s vr − � Ω ∆r mvrvr = � Ω κ fr(vr) ( � Ω fr(vr)dx)2 vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) Applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2), we obtain that � Ω ∂αr(vr) ∂s vr = � Ω ∂ [Ψ ∗ (αr(vr))] ∂s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' On another hand, by using Green’s formula, we get � Ω ∆r mvrvr = − � Ω � | ∇vr |2 +r �m − 2 2 ∇vr∇vr + � ∂Ω � | ∇vr |2 +r � ∂vr ∂ν · vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Substituting into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3), we get � Ω ∂αr(vr) ∂s vr + � Ω � | ∇vr |2 +r �m − 2 2 | ∇vr |2 = � Ω κ · fr(vr) ( � Ω fr(vr)dx)2 vr − � ∂Ω � | ∇vr |m−2 +r � ∂vr ∂ν · vr, 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES � Ω � | ∇vr |2 +r �m − 2 2 | ∇vr |2 = � Ω κ · fr(vr) (� Ω fr(vr)dx)2 vr − � Ω ∂ [Ψ ∗ (αr(vr))] ∂s − � ∂Ω � | ∇vr |m−2 +r � ∂vr ∂ν · vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, using the boundary conditions, we have � M 0 � Ω � | ∇vr |2 +r �m − 2 2 | ∇vr |2 = � M 0 � Ω κ · fr(vr) ( � Ω fr(vr)dx)2 vr − � M 0 � Ω ∂ [Ψ ∗ (αr(vr))] ∂s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='4) From Remark 2, we know that (Ψ ∗ (αr(vr)))r is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' With the aid of hypoth- esis (H3) and Lemma 9, there exists a positive constant C7 such that � M 0 � Ω κ fr(vr) ( � Ω fr(vr)dx)2 vr − � M 0 � Ω ∂ [Ψ ∗ (αr(vr))] ∂s ≤ � M 0 � Ω κ fr(vr) ( � Ω fr(vr)dx)2 vr − � Ω Ψ ∗ (αr(vr(·, M))) + � Ω Ψ ∗ (αr(vr(·, 0))) ≤ κ (σ · meas(Ω))2 � M 0 � Ω fr(vr)· | vr | + 2 · max ����� � Ω Ψ ∗ (αr(vr(·, M))) ���� , ���� � Ω Ψ ∗ (αr(vr(·, 0))) ���� � ≤ C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' It yields that � M 0 � Ω |∇vr|m ≤ � M 0 � Ω � | ∇vr |2 +r �m − 2 2 | ∇vr |2≤ C7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We deduce that vr ∈ Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2), combined with Young’s inequality, imply that \uf8eb \uf8ed� | ∇vr |2 +r �m − 2 2 ∇vr \uf8f6 \uf8f8 r is bounded in Lm′ � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m′(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' A further upper bound for vr is established in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For all r, s > 0, there exist positive constants C(t), C(t, M), and C1(t, M), such that the following inequalities hold: ||vr (s)||W 1,m(Ω) ≤ C(t), for all s ≥ t, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) � M t � Ω α′ r(vr) �∂vr ∂s �2 ≤ C(t, M), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='6) � M t � Ω �∂αr(vr) ∂s �2 ≤ C1(t, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='7) EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Multiplying the first equation of problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) by ∂vr ∂s , and integrating, we obtain that � Ω ∂αr(vr) ∂s ∂vr ∂s − � Ω ∆r mvr ∂vr ∂s = � Ω κ fr(vr) ( � Ω fr(vr)dx)2 ∂vr ∂s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) Since � Ω ∂αr(vr) ∂s ∂vr ∂s = � Ω α′ r(vr) �∂vr ∂s �2 , the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) becomes � Ω α′ r(vr) �∂vr ∂s �2 − � Ω ∆r mvr ∂vr ∂s = � Ω κ fr(vr) ( � Ω fr(vr)dx)2 ∂vr ∂s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' By applying Green’s formula, we get � Ω α′ r(vr) �∂vr ∂s �2 + 1 m ∂ ∂s � Ω � | ∇vr |2 +r �m 2 = � Ω κ fr(vr) ( � Ω fr(vr)dx)2 ∂vr ∂s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='9) Let Gr(vr) := � vr 0 gr(s)ds and gr(s) := fr(s) ( � Ω fr(s)dx)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' By using the boundedness of vr and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1), we have ∂Gr(vr) ∂s ≤ C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, it yields that � Ω gr(vr)∂vr ∂s ≤ C8 · meas(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' With this in mind, we derive � Ω α′ r(vr) �∂vr ∂s �2 + 1 m ∂ ∂s � Ω � | ∇vr |2 +r �m 2 ≤ C9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10) Then, 1 m ∂ ∂s � Ω � | ∇vr |2 +r �m 2 ≤ C9 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='11) and, by using Gronwall’s Lemma 3, we get � Ω |∇vr|m ≤ 1 m � Ω � | ∇vr |2 +r �m 2 ≤ C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='12) According to Poincar´e’s inequality, it follows that ||vr (s)||W 1,m(Ω) ≤ C(t), for all s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This, combined with inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10), yields � M t � Ω α′ r(vr) �∂vr ∂s �2 + 1 m � Ω � | ∇vr(·, M) |2 +r �m 2 ≤ 1 m � Ω � | ∇vr(·, t) |2 +r �m 2 + C9 (M − t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='13) Now, add (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='12) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='13), to obtain � M t � Ω α′ r(vr) �∂vr ∂s �2 + 1 m � Ω � | ∇vr(·, M) |2 +r �m 2 ≤ C(t, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES As a consequence, we have � M t � Ω α′ r(vr) �∂vr ∂s �2 ≤ C(t, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Since α is a locally Lipschitzian function, then there exists a positive constant L such that α′ r ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Hence, we get � M t � Ω �∂αr(vr) ∂s �2 ≤ L � M t � Ω α′ r(vr) �∂vr ∂s �2 ≤ C1(t, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Assume that hypotheses (H1)–(H3) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then there exists a weak bounded solution to problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' To achieve the proof of Theorem 15, we need to pass to the limit in problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' By virtue of Lemma 9, there exists a subsequence, still denoted (vr)r, such that vr −→ v weakly star in L∞(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Note from estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) that vr −→ v weakly in Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Since (vr)r is bounded in L∞ � t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � , then vr −→ v weakly star in L∞ � t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m 0 (Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Under the hypotheses of fr, we have fr −→ f a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This, together with Vitali’s theorem (see [20]), implies the convergence to f(v) in L1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Applying Green’s formula, ����� � M 0 � Ω ∆r mvru ����� ≤ ������ � Ω � | ∇vr |2 +r �m − 2 2 ∇vr∇u ������ , for u ∈ Lm � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m 0 (Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' By using Remark 13, the right-hand side of this inequality is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then there exists ϑ ∈ Lm′ � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W −1,m′(Ω) � such that ∆r mvr −→ ϑ weakly in Lm′ � 0, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W −1,m′(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' A classical argument (see [5]), asserts that ϑ = ∆mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) and the smoothness of function αr, yields the boundedness of the sequence (αr(vr))r in L∞ � t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' W 1,m(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' On the other hand, by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='7), we deduce that �∂αr(vr) ∂s � r is bounded in L2 � t, M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' L2(Ω) � , for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Aubin’s lemma (see [25]) allows us to claim that (αr(vr))r is relatively compact in C � ]0, M[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' L1(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Therefore, αr(vr) −→ δ strongly in C � ]0, M[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' L1(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Hence, in an entirely similar manner as in [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 1048], it can be handled that δ = α(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For the continuous of the solution at point s = 0, we proceed as in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' From Lemma 14, we deduce that αr(vr) −→ α(v) strongly in C � [0, M];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' L1(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let us consider v0 ∈ L∞(Ω) and take a smooth sequence (vr,0) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Hence, (vr,0) is bounded and convergent to v0 in L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, thanks to the dominate convergence theorem, we have α (vr,0) −→ α (v0) in L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Now, we deal with initial data v0 ∈ C1(¯Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Choosing the sequence (vr,0) bounded in the space EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 11 W 1,m(Ω) and verifying hypothesis (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2), the corresponding α(vr) are continuous at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Furthermore, we have ∥α(v(s)) − α(v(0))∥L1(Ω) ≤ ∥α(v(s)) − α (vr(s))∥L1(Ω) + ∥α (vr(s)) − α (vr,0)∥L1(Ω) + ∥α (vr,0) − α (v0)∥L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='14) In view of Lemma 16, we have ∥α(v(s)) − α(v(0))∥L1(Ω)≤ eKs ∥α (v0) − α (vr,0)∥L1(Ω) + ∥α (vr(s)) − α (vr,0)∥L1(Ω) + ∥α (vr,0) − α (v0)∥L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='15) As s goes to 0 of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='15), all terms of the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='15) tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, we deduce that α (v) ∈ C � [0, M];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' L1(Ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Finally, letting r −→ 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3), we obtain the existence of a weak bounded solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Uniqueness of solution To prove the uniqueness of the solution, we need to impose some further hypoth- esis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We assume that there exists a positive constant L2 such that | f(u) − f(v) |≤ L2 | α(u) − α(v) | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let v and u be two solutions of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) with initial data v0 and u0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, the following inequality holds: ∥α(v(s)) − α(u(s))∥L1(Ω) ≤ eKs ∥α (v0) − α (u0)∥L1(Ω), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) where K is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The proof is similar to the one in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ For the proof of our next result, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 17 (Tartar’s inequality [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' If a, b ∈ RN, then � |a|m−2a − |b|m−2b � (a − b) ≥ C(m) � |a − b|m, if m ≥ 2, |a−b|2 (|a|+|b|)2−m , if 1 < m < 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) for all m > 1, where C(m) = 22−m when m ≥ 2 and C(m) = m−1 when 1 < m < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Let us consider two solutions v and u of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) with initial data v0 and u0, respectively, such that v0 = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, v = u in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For a small positive µ, let Hµ(Y ) = min � 1, max �Y µ , 0 �� , for all Y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We use Hµ(v − u) as a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Multiplying the first equation of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1), corresponding to u and v, by Hµ(v − u) and subtracting the two equations, we derive that � s 0 � Ω ∂ ∂s (α(v) − α(u)) Hµ(v − u) − � s 0 � Ω (∆mv − ∆mu) Hµ(v − u) = � s 0 � Ω κ f(v) ( � Ω f(v)dx)2 Hµ(v − u) − � s 0 � Ω κ f(u) ( � Ω f(u)dx)2 Hµ(v − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='4) 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES Using Green’s formula and taking into account the boundary conditions, we obtain that � s 0 � Ω (∆mv) Hµ(v − u) = − � s 0 � Ω | ∇v |m−2 ∇v · ∇(v − u) · H′ µ(v − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) We easily check that � s 0 � Ω (∆mu) Hµ(v − u) = − � s 0 � Ω | ∇u |m−2 ∇u · ∇(v − u) · H′ µ(v − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='6) From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='6), it follows that � s 0 � Ω (∆mv − ∆mu) · Hµ(v − u) = − � s 0 � Ω � | ∇v |m−2 ∇v− | ∇u |m−2 ∇u � ∇(v − u) · H′ µ(v − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' By using Lemma 17, it follows that � s 0 � Ω (∆mv − ∆mu) · Hµ(v − u) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Hence, � s 0 � Ω ∂ ∂s (α(v) − α(u)) Hµ(v − u) ≤ � s 0 � Ω ∂ ∂s (α(v) − α(u)) Hµ(v − u) − � s 0 � Ω (∆mv − ∆mu) Hµ(v − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='7) Recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='7), we get � s 0 � Ω ∂ ∂s (α(v) − α(u)) · Hµ(v − u) ≤ � s 0 � Ω γ(x) · Hµ(v − u), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) where γ(x) := κ f(v) ( � Ω f(v)dx)2 − κ f(u) ( � Ω f(u)dx)2 , γ(x) · χ{v−u>0} = κf(u) � Ω [f(u) − f(v)] dx � Ω [f(u) + f(v)] dx �� Ω f(u)dx �2 �� Ω f(v)dx �2 χ{v−u>0} + κ f(v) − f(u) �� Ω f(v)dx �2 · χ{v−u>0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Adding this to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1), γ(x) · χ{v−u>0} ≤ κL2 � Ω (α(v) − α(u)) dx �� Ω(f(v) + f(u)) dx � �� Ω f(u)dx �2 �� Ω f(v)dx �2 f(u) · χ{v−u>0} + κL2 (α(v) − α(u)) �� Ω f(v) dx �2 · χ{v−u>0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='9) EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 13 On the other hand, we have κ · L2 � s 0 � Ω �� Ω (α(v) − α(u)) dx � �� Ω(f(v) + f(u)) dx � �� Ω f(u)dx �2 �� Ω f(v)dx �2 f(u) · χ{v−u>0} ≤ 2κ · L2 · meas(Ω) · sup f(a) a∈supp(f) � s 0 � Ω �� Ω (α(v) − α(u)) dx � �� Ω f(u)dx �2 �� Ω f(v)dx �2 f(u) ≤ 2κ · L2 · meas(Ω) · � sup f(a) a∈supp(f) �2 � s 0 � Ω �� Ω (α(v) − α(u)) dx � �� Ω f(u)dx �2 �� Ω f(v)dx �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10) Since � Ω (α(v) − α(u)) dx = � Ω (α(v) − α(u)) · χ{v−u>0} dx + � Ω (α(v) − α(u)) · χ{v−u≤0} dx, and α is an increasing function, we get that � Ω (α(v) − α(u)) dx ≤ � Ω (α(v) − α(u)) · χ{v−u>0} dx ≤ � Ω (α(v) − α(u))+ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='11) Keeping in mind (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='9)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='11) and hypothesis (H3) on f, it follows that � s 0 � Ω γ(x) · χ{v−u>0} dx dt ≤ κ · L2 (meas(Ω)σ)2 � s 0 � Ω (α(v) − α(u))+ dx dt + 2κ · L2 · meas(Ω) (meas(Ω) · σ)4 � sup f(a) a∈supp(f) �2 � s 0 � Ω �� Ω (α(v) − α(u))+ dx � ≤ \uf8eb \uf8ed κL2 (meas(Ω)σ)2 + 2κ · L2(meas(Ω))2 (meas(Ω)σ)4 � sup f(a) a∈supp(f) �2\uf8f6 \uf8f8 � s 0 � Ω (α(v) − α(u))+ dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='12) On the another hand, when we tend µ to zero, we get � s 0 � Ω ∂ ∂s (α(v) − α(u)) Hµ(v − u) −→ � s 0 � Ω ∂ ∂s (α(v) − α(u)) · χ{v−u>0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We also have that � s 0 � Ω γ(x) · Hµ(v − u) −→ � s 0 � Ω γ(x) · χ{v−u>0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This, combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='12), yields the existence of a positive constant C11 such that � Ω (α(v) − α(u))+ ≤ C11 · � s 0 � Ω (α(v) − α(u))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='13) Applying the usual Gronwall’s lemma, we get α(v) ≤ α(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Knowing that α is an increasing function, it follows, in particular, that α(v) = α(u) in {v − u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Keeping this and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) in mind, we obtain that ∇(v − u) = 0 in {µ > v − u > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Hence, max{0, min{v − u, µ}} = C12, where C12 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We deduce 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES that v ≤ u in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Interchanging the role of v and u, the proof of uniqueness is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Existence of an absorbing set and the universal attractor In this section we prove the existence of an universal attractor by first proving the existence of an absorbing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' To this end, let us consider (S(s))s≥0 a continuous semigroup generated by problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) such that S(s) : L∞(Ω) → L∞(Ω) v0 → α(v(s)), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) where v is the bounded weak solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' By using Theorem 8, the map (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Now, let us formulate the second main result in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For m > 2, (S(s))s≥0 possesses an universal attractor, which is bounded in W 1,m 0 (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In order to prove Theorem 19, we first show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Under assumptions (H1)–(H3), there exists a positive constant ρ such that ∥ v(s) ∥L∞(Ω)≤ ρ, for all s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Multiplying the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) by |α(v)|p α(v), and integrating over Ω, we obtain that � Ω ∂α(v) ∂s |α(v)|p α(v) − � Ω ∆mv · |α(v)|p α(v) = κ � Ω f(v) (� Ω f(v)dx)2 |α(v)|p α(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, 1 p + 2 ∂ ∂s � Ω |α(v)|p+2 − � Ω ∆mv · |α(v)|p α(v) = κ � Ω f(v) ( � Ω f(v)dx)2 |α(v)|p α(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Applying Green’s formula, and using the boundary conditions, we get 1 p + 2 ∂ ∂s � Ω |α(v)|p+2 + (p + 1) � Ω |∇v|m α′(v) |α(v)|p = κ � Ω f(v) (� Ω f(v)dx)2 |α(v)|p α(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) On the other hand, since α′(v) ≥ λ, we have � Ω |∇v|m α′(v) |α(v)|p ≥ λ � Ω |∇v|m |α(v)|p , in [0, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Now, we discuss two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' If |∇v| ≥ |α(v)|, then � Ω |∇v|m α′(v) |α(v)|p ≥ λ � Ω |α(v)|m+p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='3) Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' If |∇(v)| ≤ |α(v)|, we get � Ω |∇v|m α′(v) |α(v)|p ≥ λ � Ω |∇v|m |α(v)|p ≥ λ � Ω |∇v|m+p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 15 By using Poincar´e’s inequality, we derive that � Ω |∇v|m α′(v) |α(v)|p ≥ λ · C13 � Ω |v|m+p, for a positive constant C13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' The smoothness of the function α implies � Ω |∇v|m α′(v) |α(v)|p ≥ λ · C13 L1 � Ω |α(v)|m+p , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='4) where L1 is the Lipshitzity constant of function α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Recall from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='2) − (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='4) that 1 p + 2 ∂ ∂s � Ω |α(v)|p+2 + min �λ · C13 L1 , λ � � Ω |α(v)|m+p ≤ κ � Ω f(v) �� Ω f(v)dx �2 |α(v)|p α(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' It is easy to check that 1 p + 2 ∂ ∂s � Ω |α(v)|p+2 + min �λ · C13 L1 , λ � � Ω |α(v)|m+p ≤ C14 � Ω |α(v)|p+1 , for a positive constant C14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Set zp(s) :=∥ α(v) ∥Lp+2(Ω) and C15 := min �λ · C13 L1 , λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Making use of H¨older’s inequality and the continuous embedding of Lm+p(Ω) in Lp+2(Ω), we obtain that ∂zp(s) ∂s (zp(s))p+1 + C15 (zp(s))m+p ≤ C14 (zp(s))p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' It follows that ∂zp(s) ∂s + C15 (zp(s))m−1 ≤ C14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) This puts us in a position to employ Ghidaglia’s Lemma 4, to get zp(s) ≤ �C14 C15 � 1 m − 1 + 1 (C15 (m − 2) s) 1 m − 2 := ρs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='6) Letting p going to infinity, we obtain that ∥ α(v) ∥L∞(Ω)≤ C(η) for all s ≥ η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This implies ∥ v(s) ∥L∞(Ω)≤ max � | α−1(C(η)) |, | α−1(−C(η)) | � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='7) Let us consider ρ := max � | α−1(C(η)) |, | α−1(−C(η)) | � as the radius of the ball centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This ball is an absorbing set in L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ Remark 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Existence of an absorbing set in W 1,m(Ω) is obtained due to inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='5) together with the lower semi-continuity of the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' It yields that ||v (s)||W 1,m(Ω) ≤ C(t) := ρt, for all s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then the ball B (0, ρt) is an absorbing set in W 1,m(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' SIDI AMMI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' DAHI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' EL HACHIMI, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' TORRES Now, in order to prove Lemma 23 below, we show that the solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) is H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' To this end, we set α(v) := w and we add the following assumptions: (H4) α is a strict increasing function and α−1 ∈ C1(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (H5) i) � α−1(w) �′ is degenerate in the neighborhood of zero and there exists z ∈ [−η0, η0], η0 a positive constant, such that β0 |z|k0 ≤ � α−1(w) �′ ≤ β1 |z|k1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='8) for positive constants βj and kj, j = 0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' ii) there exists two positive constants e0 and e1 such that e0 ≤ � α−1(w) �′ ≤ e1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='9) ∂w ∂s − div ���� � α−1(w) �′��� m−2 � α−1(w) �′ |∇w|m−2∇w � = κ f(α−1(w)) �� Ω f(α−1(w))dx �2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10) w = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='11) for all z ∈] − ∞, −η0[ � ]η0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Identifying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='10) with (1) in the paper [27], and using hypotheses (H3)–(H5), we can apply the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Theorem 22 (See [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Suppose that Theorem 8 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then, under assumptions (H3)–(H5), the solution of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) is H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' In the following Lemma we prove that the operator (S(s))s≥0 is uniformly com- pact for s large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' If B is a bounded set, then � s≥s0 S(s)B is relatively compact for any s ≥ s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We can derive from Lemma 9 that the set � s≥s0 S(s)B is bounded in L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Furthermore, the approximation solution is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We are in position to invoke Theorem 22 and, consequently, we deduce, by Ascoli–Arzel`a theorem, that the set � s≥s0 S(s)B is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ Proof of Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We have to prove that (S(s))s≥0 related to problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1) possesses an universal attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We consider the following ω-limit: ω(B0) := {v ∈ L∞(Ω) : ∃sn → +∞, ∃vn ∈ B0 such that S (sn) vn → v in L∞(Ω)}, where B0 := S(t)B L∞(Ω) for some t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We apply Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1 in [26] to get that ω(B0) is a nonempty compact invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then the first condition of Definition 7 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' For the second condition of Definition 7, we proceed by absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Assume that A does not attract each bounded set in L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then there exists a bounded set B, not attracted by A, and there exists sn → ∞ and ǫ > 0 such that dist (S(sn)B, A) ≥ ǫ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='12) EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 17 from whence follows that, for every n, there exists dn ∈ B such that dist (S(sn)dn, A) ≥ ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='13) Knowing that B0 is an absorbing set for B (a bounded set), there exists s such that s ≥ s1, where s1 is a positive constant, and we have S(s)B ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Since sn → ∞, then sn ≥ s1 for large enough n and S(sn)B ⊂ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' As a consequence, we have S(sn)dn ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='14) On the other hand, recall from Lemma 23 that � s≥s0 S(s)B0 is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Consequently, the sequence (S(sn)dn)n is also relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' So, there exists a subsequence such that S(sn)dn −→ ℓ ∈ L∞(Ω), as sn −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' With the semi-group propriety, we have lim n−→∞ S(sn)dn = lim n−→∞ S(sn − s1)S(s1)dn = lim n−→∞ S(s′ n)d′ n = ℓ, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='15) where s′ n := sn − s1 and d′ n := S(s1)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We infer that ω(B0) := {v : ∃sn, dn such that S(sn)dn −→ v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='16) In view of the fact that d′ n ∈ B0, then s′ n and d′ n play the role of sn and dn, respec- tively, in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Keeping this and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='15) in mind, we obtain that ℓ ∈ ω(B0) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Then dist (ℓ, A) = 0 < ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' This is in contradiction with inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Hence, A is the universal attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Conclusions and perspectives In this paper, we proved existence and uniqueness of a bounded weak solution in Sobolev spaces for a non-local thermistor problem in the presence of triply nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' We also proved the existence of the global attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' As future work, we plan to study the regularity of the global attractor, the stability of the solution, and the optimal control for the thermistor problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Acknowledgments Torres was supported by FCT through CIDMA and project UIDB/04106/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' References [1] P.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Vespri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Manuscripta Mathematica, 75(1):65–80, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Zhou and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Westbrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Numerical solutions of the thermistor equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Journal of Computational and Applied Mathematics, 79(1):101–118, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Moulay Rchid Sidi Ammi (corresponding author) Department of Mathematics, AMNEA Group, MAIS Laboratory, Faculty of Sciences and Technics, Moulay Ismail B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 509, Errachidia, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Email address: rachidsidiammi@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='fr EXISTENCE RESULT OF THE GLOBAL ATTRACTOR 19 Ibrahim DAHI Department of Mathematics, AMNEA Group, MAIS Laboratory, Faculty of Sciences and Technics, Moulay Ismail B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' 509, Errachidia, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Email address: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='dahi@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='umi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='ma Abderrahmane El Hachimi Department of Mathematics, Faculty of Sciences, Mohammed V University of Rabat, Morocco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Email address: aelahacimi@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='fr Delfim F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Torres R&D Unit CIDMA, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content=' Email address: delfim@ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} +page_content='pt' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FAT4oBgHgl3EQfcB1A/content/2301.08561v1.pdf'} diff --git a/ldFRT4oBgHgl3EQfYzfm/content/tmp_files/2301.13551v1.pdf.txt b/ldFRT4oBgHgl3EQfYzfm/content/tmp_files/2301.13551v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a0f26925c033dba59198d5080e21b37e77ac066 --- /dev/null +++ b/ldFRT4oBgHgl3EQfYzfm/content/tmp_files/2301.13551v1.pdf.txt @@ -0,0 +1,926 @@ +arXiv:2301.13551v1 [cs.PL] 31 Jan 2023 +Designing text representations for existing data +using the TextFormats Specification Language +Giorgio Gonnella1,2 +� +1Center for Bioinformatics (ZBH), Universität Hamburg, Bundesstrasse 43, 20146 Hamburg +2Institute for Microbiology and Genetics, Georg-August-Universität Göttingen, Goldschmidtstr. 1, 37077 Göttingen +TextFormats is a software system for efficient and user-friendly creation of text format specifications, accessible from multiple pro- +gramming languages (C/C++, Python, Nim) and the Unix command line. To work with a format, a specification written in the +TextFormats Specification Language (TFSL) must be created. The specification defines datatypes for each part of the format. +The syntax for datatype definitions in TextFormats specifications is based on the text representation. Thus this system is well suited +for the description of existing formats. However, when creating a new text format for representing existing data, the user may use +different possible definitions, based on the type of value and the representation choices. +This study explores the possible definition syntax in the TextFormats Specification Language to be used for creating text represen- +tations of scalar values (e.g. string, numeric value, boolean) and compound data structures (e.g. array, mapping). The results of the +analysis are presented systematically, together with examples for each each type of different values that can be represented, and usage +advices. +TextFormats | Datatype definition | Parser | Text representation | Data format | Domain specific language | Software library | Format specification | File format +| Data type | Text format | Tutorial +Correspondence: giorgio.gonnella@uni-goettingen.de +TextFormats (Gonnella, 2022) is a software library and a set +of tools for defining text formats. Although it was initially de- +veloped for the representation of bioinformatics formats, it is +a generic software which can be applied to a variety of fields. +Once a format has been defined using the domain-specific +language TFSL (TextFormats Specification Language), the +TextFormats specification can be used for parsing the for- +mat, as well as writing data in the format, using the APIs +for several programming languages (Nim, Python, C/C++) or +in scripts, using the provided Unix command-line tools. +In TFSL, various kinds of definitions are used to describe +different types of data textual representations, as shown in +Table 1. These definitions are dependent not only on the type +of value but also on other characteristics of the data and its +representation. For example they depend on the set of possi- +ble values: e.g. for a string, one uses constant for a single +value, values for a small set of values and regex for all +strings matching a regular expression. Also different kind of +definitions can sometimes be used for the same set of strings +(e.g. +values, regex and regexes). +For some com- +pound data values, the kind of definition to use depends on +if and how semantic and datatype of the composing elements +is coded in the text representation (e.g. composed_of vs. +labeled_list or tagged_list). Thus it is interesting +to investigate the use cases of these different definition kinds, +depending on the type of represented value. +It is important to note that there is not always a one-to-one +correspondence between the type of represented value and its +text representation. For instance, the text representation "I" +could represent the string "I" (e.g. as the first person singular +pronoun in English) or represent the integer value expressed +as a Roman numeral, which is more commonly represented +using Arabic numerals (“1”). +This paper presents a systematic exploration of the different +types of values that can be represented in portions of a text +format and reports the corresponding TextFormats definitions +required to accurately describe the data in a specification. For +each type of value it includes practical examples of TFSL def- +initions to illustrate the concepts presented. +Types of Data Values +In the context of text formats, various type of values can be +represented. One important distinction to consider is whether +a data point logically represents an indivisible, atomic unit or +can be broken down into multiple components. If it is the for- +mer, it is considered a scalar value, and if it is the latter, it is +considered a compound value. +Scalar values are single, atomic units of data. This category +encompasses several subtypes, including numerical data, cat- +egorical data, boolean values, and strings. Numerical data +can include integer and floating-point values, and is used to +represent mathematical values. Categorical data, on the other +hand, assigns each data point to a predefined, finite set of +Gonnella +| +arXiv +| +February 1, 2023 +| +1–8 + +Definition key +Description +for scalar data +constant +Single, invariant value +values +One of a set of values +regex +Matches provided regular expression +regexes +Matches one of the provided regular expressions +integer +Signed integer, optionally with range validation +unsigned_integer +Unsigned integer, optionally with range validation +float +Floating point value, optionally with range validation +for compound data +list_of +Ordered set; element datatype and semantic independent of position +composed_of +Ordered set; element datatype and semantic dependent of position +labeled_list +Key/value pairs set; element datatype and semantic dependent on the key +tagged_list +Tagname/typecode/value triples; semantic dependent on tagname, datatype on typecode +multiple choices +one_of +One of different formats, described by separate definitions +Table 1. Datatype definition keys available in the TextFormats Specification Language, according to the class of data type to be defined (scalar or compound), as well as for +the definition of multiple choices for a single element (one_of). +possible values. Boolean values are an example of categori- +cal data, for representing a binary condition, and can be either +true or false. Finally, strings are sequences of characters, and +can be used to represent a variety of values. +Compound values consist of multiple components, each of +which can be a scalar or itself also a compound value. Com- +pound data types provide a way to organize data into a com- +plex structure, allowing for the representation of structured, +multi-layered information. For instance, a compound data +type could consist of several numerical values, each repre- +senting a different data point, or it could consist of multiple +components, each representing a different aspect of the data. +The important feature of compound data types is that they al- +low for the representation of multi-faceted information in a +way that can be parsed and analyzed in a meaningful man- +ner. Thus the semantics and data type of each component +must be known to the parser, either as an external convention, +or through metadata included in the text representation itself +(e.g. tags). +Definitions for Scalar Values +Numeric values. Several kind of datatype definitions are +dedicated to numerical variables. In order to define numer- +ical values, the following criteria are to be considered. First, +if the value is an integer or not. Second, what is the range of +the possible values. Third, because they are internally repre- +sented differently in many contexts (e.g. C), if a value is an +integer, it shall be considered if it is signed or not. +Any value. If every valid value is acceptable, as long as it +fits in the range of the data type in the programming lan- +guage or environment in which the data is used, then the pre- +defined integer, unsigned_integer and float are +used. Since they are predefined, they do not consist in a defi- +nition, but the key (e.g. integer) is simply inserted in the +specification in the position where the definition would be re- +quired. This is done usually, in a compound datatype or an +alias, as in the following examples: +datatypes: +num1: unsigned_integer +list1: {list_of: integer} +Values in a range. If only values in a given range are accepted, +the options min and/or max can be used: +datatypes: +num2: integer: {min: -1, max: 1} +num3: unsigned_integer: +{min: 0, max: 100} +num4: float: {min: 0, max: 1} +The default is to include the limits in the accepted values, but +these can be excluded using (min|max)_excluded. This +is mostly useful only for floating point values, since for inte- +gers it suffices to increase or decrease the limit by 1. +datatypes: +num5: float: {min: 0, max: 100, +min_excluded: true, +max_excluded: true} +Element of a set of values. If only elements of a given set of +values shall be representable, the values definition key is +used, as in the following example: +datatypes: +num6: {values: [1,2,3]} +2 +| +arXiv +Gonnella +| +TFSL definitions by value type + +A limitation is that since all possible values must be enumer- +ated, the performance of this definition is very bad, if the +number of elements of the set is too high. +Single possible value. If only a single value shall be accepted +a constant definition is used: +datatypes: +num7: {constant: 3} +Roman numerals. The support in TFSL of numerical values +in non-conventional representations is very limited. The only +case which can be represented with ease is the case in which +all possible values can be enumerated - and thus their number +is limited, otherwise the performance would be negatively af- +fected. In this case a values definition can be used, by +specifying conversions for each element to the corresponding +numerical value. An example is given here, for representing +the roman numbers from 1 to 3: +datatypes: +num8: +values: +- "I": 1 +- "II": 2 +- "III": 3 +In other cases (such as if any value in a range or all numeric +value must be representable), the parsing or formatting of the +values cannot be directly defined in TextFormats. Thus the +data type must be defined as a string (e.g. by a regex) and +handled by the calling code, as in the following example: +datatypes: +num9: {regex: "^[MDCLXVI]+$"} +Boolean variables. Booleans are variables that can only +contain one of two values: true or false. Their representa- +tion is usually a pair of strings such as "T" and "F", or "0" +and "1". +Single representations. The following definition shows how +to define a type for a boolean variable, with a string value for +each of the two decoded values (true or false). These repre- +sentations can be provided using an values definition and +a mapping: +datatypes: +boolean1: +values: ["T": true, "F": false} +Multiple representations. Sometimes multiple string repre- +sentations are accepted in a format for each of the two values +of a boolean. In this case a values or regex definition is +used and canonical representations must be specified: +datatypes: +boolean2: +regexes: +- "[Tt](rue)?": true +- "[Ff](alse)?": false +canonical: {"T": true, "F": false} +Presence or absence. In other cases, the value of a boolean +variable is encoded as the presence or absence of a given el- +ement in the text representation. In this case a constant +definition with a default value (empty key) can be used: +datatypes: +boolean3: +constant: {"$": true} +empty: false +Handling the undefined state. In some cases, a three-way +variable state is possible. For example, a variable could take +a boolean value or a special undefined value. In such cases, +further options are simply added to the values definition +mapping: +datatypes: +boolean4: +values: ["T": true, "F": false, +"NA": null] +String values. The present section describe how to define +the data types for values, which are stored in string variables. +Any string. If a component of a format can contain any string, +the predefined string datatype can be used for it. This is +used in aliases or compound definitions: +datatypes: +str1: string +list2: {list_of: string} +Categorical labels. Categorical variables have values, which +can only take a limited number of different values, which can +be enumerated. Some cases of categorical values have al- +ready been handled above, i.e. boolean values and numerical +variables, with a fixed number of possible values. These were +defined using a values definition. Also if the categories are +stored in strings, the same kind of definition is used: +datatypes: +str2: {values: ["ABC", "DEF"]} +Single possible value. If a string variable may only contain a +single value, then a constant definition is used: +datatypes: +str3: {constant: "XYZ"} +Regular expressions. If a string must match a given regular +expression, a definition of kind regex is used: +datatypes: +str4: {regex: "[ABC]{1,3}"} +In some cases, it is easier to split a regex into multiple pieces, +which are defined separately. While the universe of match- +ing strings remains the same, if a single or multiple regexes +are used, the use of multiple regexes has the advantage that +the single regexes could be more readable, but also that each +piece can be treated differently, when the strings are mapped +to given values: +Gonnella +| +TFSL definitions by value type +arXiv +| +3 + +datatypes: +str5: +regexes: +- "[ABC]{1,3}" +- "[DEF]{5,7}" +Empty strings. If the empty string shall also be handled, its +value can be provided using empty (this option is available +for any kind of definition). The option has the highest pri- +ority, thus also if e.g. a regular expression matching also an +empty string is provided, the empty string case is handled as +defined in the empty option. E.g. in the following case, the +empty string results in a decoded value 0: +datatypes: +str6: {regex: "\d*", empty: "0"} +Structured strings. In some cases, although a value shall be +decoded as string, and not further parsed into smaller ele- +ments, it has an internal structure. In this case it can be use- +ful to create a definition (e.g. for a compound datatype, as +explained below) and then let TextFormats know that the def- +inition shall only be used for validation, but not for parsing, +using the as_string: true option. +For example the following definition of a string (containing +unsigned integers and ‘.’ separating them) uses a relatively +complex regular expression: +datatypes: +str7: +regex: "(0|[1-9][0-9]*) +(\.(0|[1-9][0-9]*))*" +The following definition is equivalent, but more readable: +datatypes: +str8: +list_of: {regex: unsigned_integer} +splitted_by: "." +min_length: 1 +as_string: true +Acronyms. By default, string are just encoded as the string +itself, i.e. parsing involves validation, but no modification of +the value itself. In some cases a different string should be +present in the string representation compared to the decoded +value: e.g. one could want to expand an acronym. For this a +decoded mapping is used: +datatypes: +str9: +values: +- "USA": "United States" +- "UK": "United Kingdom" +If multiple encoded values are decoded to the same decoded +value (always for regular expressions), then canonical en- +coded forms must be specified, so that the encoder knows +which one shall be used. E.g.: +datatypes: +str10: +regexes: +- "U(SA?|sa)": "United States" +- "U[Kk]": "United Kingdom" +canonical: +- "USA": "United States" +- "UK": "United Kingdom" +Definitions for Compound Values +There are several different types of compound data. Hereby +we distinguish 3 cases, which are handled separately: +1. ordered lists of elements, which are semantically equiv- +alent +2. dictionaries or mappings, i.e. open associative data +structures in which different elements may have dis- +tinct semantics +3. objects or structures, i.e. which generally contain dif- +ferent, semantically distinct attributes +It is worth noticing that the names of the data types for these +kinds of compound values depend on the context, such as pro- +gramming language, and on the underlying data structure, i.e. +how the data is stored in memory. +Compound Data Compatibility in TextFormats. Compound +values can include other compound values as components. +This hierarchical structure can be represented using a tree, +with the depth potentially being indefinite and marked in the +text representation using indentation or nested pairs of paren- +theses. However, for a format to be compatible with the cur- +rent implementation of TextFormats, it must represent a reg- +ular language (with the possible exception of parts of the for- +mat handled by an external library, i.e. currently JSON). As +a result, the definition tree in compound data types must be +known at definition time and circular definitions are not al- +lowed in TFSL. +Lists, Arrays, Sequences, Sets. We consider here com- +pound data values, where the elements are ordered (the order +may, but must not, be meaningful), but they are all considered +semantically equivalent and the element data type and set of +representable values is not dependent on their position in the +list. +4 +| +arXiv +Gonnella +| +TFSL definitions by value type + +If the order matters, the compound values are often stored in +an array or list data structure (e.g. a linked list) depending on +the underlying data structure and the context (e.g. program- +ming language), and are thus called lists (e.g. Python), arrays +(e.g. C, Python) or sequences (e.g. Nim, YAML). Values +of this kind are described in TextFormats using list_of +datatype definitions. +The definition of compound values where the semantics of +the elements does not differ among the elements is done in +TextFormats using the list_of key, as in the following ex- +ample: +datatypes: +list3: +list_of: {regex: [A-Z]} +Element separators. Often the parsing of the single elements +is made possible by a separator string between the elements, +which does not occur in the elements itself. This is specified +using the option splitted_by: +datatypes: +list4: +list_of: unsigned_integer +splitted_by: "," +Separator escaping. In some cases, however, the separating +string can also be present in the elements itself, e.g. by es- +caping it. In this case the separator option is used in the +definition, and a e.g. a regular expression is used for defining +the elements: +datatypes: +list5: +list_of: +regex: "(\\\:|[A-Za-z0-9 _])*" +# allows : escaped by \ +separator: ":" +Given the definition above, for example, the string +elem 1:elem2:elem_3:elem\:\:4 +would be parsed into the four elements elem1, elem2, +elem_3 and elem\:\:4. +Fixed length elements. In case the elements of a list have all +the same length, a separator is generally not necessary. How- +ever, if one is present, it may be also present in the element +text itself, since there is no risk of confusion. In this case the +separator option is used, e.g.: +datatypes: +list6: +list_of: +regex: "[:0-9]{3}" +separator: ":" +According to the previous definition, each element has the +size 3, thus it does not matter that the separator is pos- +sibly included in the elements. +For example the string +001:0:::002:2:1:112:::: would be parsed into the +six elements 001, 0::, 002, 2:1, 112 and :::. +Enclosing strings. In some cases, constant strings are present +before the first and/of after the last element of a list, for ex- +ample an opening and a closing bracket: +datatypes: +list7: +list_of: unsigned_integer +splitted_by: "," +prefix: "(" +suffix: ")" +This definition allows to parse a string representation like +(1,2,3,4). +Number of elements. In some cases there is a minimum +and/or maximum or a fixed number elements of the list. This +can be enforced using validation rules, as in the following +examples: +datatypes: +list8: +# e.g. 0;-1;32 +list_of: integer +splitted_by: ";" +length: 3 +list9: +list_of: integer +splitted_by: ";" +min_length: 5 +max_length: 7 +Empty lists. Empty lists are also supported. If there is a prefix +and suffix, the empty list is recognizable and does not require +a special handling. If there are no enclosing strings, then the +representation of an empty list is an empty string. This case +can be handled, by using the empty option: +datatypes: +list9: +list_of: {regex: "[A-Z]"} +empty: [] +`` +Predefined +representations. Using +decoding +mappings +and/or a default decoded value (see below) is it possible +to decode given strings to predefined lists. In case multiple +representations of the same value are given, the canonical +option must define which one shall be used for encoding, as +in the following example: +datatypes: +list10: +values: +"a": ["a"] +"1a": ["a"] +"2a": ["a", "a"] +"3a": ["a", "a", "a"] +empty: [] +canonical: {"1a": ["a"]} +Gonnella +| +TFSL definitions by value type +arXiv +| +5 + +Sets and Multisets. Sometimes a different kind of collection +is available in case the order of the elements is not important, +and different kind of data structures are used for representing +them in memory, such as hash tables. This kind of collections +are available as e.g. sets in Python and hash sets in Nim. +There is no special handling for sets in TextFormats. This +means that the elements of sets are regarded as a list. Thus, +equivalence operations which disregard the order of the ele- +ments must be implemented externally. Similarly, the unique- +ness of the set elements (vs. multisets) must be validated ex- +ternally. +Heterogeneous lists. A one_of definition can be combined +with a list_of definition to implement lists of elements +which have different types, which is not dependent on the po- +sitional order of the element and is not explicitly annotated +by a key or typecode. Instead, it is the formatting of the ele- +ment itself which reveals the type. For example, the follow- +ing defines a list containing either integers or single upcase +characters: +datatypes: +list11: +list_of: +one_of: +- integer +- regex: "[A-Z]" +splitted_by: "," +An example of string which can be handled by the previous +definition is: 1,-3,A,5,B,-2. +Mappings, Dictionaries, Associative arrays. In another +type of compound values, the semantics and data type of the +elements can differ, and different instances may contain or +not some of the elements, or sometimes contain multiple el- +ements of the same type. Such compound values are usu- +ally stored in open associative data structures, with different +names and underlying data structures, such as dictionaries +(Python), tables (Nim), hash tables (C), objects (JSON) or +maps (YAML). +In a text format, this kind of data can be represented in +different ways. +For example, the semantic and data type +can be specified explicitly in the text representation, or be +implicit by the format of the text itself. +Depending on +this, the kind of definition to be used in TextFormats differs +(e.g. list_of with one_of elements, labeled_list +or tagged_list). +Semantics by format. If the semantics of the elements of a +list is determined by the format, a list_of definition can +be given, in which the element is defined using a one_of +definition. This is the same case illustrated above under the +paragraph heterogeneous lists. In this case, which does not +occur very often in practice, the result of parsing and the data +to be passed to the encoding function must be a list. Thus +some external preprocessing or postprocessing will be neces- +sary to transform the data to or from a mapping. +Key/value pairs. In many cases, a collection contains ele- +ments of different type, and the semantics of each elements +is given explicitly, alongside the value of the element. Thus, +each element is present as a tuple of keys and values. Al- +though a list_of could be used also for this case, TextFor- +mats offers a specialized kind of list definition for it, in case +the set of possible keys and their associated data types are +known in advance. +In such cases a labeled_list definition is used, as in the +following example: +datatypes: +map1: +labeled_list: +rank: unsigned_integer +name: string +splitted_by: ";" +The set of possible keys and the datatypes of the values for +each of the keys are given under the labeled_list key, +as a mapping. An internal_separator string can be +specified, separating the key from the value (the default is +:). The internal separator cannot be empty and cannot occur +in the keys, but it can occur in the values. This condition is +generally met in formats which implement key/values lists. +Single-instance keys. Names are by default allowed to +present multiple times in the list. For this reason, the ele- +ments values are always given in the decoded value as lists. +In some cases, all or some of names can only be present once. +This can be enforced by listing them under the single key: +datatypes: +map2: +labeled_list: +rank: unsigned_integer +name: string +splitted_by: ";" +single: [rank, name] +Required keys. Also, by default, some names may be absent +in the set of elements. If some of the names must be present, +they are listed under the required key: +datatypes: +map3: +labeled_list: +rank: unsigned_integer +name: string +splitted_by: ";" +internal_separator: "=" +required: [name] +SAM-style tags. In some cases the values of a collec- +tion are each accompanied by a name and typecode, +i.e. as triples value/name/typecode. +A prominent ex- +ample for this are SAM-style tags, +which often in- +cluded in recent bioinformatics formats, +e.g. +GFA +(GFA Format Specification Working Group, 2016, 2018) and +6 +| +arXiv +Gonnella +| +TFSL definitions by value type + +VCF (Danecek et al., 2011), after their original definition for +use in the SAM format (Li et al., 2009). +The difference with the key/value case is that the name de- +fines the semantic of the value, but not all names (differently +from labeled values lists) must be defined in advance. Since +the name is not necessarily predefined, the type must be ex- +plicitly given, thus a typecode is present in the text represen- +tation. Each typecode is associated to a datatype definition. +For these case, the tagged_list definition key is used, +under which all available datatypes codes and the associated +datatype definitions are given under the tagged_list key +as a mapping. +An example of tagged list definition is given here: +datatypes: +map4: +tagged_list: +i: integer +f: float +tagname: "[A-Z]" +internal_separator: "." +splitted_by: ";" +The definition given above can e.g. handle the string repre- +sentation A.i.12;B.f.1.3, which is parsed into the two +elements A with the value 12 and B with the value 1.3. +Generalized tags. The valid names and their formatting is +specified using a regular expression. The internal separator +key has a default value (colon, :) and it must be a non-empty +string. It cannot be present in tagnames and type codes (but +can be present in values). This restriction is reasonable and +e.g. met in the in SAM tags. +Predefined +representations. Using +decoding +mappings +and/or a default decoded value (see below) is it possible +to decode given strings to predefined mappings/dictionaries. +In case a data value has multiple string representations, the +canonical one must be specified, which is then used for en- +coding. +datatypes: +map5: +values: +"ax": {"a": "x", "b": "y"} +"a": {"a": "x", "b": "y"} +"ay": {"a", "y", "b": "y"} +"bx": {"a": "x", "b": "x"} +canonical: +"ax": {"a": "x", "b": "y"} +"ay": {"a", "y", "b": "y"} +"bx": {"a": "x", "b": "x"} +empty: +{"a": "z", "b": "z"} +Implicit entries. In some cases, the decoded dictionary +shall contain a given constant key/value pair, which is +not explicitely encoded. +These are specified using the +implicit mapping, which is available for composed_of, +labeled_list and tagged_list definitions: +datatypes: +map6: +composed_of: +- name: string +- copies: unsigned_integer +splitted_by: "," +implicit: {type: "rRNA"} +This definition can e.g. +handle the string representa- +tion "16S,2", which is parsed into the mapping with +3 keys {"name": +"16S", copies: +2, type: +"rRNA"}. +Objects, Structs. In this section we handle collections of +values where each instance contain the same set of elements +(with possible exceptions), representing different aspects of +the data. Each of the element has its own data type and se- +mantics. +This kind of compound values is represented in structs (C, +Python) or instances of classes (Python, Nim). Other possi- +ble representations are those mentioned in the previous para- +graph (mappings, hash tables), eventually adding validations, +making sure that all and only the correct elements are present. +In TextFormats the description of this kind of data is per- +fomed using composed_of definitions. Under the defini- +tion key, a list is of tuples is given, each one as name and +definition. Note that this is a list (thus the - in YAML) and +not a mapping, since the order of the elements is important, +as it defines which element is which: +datatypes: +obj1: +composed_of: +- first: unsigned_integer +- second: float +- third: {regex: "[A-Za-z0-9]"} +splitted_by: " " +Enclosing strings. As for lists, composed_of definitions +can include a prefix and/or a suffix option, which de- +fine enclosing strings (e.g. parentheses) before the first ele- +ment and/or after the last element. +Element separators. The splitted_by and separator +options are used for describing how to separate the single +elements of the compound value. +These have the same +kind of usage already illustrated in the Lists section, i.e. +splitted_by is used for separators which cannot occur in +the elements text, while separator is used otherwise, e.g. +when the escaped separator can occur in the elements or the +element size is recognized by their format, e.g. fixed-length +elements. +Multiple separators. In some cases, different separators are +used between different pairs of elements. In this case, they +can be specified as additional constant elements and hidden +Gonnella +| +TFSL definitions by value type +arXiv +| +7 + +in the decoded dictionary using the hide_constants op- +tion, as in the following example: +datatypes: +obj2: +composed_of: +- x: unsigned_integer +- sep1: {constant: ";"} +- y: float +- sep2: {constant: "|"} +- z: {regex: "[A-Za-z]"} +hide_constants: true +An example of string representation which is parsed by the +previous definition is 1;2.0|A. Hereby the resulting map- +ping has three elements, as the constant separators are only +considered for parsing: {x:1, y:2.0, z:A}. +Optional separated elements. Some of the elements can be +optional, i.e. sometimes absent from the sequence of ele- +ments. In case the sequence is splitted by a non-empty sepa- +rator string, an empty element can be recognized by the pres- +ence of this separator. In this case the empty option is used +in the elements definitions, as in the following example: +datatypes: +obj3: +composed_of: +- first: {values: [1,2], empty: 0} +- second: {values: [A,B], empty: C} +splitted_by: ";" +Optional trailing elements. In some cases, a given number of +elements at the beginning of the text representation may be +mandatory, while the following can be present or not (and +are mandatory only if elements after them in the order are +present). In this case, the required option is used (number +of required elements): +datatypes: +obj4: +composed_of: +- first: {values: [1,2], empty: 0} +- second: {values: [A,B], empty: C} +splitted_by: ";" +required: 1 +Optional internal elements. In the case, some internal ele- +ment can be missing (together with its associated separator, +or when no separator is used) but there is no ambuiguity, +e.g. because the following element of the sequence has a type +that allows it to be distinguished from the optional element, +or because the total number of elements changes depending +on the presence or absence of the optional element. +In this case the user must provide multiple alternative defini- +tions of the structure (i.e. with and without the optional ele- +ment) using a one_of definition. Furthermore, in order to +provide the same set of values for all instances of the object +or struct, the implicit option can be used. +datatypes: +obj5: +one_of: +- composed_of: +- name: string +- expressed: +values: {"+": true, "-": false} +splitted_by: "," +implicit: {"copies": 1} +- composed_of: +- name: string +- copies: unsigned_integer +- expressed: +values: {"+": true, "-": false} +splitted_by: "," +The above definition would parse the string representa- +tion "X,+" to the mapping {name: +X, expressed: +true, copies: +1}, while "X,2,+" would be parsed +to +{name: +X, expressed: +true, copies: +2}. +Unions. In some cases an element of a format can be ex- +pressed in multiple different ways. In dynamically typed lan- +guages such as Python, any variable can store this kind of +values. In C, such values could be e.g. stored as unions, and +in Nim as variant objects. +In TextFormats the type of such values can be defined defined +using definitions of type one_of. E.g. the following allows +to represent an unsigned integer value, as a number, if it is >= +1, otherwise a floating point: +datatypes: +num10: +one_of: +- unsigned_integer: {min: 1} +- float +Note that the content of the one_of key is a YAML list; the +order of the elements in the list defines the order or prece- +dence of the definitions (the first which applies is used). +Conclusions +In this paper, the representation of different kind of data in +text formats, as specified using the library TextFormats, has +been analysed. Thereby it was demonstrated that most types +of values that can be used in programming languages such as +C and Python, can also be represented in TextFormats. +In TextFormats specifications, the same type of value is some- +times represented using a different kind of datatype definition. +This is true for both scalar values and compound values. For +example, in the above examples, boolean values are some- +times represented using definitions of kind values, some- +times regexes or even constant, depending on what is +their representation in the format. Among the examples for +compound values, e.g., collections of tagged elements are +sometimes represented using list_of, but in other cases +8 +| +arXiv +Gonnella +| +TFSL definitions by value type + +using the specialized list definition keys tagged_list and +labeled_list. +The reason for this is that the TextFormat syntax for datatype +definitions is oriented to the text representation and not to +the type of represented value. The systematic review of the +definition types based on the type of value is particularly use- +ful when defining a new format and complements the TFSL +syntax manual included in the library documentation, which +is more useful when a specification is written for a format +which already exists. +References +Giorgio Gonnella. Textformats: Simplifying the definition +and parsing of text formats in bioinformatics. PLOS ONE, +17(5):1–17, 05 2022. doi: 10.1371/journal.pone.0268910. +URL https://doi.org/10.1371/journal.pone.0268910. +GFA +Format +Specification +Working +Group. +The +GFA +format +specification, +2016. +URL +http://gfa-spec.github.io/GFA-spec/GFA1.html. +GFA Format Specification Working Group. Graphical frag- +ment assembly (GFA) 2.0 format specification, 2018. URL +http://gfa-spec.github.io/GFA-spec/GFA2.html. +Petr Danecek, Adam Auton, Goncalo Abecasis, Cornelis A. +Albers, Eric Banks, Mark A. DePristo, Robert E. Hand- +saker, Gerton Lunter, Gabor T. Marth, Stephen T. Sherry, +Gilean McVean, Richard Durbin, and 1000 Genomes +Project Analysis Group. +The variant call format and +VCFtools. +Bioinformatics, 27(15):2156–2158, 06 2011. +ISSN 1367-4803. doi: 10.1093/bioinformatics/btr330. +URL https://doi.org/10.1093/bioinformatics/btr330. +Heng Li, Bob Handsaker, Alec Wysoker, Tim Fennell, +Jue Ruan, Nils Homer, Gabor Marth, Goncalo Abecasis, +Richard Durbin, and 1000 Genome Project Data Process- +ing Subgroup. The Sequence Alignment/Map format and +SAMtools. Bioinformatics, 25(16):2078–2079, 06 2009. +ISSN 1367-4803. doi: 10.1093/bioinformatics/btp352. +URL https://doi.org/10.1093/bioinformatics/btp352. +ACKNOWLEDGEMENTS +Giorgio Gonnella has been supported by the DFG Grant GO 3192/1-1 “Automated +characterization of microbial genomes and metagenomes by collection and verifica- +tion of association rules”. The funders had no role in study design, data collection +and analysis, decision to publish, or preparation of the manuscript. +AUTHOR CONTRIBUTIONS +These +contributions +follow +the +Contributor +Roles +Taxonomy +guidelines: +https://casrai.org/credit/. +Conceptualization: G.G.; Data curation: G.G.; Formal +analysis: G.G.; Funding acquisition: G.G.; Investigation: G.G.; Methodology: G.G.; +Project administration: G.G.; Resources: G.G.; Software: G.G.; Supervision: G.G.; +Validation: G.G.; Visualization: G.G.; Writing – original draft: G.G.; Writing – review +& editing: G.G. +COMPETING FINANCIAL INTERESTS +The authors declare no competing financial interests. +Gonnella +| +TFSL definitions by value type +arXiv +| +9 + diff --git a/ldFRT4oBgHgl3EQfYzfm/content/tmp_files/load_file.txt b/ldFRT4oBgHgl3EQfYzfm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..acea1cf30c6a05f58364b077a03ab59f3af55f1e --- /dev/null +++ b/ldFRT4oBgHgl3EQfYzfm/content/tmp_files/load_file.txt @@ -0,0 +1,458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf,len=457 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='13551v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='PL] 31 Jan 2023 Designing text representations for existing data using the TextFormats Specification Language Giorgio Gonnella1,2 � 1Center for Bioinformatics (ZBH), Universität Hamburg, Bundesstrasse 43, 20146 Hamburg 2Institute for Microbiology and Genetics, Georg-August-Universität Göttingen, Goldschmidtstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' 1, 37077 Göttingen TextFormats is a software system for efficient and user-friendly creation of text format specifications, accessible from multiple pro- gramming languages (C/C++, Python, Nim) and the Unix command line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' To work with a format, a specification written in the TextFormats Specification Language (TFSL) must be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The specification defines datatypes for each part of the format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The syntax for datatype definitions in TextFormats specifications is based on the text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus this system is well suited for the description of existing formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' However, when creating a new text format for representing existing data, the user may use different possible definitions, based on the type of value and the representation choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This study explores the possible definition syntax in the TextFormats Specification Language to be used for creating text represen- tations of scalar values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' string, numeric value, boolean) and compound data structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' array, mapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The results of the analysis are presented systematically, together with examples for each each type of different values that can be represented, and usage advices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' TextFormats | Datatype definition | Parser | Text representation | Data format | Domain specific language | Software library | Format specification | File format | Data type | Text format | Tutorial Correspondence: giorgio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='gonnella@uni-goettingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='de TextFormats (Gonnella, 2022) is a software library and a set of tools for defining text formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Although it was initially de- veloped for the representation of bioinformatics formats, it is a generic software which can be applied to a variety of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Once a format has been defined using the domain-specific language TFSL (TextFormats Specification Language), the TextFormats specification can be used for parsing the for- mat, as well as writing data in the format, using the APIs for several programming languages (Nim, Python, C/C++) or in scripts, using the provided Unix command-line tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In TFSL, various kinds of definitions are used to describe different types of data textual representations, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' These definitions are dependent not only on the type of value but also on other characteristics of the data and its representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example they depend on the set of possi- ble values: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' for a string, one uses constant for a single value, values for a small set of values and regex for all strings matching a regular expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Also different kind of definitions can sometimes be used for the same set of strings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' values, regex and regexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For some com- pound data values, the kind of definition to use depends on if and how semantic and datatype of the composing elements is coded in the text representation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' composed_of vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' labeled_list or tagged_list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus it is interesting to investigate the use cases of these different definition kinds, depending on the type of represented value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' It is important to note that there is not always a one-to-one correspondence between the type of represented value and its text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For instance, the text representation "I" could represent the string "I" (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' as the first person singular pronoun in English) or represent the integer value expressed as a Roman numeral, which is more commonly represented using Arabic numerals (“1”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This paper presents a systematic exploration of the different types of values that can be represented in portions of a text format and reports the corresponding TextFormats definitions required to accurately describe the data in a specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For each type of value it includes practical examples of TFSL def- initions to illustrate the concepts presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Types of Data Values In the context of text formats, various type of values can be represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' One important distinction to consider is whether a data point logically represents an indivisible, atomic unit or can be broken down into multiple components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If it is the for- mer, it is considered a scalar value, and if it is the latter, it is considered a compound value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Scalar values are single, atomic units of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This category encompasses several subtypes, including numerical data, cat- egorical data, boolean values, and strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Numerical data can include integer and floating-point values, and is used to represent mathematical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Categorical data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' on the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' assigns each data point to a predefined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' finite set of Gonnella | arXiv | February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' 2023 | 1–8 Definition key Description for scalar data constant Single,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' invariant value values One of a set of values regex Matches provided regular expression regexes Matches one of the provided regular expressions integer Signed integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' optionally with range validation unsigned_integer Unsigned integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' optionally with range validation float Floating point value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' optionally with range validation for compound data list_of Ordered set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' element datatype and semantic independent of position composed_of Ordered set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' element datatype and semantic dependent of position labeled_list Key/value pairs set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' element datatype and semantic dependent on the key tagged_list Tagname/typecode/value triples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' semantic dependent on tagname, datatype on typecode multiple choices one_of One of different formats, described by separate definitions Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Datatype definition keys available in the TextFormats Specification Language, according to the class of data type to be defined (scalar or compound), as well as for the definition of multiple choices for a single element (one_of).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Boolean values are an example of categori- cal data, for representing a binary condition, and can be either true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Finally, strings are sequences of characters, and can be used to represent a variety of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Compound values consist of multiple components, each of which can be a scalar or itself also a compound value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Com- pound data types provide a way to organize data into a com- plex structure, allowing for the representation of structured, multi-layered information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For instance, a compound data type could consist of several numerical values, each repre- senting a different data point, or it could consist of multiple components, each representing a different aspect of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The important feature of compound data types is that they al- low for the representation of multi-faceted information in a way that can be parsed and analyzed in a meaningful man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus the semantics and data type of each component must be known to the parser, either as an external convention, or through metadata included in the text representation itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' tags).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Definitions for Scalar Values Numeric values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Several kind of datatype definitions are dedicated to numerical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In order to define numer- ical values, the following criteria are to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' First, if the value is an integer or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Second, what is the range of the possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Third, because they are internally repre- sented differently in many contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' C), if a value is an integer, it shall be considered if it is signed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Any value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If every valid value is acceptable, as long as it fits in the range of the data type in the programming lan- guage or environment in which the data is used, then the pre- defined integer, unsigned_integer and float are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Since they are predefined, they do not consist in a defi- nition, but the key (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' integer) is simply inserted in the specification in the position where the definition would be re- quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This is done usually, in a compound datatype or an alias, as in the following examples: datatypes: num1: unsigned_integer list1: {list_of: integer} Values in a range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If only values in a given range are accepted, the options min and/or max can be used: datatypes: num2: integer: {min: -1, max: 1} num3: unsigned_integer: {min: 0, max: 100} num4: float: {min: 0, max: 1} The default is to include the limits in the accepted values, but these can be excluded using (min|max)_excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This is mostly useful only for floating point values, since for inte- gers it suffices to increase or decrease the limit by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' datatypes: num5: float: {min: 0, max: 100, min_excluded: true, max_excluded: true} Element of a set of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If only elements of a given set of values shall be representable, the values definition key is used, as in the following example: datatypes: num6: {values: [1,2,3]} 2 | arXiv Gonnella | TFSL definitions by value type A limitation is that since all possible values must be enumer- ated, the performance of this definition is very bad, if the number of elements of the set is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Single possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If only a single value shall be accepted a constant definition is used: datatypes: num7: {constant: 3} Roman numerals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The support in TFSL of numerical values in non-conventional representations is very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The only case which can be represented with ease is the case in which all possible values can be enumerated - and thus their number is limited, otherwise the performance would be negatively af- fected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case a values definition can be used, by specifying conversions for each element to the corresponding numerical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' An example is given here, for representing the roman numbers from 1 to 3: datatypes: num8: values: "I": 1 "II": 2 "III": 3 In other cases (such as if any value in a range or all numeric value must be representable), the parsing or formatting of the values cannot be directly defined in TextFormats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus the data type must be defined as a string (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' by a regex) and handled by the calling code, as in the following example: datatypes: num9: {regex: "^[MDCLXVI]+$"} Boolean variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Booleans are variables that can only contain one of two values: true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Their representa- tion is usually a pair of strings such as "T" and "F", or "0" and "1".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Single representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The following definition shows how to define a type for a boolean variable, with a string value for each of the two decoded values (true or false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' These repre- sentations can be provided using an values definition and a mapping: datatypes: boolean1: values: ["T": true, "F": false} Multiple representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Sometimes multiple string repre- sentations are accepted in a format for each of the two values of a boolean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case a values or regex definition is used and canonical representations must be specified: datatypes: boolean2: regexes: "[Tt](rue)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ": true "[Ff](alse)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ": false canonical: {"T": true, "F": false} Presence or absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In other cases, the value of a boolean variable is encoded as the presence or absence of a given el- ement in the text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case a constant definition with a default value (empty key) can be used: datatypes: boolean3: constant: {"$": true} empty: false Handling the undefined state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, a three-way variable state is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example, a variable could take a boolean value or a special undefined value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In such cases, further options are simply added to the values definition mapping: datatypes: boolean4: values: ["T": true, "F": false, "NA": null] String values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The present section describe how to define the data types for values, which are stored in string variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Any string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If a component of a format can contain any string, the predefined string datatype can be used for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This is used in aliases or compound definitions: datatypes: str1: string list2: {list_of: string} Categorical labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Categorical variables have values, which can only take a limited number of different values, which can be enumerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Some cases of categorical values have al- ready been handled above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' boolean values and numerical variables, with a fixed number of possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' These were defined using a values definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Also if the categories are stored in strings, the same kind of definition is used: datatypes: str2: {values: ["ABC", "DEF"]} Single possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If a string variable may only contain a single value, then a constant definition is used: datatypes: str3: {constant: "XYZ"} Regular expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If a string must match a given regular expression, a definition of kind regex is used: datatypes: str4: {regex: "[ABC]{1,3}"} In some cases, it is easier to split a regex into multiple pieces, which are defined separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' While the universe of match- ing strings remains the same, if a single or multiple regexes are used, the use of multiple regexes has the advantage that the single regexes could be more readable, but also that each piece can be treated differently, when the strings are mapped to given values: Gonnella | TFSL definitions by value type arXiv | 3 datatypes: str5: regexes: "[ABC]{1,3}" "[DEF]{5,7}" Empty strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If the empty string shall also be handled, its value can be provided using empty (this option is available for any kind of definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The option has the highest pri- ority, thus also if e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' a regular expression matching also an empty string is provided, the empty string case is handled as defined in the empty option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' in the following case, the empty string results in a decoded value 0: datatypes: str6: {regex: "\\d*", empty: "0"} Structured strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, although a value shall be decoded as string, and not further parsed into smaller ele- ments, it has an internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case it can be use- ful to create a definition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' for a compound datatype, as explained below) and then let TextFormats know that the def- inition shall only be used for validation, but not for parsing, using the as_string: true option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example the following definition of a string (containing unsigned integers and ‘.’ separating them) uses a relatively complex regular expression: datatypes: str7: regex: "(0|[1-9][0-9]*) (\\.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' (0|[1-9][0-9]*))*" The following definition is equivalent, but more readable: datatypes: str8: list_of: {regex: unsigned_integer} splitted_by: ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" min_length: 1 as_string: true Acronyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' By default, string are just encoded as the string itself, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' parsing involves validation, but no modification of the value itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases a different string should be present in the string representation compared to the decoded value: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' one could want to expand an acronym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For this a decoded mapping is used: datatypes: str9: values: "USA": "United States" "UK": "United Kingdom" If multiple encoded values are decoded to the same decoded value (always for regular expressions), then canonical en- coded forms must be specified, so that the encoder knows which one shall be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' : datatypes: str10: regexes: "U(SA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='|sa)": "United States" "U[Kk]": "United Kingdom" canonical: "USA": "United States" "UK": "United Kingdom" Definitions for Compound Values There are several different types of compound data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Hereby we distinguish 3 cases, which are handled separately: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ordered lists of elements, which are semantically equiv- alent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' dictionaries or mappings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' open associative data structures in which different elements may have dis- tinct semantics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' objects or structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' which generally contain dif- ferent, semantically distinct attributes It is worth noticing that the names of the data types for these kinds of compound values depend on the context, such as pro- gramming language, and on the underlying data structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' how the data is stored in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Compound Data Compatibility in TextFormats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Compound values can include other compound values as components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This hierarchical structure can be represented using a tree, with the depth potentially being indefinite and marked in the text representation using indentation or nested pairs of paren- theses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' However, for a format to be compatible with the cur- rent implementation of TextFormats, it must represent a reg- ular language (with the possible exception of parts of the for- mat handled by an external library, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' currently JSON).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' As a result, the definition tree in compound data types must be known at definition time and circular definitions are not al- lowed in TFSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Lists, Arrays, Sequences, Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' We consider here com- pound data values, where the elements are ordered (the order may, but must not, be meaningful), but they are all considered semantically equivalent and the element data type and set of representable values is not dependent on their position in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' 4 | arXiv Gonnella | TFSL definitions by value type If the order matters, the compound values are often stored in an array or list data structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' a linked list) depending on the underlying data structure and the context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' program- ming language), and are thus called lists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Python), arrays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' C, Python) or sequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Nim, YAML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Values of this kind are described in TextFormats using list_of datatype definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The definition of compound values where the semantics of the elements does not differ among the elements is done in TextFormats using the list_of key, as in the following ex- ample: datatypes: list3: list_of: {regex: [A-Z]} Element separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Often the parsing of the single elements is made possible by a separator string between the elements, which does not occur in the elements itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This is specified using the option splitted_by: datatypes: list4: list_of: unsigned_integer splitted_by: "," Separator escaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, however, the separating string can also be present in the elements itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' by es- caping it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case the separator option is used in the definition, and a e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' a regular expression is used for defining the elements: datatypes: list5: list_of: regex: "(\\\\\\:|[A-Za-z0-9 _])*" # allows : escaped by \\ separator: ":" Given the definition above, for example, the string elem 1:elem2:elem_3:elem\\:\\:4 would be parsed into the four elements elem1, elem2, elem_3 and elem\\:\\:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Fixed length elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In case the elements of a list have all the same length, a separator is generally not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' How- ever, if one is present, it may be also present in the element text itself, since there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case the separator option is used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' : datatypes: list6: list_of: regex: "[:0-9]{3}" separator: ":" According to the previous definition, each element has the size 3, thus it does not matter that the separator is pos- sibly included in the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example the string 001:0:::002:2:1:112:::: would be parsed into the six elements 001, 0::, 002, 2:1, 112 and :::.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Enclosing strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, constant strings are present before the first and/of after the last element of a list, for ex- ample an opening and a closing bracket: datatypes: list7: list_of: unsigned_integer splitted_by: "," prefix: "(" suffix: ")" This definition allows to parse a string representation like (1,2,3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases there is a minimum and/or maximum or a fixed number elements of the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This can be enforced using validation rules, as in the following examples: datatypes: list8: # e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='32 list_of: integer splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" length: 3 list9: list_of: integer splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" min_length: 5 max_length: 7 Empty lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Empty lists are also supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If there is a prefix and suffix, the empty list is recognizable and does not require a special handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If there are no enclosing strings, then the representation of an empty list is an empty string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This case can be handled, by using the empty option: datatypes: list9: list_of: {regex: "[A-Z]"} empty: [] `` Predefined representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Using decoding mappings and/or a default decoded value (see below) is it possible to decode given strings to predefined lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In case multiple representations of the same value are given, the canonical option must define which one shall be used for encoding, as in the following example: datatypes: list10: values: "a": ["a"] "1a": ["a"] "2a": ["a", "a"] "3a": ["a", "a", "a"] empty: [] canonical: {"1a": ["a"]} Gonnella | TFSL definitions by value type arXiv | 5 Sets and Multisets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Sometimes a different kind of collection is available in case the order of the elements is not important, and different kind of data structures are used for representing them in memory, such as hash tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This kind of collections are available as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' sets in Python and hash sets in Nim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' There is no special handling for sets in TextFormats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This means that the elements of sets are regarded as a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus, equivalence operations which disregard the order of the ele- ments must be implemented externally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Similarly, the unique- ness of the set elements (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' multisets) must be validated ex- ternally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Heterogeneous lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' A one_of definition can be combined with a list_of definition to implement lists of elements which have different types, which is not dependent on the po- sitional order of the element and is not explicitly annotated by a key or typecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Instead, it is the formatting of the ele- ment itself which reveals the type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example, the follow- ing defines a list containing either integers or single upcase characters: datatypes: list11: list_of: one_of: integer regex: "[A-Z]" splitted_by: "," An example of string which can be handled by the previous definition is: 1,-3,A,5,B,-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Mappings, Dictionaries, Associative arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In another type of compound values, the semantics and data type of the elements can differ, and different instances may contain or not some of the elements, or sometimes contain multiple el- ements of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Such compound values are usu- ally stored in open associative data structures, with different names and underlying data structures, such as dictionaries (Python), tables (Nim), hash tables (C), objects (JSON) or maps (YAML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In a text format, this kind of data can be represented in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example, the semantic and data type can be specified explicitly in the text representation, or be implicit by the format of the text itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Depending on this, the kind of definition to be used in TextFormats differs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' list_of with one_of elements, labeled_list or tagged_list).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Semantics by format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If the semantics of the elements of a list is determined by the format, a list_of definition can be given, in which the element is defined using a one_of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This is the same case illustrated above under the paragraph heterogeneous lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case, which does not occur very often in practice, the result of parsing and the data to be passed to the encoding function must be a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus some external preprocessing or postprocessing will be neces- sary to transform the data to or from a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Key/value pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In many cases, a collection contains ele- ments of different type, and the semantics of each elements is given explicitly, alongside the value of the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thus, each element is present as a tuple of keys and values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Al- though a list_of could be used also for this case, TextFor- mats offers a specialized kind of list definition for it, in case the set of possible keys and their associated data types are known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In such cases a labeled_list definition is used, as in the following example: datatypes: map1: labeled_list: rank: unsigned_integer name: string splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" The set of possible keys and the datatypes of the values for each of the keys are given under the labeled_list key, as a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' An internal_separator string can be specified, separating the key from the value (the default is :).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The internal separator cannot be empty and cannot occur in the keys, but it can occur in the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This condition is generally met in formats which implement key/values lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Single-instance keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Names are by default allowed to present multiple times in the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For this reason, the ele- ments values are always given in the decoded value as lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, all or some of names can only be present once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This can be enforced by listing them under the single key: datatypes: map2: labeled_list: rank: unsigned_integer name: string splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" single: [rank, name] Required keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Also, by default, some names may be absent in the set of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' If some of the names must be present, they are listed under the required key: datatypes: map3: labeled_list: rank: unsigned_integer name: string splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" internal_separator: "=" required: [name] SAM-style tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases the values of a collec- tion are each accompanied by a name and typecode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' as triples value/name/typecode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' A prominent ex- ample for this are SAM-style tags, which often in- cluded in recent bioinformatics formats, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' GFA (GFA Format Specification Working Group, 2016, 2018) and 6 | arXiv Gonnella | TFSL definitions by value type VCF (Danecek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=', 2011), after their original definition for use in the SAM format (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The difference with the key/value case is that the name de- fines the semantic of the value, but not all names (differently from labeled values lists) must be defined in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Since the name is not necessarily predefined, the type must be ex- plicitly given, thus a typecode is present in the text represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Each typecode is associated to a datatype definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For these case, the tagged_list definition key is used, under which all available datatypes codes and the associated datatype definitions are given under the tagged_list key as a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' An example of tagged list definition is given here: datatypes: map4: tagged_list: i: integer f: float tagname: "[A-Z]" internal_separator: ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" The definition given above can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' handle the string repre- sentation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='3, which is parsed into the two elements A with the value 12 and B with the value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Generalized tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The valid names and their formatting is specified using a regular expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The internal separator key has a default value (colon, :) and it must be a non-empty string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' It cannot be present in tagnames and type codes (but can be present in values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This restriction is reasonable and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' met in the in SAM tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Predefined representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Using decoding mappings and/or a default decoded value (see below) is it possible to decode given strings to predefined mappings/dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In case a data value has multiple string representations, the canonical one must be specified, which is then used for en- coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' datatypes: map5: values: "ax": {"a": "x", "b": "y"} "a": {"a": "x", "b": "y"} "ay": {"a", "y", "b": "y"} "bx": {"a": "x", "b": "x"} canonical: "ax": {"a": "x", "b": "y"} "ay": {"a", "y", "b": "y"} "bx": {"a": "x", "b": "x"} empty: {"a": "z", "b": "z"} Implicit entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, the decoded dictionary shall contain a given constant key/value pair, which is not explicitely encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' These are specified using the implicit mapping, which is available for composed_of, labeled_list and tagged_list definitions: datatypes: map6: composed_of: name: string copies: unsigned_integer splitted_by: "," implicit: {type: "rRNA"} This definition can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' handle the string representa- tion "16S,2", which is parsed into the mapping with 3 keys {"name": "16S", copies: 2, type: "rRNA"}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Objects, Structs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this section we handle collections of values where each instance contain the same set of elements (with possible exceptions), representing different aspects of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Each of the element has its own data type and se- mantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This kind of compound values is represented in structs (C, Python) or instances of classes (Python, Nim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Other possi- ble representations are those mentioned in the previous para- graph (mappings, hash tables), eventually adding validations, making sure that all and only the correct elements are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In TextFormats the description of this kind of data is per- fomed using composed_of definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Under the defini- tion key, a list is of tuples is given, each one as name and definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Note that this is a list (thus the - in YAML) and not a mapping, since the order of the elements is important, as it defines which element is which: datatypes: obj1: composed_of: first: unsigned_integer second: float third: {regex: "[A-Za-z0-9]"} splitted_by: " " Enclosing strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' As for lists, composed_of definitions can include a prefix and/or a suffix option, which de- fine enclosing strings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' parentheses) before the first ele- ment and/or after the last element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Element separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The splitted_by and separator options are used for describing how to separate the single elements of the compound value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' These have the same kind of usage already illustrated in the Lists section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' splitted_by is used for separators which cannot occur in the elements text, while separator is used otherwise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' when the escaped separator can occur in the elements or the element size is recognized by their format, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' fixed-length elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Multiple separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, different separators are used between different pairs of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case, they can be specified as additional constant elements and hidden Gonnella | TFSL definitions by value type arXiv | 7 in the decoded dictionary using the hide_constants op- tion, as in the following example: datatypes: obj2: composed_of: x: unsigned_integer sep1: {constant: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='"} y: float sep2: {constant: "|"} z: {regex: "[A-Za-z]"} hide_constants: true An example of string representation which is parsed by the previous definition is 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='0|A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Hereby the resulting map- ping has three elements, as the constant separators are only considered for parsing: {x:1, y:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='0, z:A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Optional separated elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Some of the elements can be optional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' sometimes absent from the sequence of ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In case the sequence is splitted by a non-empty sepa- rator string, an empty element can be recognized by the pres- ence of this separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case the empty option is used in the elements definitions, as in the following example: datatypes: obj3: composed_of: first: {values: [1,2], empty: 0} second: {values: [A,B], empty: C} splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" Optional trailing elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases, a given number of elements at the beginning of the text representation may be mandatory, while the following can be present or not (and are mandatory only if elements after them in the order are present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case, the required option is used (number of required elements): datatypes: obj4: composed_of: first: {values: [1,2], empty: 0} second: {values: [A,B], empty: C} splitted_by: ";' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='" required: 1 Optional internal elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In the case, some internal ele- ment can be missing (together with its associated separator, or when no separator is used) but there is no ambuiguity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' because the following element of the sequence has a type that allows it to be distinguished from the optional element, or because the total number of elements changes depending on the presence or absence of the optional element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In this case the user must provide multiple alternative defini- tions of the structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' with and without the optional ele- ment) using a one_of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Furthermore, in order to provide the same set of values for all instances of the object or struct, the implicit option can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' datatypes: obj5: one_of: composed_of: name: string expressed: values: {"+": true, "-": false} splitted_by: "," implicit: {"copies": 1} composed_of: name: string copies: unsigned_integer expressed: values: {"+": true, "-": false} splitted_by: "," The above definition would parse the string representa- tion "X,+" to the mapping {name: X, expressed: true, copies: 1}, while "X,2,+" would be parsed to {name: X, expressed: true, copies: 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In some cases an element of a format can be ex- pressed in multiple different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In dynamically typed lan- guages such as Python, any variable can store this kind of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In C, such values could be e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' stored as unions, and in Nim as variant objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In TextFormats the type of such values can be defined defined using definitions of type one_of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' the following allows to represent an unsigned integer value, as a number, if it is >= 1, otherwise a floating point: datatypes: num10: one_of: unsigned_integer: {min: 1} float Note that the content of the one_of key is a YAML list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' the order of the elements in the list defines the order or prece- dence of the definitions (the first which applies is used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Conclusions In this paper, the representation of different kind of data in text formats, as specified using the library TextFormats, has been analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Thereby it was demonstrated that most types of values that can be used in programming languages such as C and Python, can also be represented in TextFormats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' In TextFormats specifications, the same type of value is some- times represented using a different kind of datatype definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' This is true for both scalar values and compound values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' For example, in the above examples, boolean values are some- times represented using definitions of kind values, some- times regexes or even constant, depending on what is their representation in the format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Among the examples for compound values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=', collections of tagged elements are sometimes represented using list_of, but in other cases 8 | arXiv Gonnella | TFSL definitions by value type using the specialized list definition keys tagged_list and labeled_list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The reason for this is that the TextFormat syntax for datatype definitions is oriented to the text representation and not to the type of represented value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The systematic review of the definition types based on the type of value is particularly use- ful when defining a new format and complements the TFSL syntax manual included in the library documentation, which is more useful when a specification is written for a format which already exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' References Giorgio Gonnella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Textformats: Simplifying the definition and parsing of text formats in bioinformatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' PLOS ONE, 17(5):1–17, 05 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='0268910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1371/journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='pone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='0268910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' GFA Format Specification Working Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The GFA format specification, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' URL http://gfa-spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='io/GFA-spec/GFA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' GFA Format Specification Working Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Graphical frag- ment assembly (GFA) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='0 format specification, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' URL http://gfa-spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='io/GFA-spec/GFA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Petr Danecek, Adam Auton, Goncalo Abecasis, Cornelis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Albers, Eric Banks, Mark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' DePristo, Robert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Hand- saker, Gerton Lunter, Gabor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Marth, Stephen T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Sherry, Gilean McVean, Richard Durbin, and 1000 Genomes Project Analysis Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The variant call format and VCFtools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Bioinformatics, 27(15):2156–2158, 06 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ISSN 1367-4803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1093/bioinformatics/btr330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1093/bioinformatics/btr330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Heng Li, Bob Handsaker, Alec Wysoker, Tim Fennell, Jue Ruan, Nils Homer, Gabor Marth, Goncalo Abecasis, Richard Durbin, and 1000 Genome Project Data Process- ing Subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The Sequence Alignment/Map format and SAMtools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Bioinformatics, 25(16):2078–2079, 06 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ISSN 1367-4803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1093/bioinformatics/btp352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='1093/bioinformatics/btp352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Giorgio Gonnella has been supported by the DFG Grant GO 3192/1-1 “Automated characterization of microbial genomes and metagenomes by collection and verifica- tion of association rules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS These contributions follow the Contributor Roles Taxonomy guidelines: https://casrai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='org/credit/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Conceptualization: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Data curation: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Formal analysis: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Funding acquisition: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Investigation: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Methodology: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Project administration: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Resources: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Software: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Supervision: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Validation: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Visualization: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Writing – original draft: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Writing – review & editing: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' COMPETING FINANCIAL INTERESTS The authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} +page_content=' Gonnella | TFSL definitions by value type arXiv | 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFRT4oBgHgl3EQfYzfm/content/2301.13551v1.pdf'} diff --git a/nNE1T4oBgHgl3EQfhQT4/content/tmp_files/2301.03240v1.pdf.txt b/nNE1T4oBgHgl3EQfhQT4/content/tmp_files/2301.03240v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a7334f74e7cd7d270bded95d04b452e43b386ed --- /dev/null +++ b/nNE1T4oBgHgl3EQfhQT4/content/tmp_files/2301.03240v1.pdf.txt @@ -0,0 +1,760 @@ +1 +Superconductivity in an Orbital-reoriented SnAs Square Lattice: +a Case Study of Li0.6Sn2As2 and NaSnAs +Junjie Wang†[a], Tianping Ying*†[a], Jun Deng†[a], Cuiying Pei†[b], Tongxu Yu[c], +Xu Chen[a], Yimin Wan[d], Mingzhang Yang[a], Weiyi Dai[c], Dongliang Yang[e], +Yanchun Li[e], Shiyan Li[d], Soshi Iimura[f], Shixuan Du[a], Hideo Hosono[f], +Yanpeng Qi*[b], Jian-gang Guo*[a,g] +[a] J.J. Wang, Dr. J. Deng, Dr. X. Chen, M. Z. Yang, Prof. Dr. S. X. Du, Prof. +Dr. T. P. Ying, Prof. Dr. J.-G. Guo +Institute of Physics and University of Chinese Academy of Sciences, +Chinese Academy of Sciences, Beijing 100190, China +E-mail: ying@iphy.ac.cn; jgguo@iphy.ac.cn +[b] Dr. C. Y. Pei, Prof. Dr. Y. P. Qi +School of Physical Science and Technology, ShanghaiTech University, +Shanghai 201210, China + +ShanghaiTech Laboratory for Topological Physics, ShanghaiTech +University, Shanghai 201210, China + +Shanghai Key Laboratory of High-resolution Electron Microscopy, +ShanghaiTech University, Shanghai 201210, China +E-mail: qiyp@shanghaitech.edu.cn +[c] Dr. T. X. Yu, Dr. W. Y. Dai + +Gusu Laboratory of Materials, Jiangsu 215123, China + +Suzhou Laboratory, Jiangsu 215123, China +[d] Y. M. Wan, Prof. Dr. S. Y. Li, State Key Laboratory of Surface Physics +and Department of Physics, Department of Physics, Fudan University, +Shanghai 200438, China +[e] Dr. D. L. Yang, Prof. Dr. Y. C. Li +Beijing Synchrotron Radiation Facility and Institute of High Energy +Physics, Chinese Academy of Sciences, Beijing, 100049, China +[f] Prof. Dr. S. Iimura, Prof. Dr. H. Hosono +National Institute for Materials Science (NIMS), Tsukuba, Ibaraki, 305- +0047, Japan +Materials Research Center for Element Strategy, Tokyo Institute of +Technology, Yokohama, 226-8503, Japan +[g] Prof. Dr. J.-G. Guo +Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, +China +† +These authors contribute equally. + + + + + +2 +Abstract: Searching for functional square lattices in layered superconductor +systems offers an explicit clue to modify the electron behavior and find exotic +properties. The trigonal SnAs3 structural units in SnAs-based systems are +relatively conformable to distortion, which provides the possibility to achieve +structurally topological transformation and higher superconducting transition +temperatures. In the present work, the functional As square lattice was realized +and activated in Li0.6Sn2As2 and NaSnAs through a topotactic structural +transformation of trigonal SnAs3 to square SnAs4 under pressure, resulting in a +record-high Tc among all synthesized SnAs-based compounds. Meanwhile, the +conductive channel transfers from the out-of-plane pz orbital to the in-plane +px+py orbitals, facilitating electron hopping within the square 2D lattice and +boosting the superconductivity. The reorientation of p-orbital following a +directed local structure transformation provides an effective strategy to modify +layered superconductors. +Introduction +Attributing the emergent properties of compounds to structural units is +naturally a reductionism standpoint, but is highly desirable and easily operable +to understand, manipulate, and predict the performance of the systems under +various physical conditions. In this aspect, the well-defined square lattice plane +in high transition temperature (Tc) superconductors, e.g., cuprates[1],[2] and iron- +based[3],[4] systems is deemed as an indispensable ingredient and plays a +unique role in penetrating the puzzle of underlying pairing mechanisms. The +cuprates share the square Cu-O plane where the electrons propagate through +the Cu d-orbitals with the super-exchange interaction of oxygen p-orbitals[5],[6],[7]. +Adding an apical oxygen to enhance the out-of-plane dispersion of the electron +will localize the in-plane electronic wave function, and ultimately lower the +maximum Tc[8]. Similarly, the key ingredient of iron-based superconductors is +the spatial repetition of FeAs/FeSe tetrahedron units to form a layered structure +with alternating square lattices of iron and As/Se planes[9],[10],[11],[12],[13], in which +the distortion of the FeSe from planar-tetragonal to 3D-trigonal connections +under high pressure is accompanied by the complete loss of superconductivity +(SC)[14]. From a stereochemistry perspective, the square tiling can make full +use of the orthogonal d- and p-orbitals, enabling easy charge carrier hopping +between adjacent lobes[15]. Thus, searching novel layered systems with square +motifs is a promising strategy for exploring high-Tc superconductors. +SnAs-based compounds are such auspicious candidates, where the Tcs of +layered compounds are surprisingly lower than their 3D counterparts. Layered +SnAs-based compounds have diverse combinations with alkali[16],[17],[18],[19], +alkaline-earth[20], rare-earth metals[21], and even molecule clusters[22], resulting +in highly tunable carrier concentrations and stacking sequence. Despite the +structural flexibility, however, they are often semiconducting or barely +superconducting at low temperatures of 1.1~1.3 K[16],[17],[18],[19],[20],[21],[22]. The + +3 +binary SnAs with a rock-salt structure (Fm-3m), in contrast, has a relatively +higher Tc of 4 K[23],[24]. A naive interpretation of the lowered Tc in layered-SnAs +compounds is related to the trigonal pyramid SnAs3 unit. It is the steric effect of +the misaligned p-orbitals that somehow impedes electron flow within the +superconducting layer. +High pressure can create atypical materials by repopulating atomic orbitals +and +reconstructing +functional +units +without +altering +the +chemical +composition[25],[26],[27],[28]. Herein, we explore the possibility of obtaining higher- +Tc superconductor by constructing square-lattice configurations under pressure, +employing layered metallic Li0.6Sn2As2 and semiconducting NaSnAs as +examples. Theoretical calculations predict a topotactic transformation of the +functional units from SnAs3 to SnAs4, accompanied by a trigonal-tetragonal +phase transition, which is verified by our in-situ synchrotron diffraction +measurements. Prominently, the tetragonal Li0.6Sn2As2 shows the highest Tc of +7.5 K among all the synthesized SnAs-based compounds that is six times +higher than its ambient value. Pressurized NaSnAs exhibits a similar SnAs3- +SnAs4 transformation and enhanced SC, but with a more complex three-stage +phase transformation from semiconductor to two successive superconducting +phases. Theoretical analyses of their orbital components reveal an intriguing +charge redistribution from out-of-plane to in-plane, which enhances electron +hopping and contributes to the record high Tc in the SnAs square lattice. +Results and Discussion +Ambient structures (α phase) of Li0.6Sn2As2 (R3̅m, Fig. 1a) and NaSnAs +(P63mc, Fig. S1) share the similar trigonal pyramid of SnAs3 building blocks, +with the intercalation of alkali metals in between one or two SnAs layers. As +shown in Fig. 1c, the electrons from 5p of Sn, 4p of As and one extra electron +from the Li+/Na+ construct the bonding state of SnAs3, leaving a lone pair from +5s2 dangling along the c axis, away from the SnAs3 pyramid to create the +canonical lone-pair electrons[26],[29],[30]. Due to the fully occupied Sn-As bonds +and the repulsive interaction of lone-pair electrons, NaSnAs turns out to be a +semiconductor with a band gap of 0.31 eV[31]. It is therefore easy to understand +the metallic nature of Li0.6Sn2As2 (0.3 e/SnAs) and NaSn2As2 (0.5 e/SnAs) with +reduced electron doping, which further become superconducting with similar Tc +of 1.5 K (Note that Li0.6Sn2As2 is a newly discovered superconductor, and its +properties are shown in Figs. S2 and S3). Given that the electrons move within +the SnAs plane in a zigzag manner (illustrated as dashed curve in Fig. 1c), it is +possible to alter the SnAs3 configuration by pressure, which may facilitate the +flow of electrons. Therefore, we begin with determining the structure under +pressure through theoretical structure search package CALYPSO[32],[33]. During +the structure searching, the size of unit cells is limited from 2 up to 4 formulas +for each stoichiometry (see Fig. S4). Both the generation sizes and the number +of generations were set to 30 for getting a converged result. + +4 +Our structural searches up to 30 GPa reveal that a new phase with a space +group of I4/mmm (β phase) is the most stable one. We superimposed the β +phase on the α phase as open circles to illustrate the structural transformation +(Fig. 1b and Fig. S5). The SnAs3 building blocks only need to bear a mild +distortion under this transformation. This is reasonable if considering that the +coordination numbers generally increase under high pressure to homogenize +the electron distribution [34],[35]. We check the pressure-dependent enthalpy of +the β phase with respect to that of the α phase from 0 to 50 GPa in Fig. 1(d). +Beyond a critical pressure of around 25 GPa, the β phase overwhelms the α +phase in energy. Figure S6 shows the phonon dispersion of the β phase at 30 +GPa, which does not have any imaginary frequencies, indicating the stability of +the high-pressure phase. It is worth noting that the tetragonal pyramid +configuration of SnAs4 has not been previously reported. +We also did the structural search of NaSnAs under pressure. At low pressure, +NaSnAs undergoes a structural transformation from the α phase to the β phase +at 15 GPa, with the lowest enthalpy in the Na-Sn-As system. However, +calculations at 30 GPa predict that another tetragonal phase with the space +group P4/mmm (γ phase) is the most stable. Figure S7 depicts the successive +structural transformation of the three phases. The γ phase can be viewed as a +change in the stacking sequence of the Na spacer layer from the β phase, +requiring every other Na layer transmits through its adjacent SnAs layers. The +β phase is predicted to be stable in a pressure window of 15-20 GPa, as +indicated by the pressure-dependent formation enthalpy shown in Fig. 1e. +Obviously, the interlayer transmission of Na from the β phase to γ phase should +be more energetically costly than the intralayer SnAs3 to SnAs4 transformation. +Nonetheless, both β phase and γ phase share the identical SnAs4 building block. +To verify the SnAs3-SnAs4 transformation, we performed in-situ synchrotron +diffraction experiments on Li0.6Sn2As2 at various pressures ranging from +ambient pressure to 48.0 GPa. All diffraction peaks show a systematic shift +below 25.0 GPa and can be indexed into the α phase (Fig. S8). An additional +peak at 15.5° gradually arises as pressure above 25.0 GPa, signaling a +pressure-driven phase transition. When the pressure is increased to 40 GPa, +the initial peaks corresponding to the α phase become almost indistinguishable. +We carried out Rietveld refinements on each synchrotron x-ray diffraction +pattern, and put three typical profiles of 48.0 GPa, 32.8 GPa, and 1.88 GPa in +Fig. 2a. We refined the pattern of 48.0 GPa based on the theoretically predicted +β phase and obtained the agreement factors of Rp = 0.84% and Rwp = 5.3%, +which are comparable to the ambient refinement values of Rp = 1.21% and Rwp += 5.43%, respectively. A two-phase refinement against the pattern of 32.8 GPa +produces the best fitting result. We note that the diffraction pattern +depressurized phase is almost identical to the initial one except mild peak +broadening (Fig. S9), indicating the reversibility of the SnAs3-SnAs4 +transformation. + +5 +The contour plot of the diffraction patterns reveals more details of the +structural transition (Fig. 2b). The critical pressure of 25.0 GPa is clearly with +the emergence of new diffraction peaks at 15.5° and 20.2°. An interesting +observation is that the diffraction peaks corresponding to the α phase (102, 105, +110, 025, and 207 in particular) exhibit a consistent kink at 25.0 GPa, indicating +substantial lattice distortion in the α phase as an intermediate transition state. +A qualitative analysis of the volume fractions of α and β phase extracted from +Rietveld refinements are shown in Fig. 2c. +Another notable thing is the dramatic change in bond lengths under pressure. +Figure S10 labels the Sn-As and As-As distances at 0 and 30 GPa. The out-of- +plane As-As distance is reduced from 4.04 Å to 2.58 Å and the in-plane As-As +distance shrinks abruptly from 4.01 Å to 3.51 Å along with the SnAs3-SnAs4 +transformation. However, the Sn-As bond is increased to 2.97 Å, which is even +larger than the ambient value of 2.72 Å. The elongation of the Sn-As bond +suggests that the prior conduction channel through As-Sn-As is greatly +impeded, whereas direct hopping between As-As anions within the square +lattice should be much favored. +The increased coordination numbers, elongated Sn-As bonds, and collapsed +As-As distance induced by the SnAs3-SnAs4 transformation significantly affect +the electronic transport properties. The α-Li0.6Sn2As2 exhibits only one +superconducting transition below 27.7 GPa, with its Tc gradually increasing +from 1.3 K to 5.3 K (blue arrows in Fig.3a). Noticeably, a drop in resistivity at +7.5 K is observed at 27.7 GPa (enlarged rectangle shown in Fig. 3a), which is +consistent with the emergence of the β phase around 25.0 GPa. To the best of +our knowledge, this is the highest Tc experimentally realized in the SnAs-based +compounds, six times higher than the ambient value. As pressure increasing, +the decline in resistivity gets more pronounced, yet its transition temperature +remains nearly constant (red arrows in Fig. 3a). All the raw data are shown in +Fig. S11. We confirm that this newly discovered kink in resistivity is +superconductive according to the gradual suppression of the Tc under magnetic +fields (Fig. 3c). The superconducting drop corresponding to the α phase is seen +at 53.1 GPa, indicating the survival of the trace α phase. This is also in line with +our estimation of the volume fraction shown in Fig. 2c. +Figure 3b reveals the Tc evolutions of the α phase and β phase in response +to pressure. The Tc in the β phase remains nearly constant at moderate +pressure (25.0~45.0 GPa) and then decreases somewhat. Their Tcs become to +be one transition above 53.0 GPa. Given that the structure at high pressure is +dominated by the β phase, the remaining superconducting transition should be +attributed to the SnAs4 arrangement. Note that the maximum Tc is achieved in +the mixture phase rather than the pure β phase. A plausible reason for this +could be that the Tc-pressure diagram of the β phase has a dome-like shape, +within which the maximum Tc occurring ~35 GPa. Figure 3d shows the Hc2(0) +of α (0 GPa) and β (73 GPa) phases. Linear extrapolation shows a higher Hc2(0) +in the β phase. + +6 +The ensuing transition from α to β to γ phases in NaSnAs is investigated by +transport measurements. NaSnAs undergoes a semiconductor-metal transition +under lower pressure, then becomes SC at ~10 GPa (shown in Fig. S12). Its Tc +slowly increases to 3.6 K at 35 GPa, then suddenly jumps to 6.8 K. While the +onset pressure of the first superconducting phase at ~10 GPa is qualitatively +consistent with the theoretical pressure of α-β phase transition, the pressure of +Tc’s jump is higher than our theoretical prediction of 20 GPa (Fig. 1e). This is +understandable because the predicted γ phase is a thermally equivalent ground +state. Since our in-situ high-pressure measurements are carried out in a DAC +cell at room temperature, the energy barrier for the inter-plane migration (β-γ) +should be much higher than the intra-plane ion displacement (α-β). Despite +their different carrier doping levels, the charge-balanced NaSnAs achieves a +comparable Tc to that of the β-Li0.6Sn2As2, indicating the highest Tc in the SnAs +system is determined the common building block of SnAs4 in terms of structure. +The structural transformation of SnAs3-SnAs4 can lead to charge +rearrangements of conducting electrons. As shown in Fig. S13, the Fermi level +(EF) in LiSn2As2 is mostly composed of As’s 4p and Sn’s 5p orbitals, with the +4s and 5s orbitals lying far below the EF. We plot the orbital projected density +of states (PDOS) in Fig. 4a. At ambient pressure, the PDOS of As’s pz +component at 0.98 states/eV is higher than the sum of its px and py orbital (0.76 +states/eV). This feature is more prominent for Sn because the intensity ratio of +pz/(px+py) reaches 3.6 at the EF. Specifically, the spectra weights of pz and px+py +are reversed for both As and Sn at high pressure. Specifically, the +enhancement of the PDOS of As’s px+py is 4 times higher than that of the pz +component. The SnAs3-SnAs4 transformation induces a prominent charge +redistribution from the out-of-plane (pz) orbital to the in-plane (px, py) ones. We +perform Crystal Orbital Hamilton Population (COHP) calculations for the α and +β phases. As shown in Fig. 4b, the α phase is dominated by Sn-As bonding +states below -1 eV, with a small component at the EF. In the β phase, however, +the out-of-plane As-As bonding component of inter SnAs4 units overwhelms +that of Sn-As bonding of intra unit, and dominates the EF by As-As anti-bonding. +Moreover, the integrals of COHP (ICOHP) up to the EF for Sn-As and As-As are +-3.16 and -0.08 eV/pair in the α phase, and change to -1.48 and -3.89 eV/pair +in the β phase, respectively. This suggests that the interaction of Sn-As +dominates in the α phase, while that of As-As prevails in the β phase. +We plot the partial electron distribution near the EF (-0.2~0.2 eV) to see the +charge distribution. For the α-LiSn2As2, the valence electrons mainly gather +around Sn and As atoms and are oriented along the c axis (Fig. 4c, left). Thus +the electrons can mainly migrate along As-Sn-As via zigzagging through the +shared 5pz-4pz orbitals (Fig. 4d left panel and Fig. 1c right panel). Once the +SnAs3 is transformed into SnAs4, the charge redistributions from pz to px+py are +clearly seen in the right panels of Fig. 4c and 4d. We examined the partial +electron distribution of the β-LiSn2As2 in each energy sector. With the electron + +7 +filling from the lowest occupied σ bond to the σ* bond, their respective electron +dispersion in real space is shown in Fig. S14. +Summarizing all the analyses above, we provide a simple molecular orbital +diagram to understand the evolution of bond connections in LiSn2As2 (Fig. 4f +and 4g). The out-of-plane As-As interaction drives the EF sitting on the πxy* band +with the extra electron donation from Li and Sn, which well explains the +calculated dominating px+py PDOS (lower panel in Fig. 4a) and the As-As anti- +bonding (lower panel in Fig. 4b). Considering the orthogonal configuration of px +and py orbitals, the electron is more likely to migrate within this As-square lattice +than in the triangle ones (Fig. 4e). The abrupt shrinkage in the As-As distance +from 4 Å to 3.5 Å, on the other hand, reinforces in-plane hopping via overlapping +of the orthogonal px+py lobes. This reminds us of the highly tunable Bi-Bi +distance from 3.85 to 4.08 Å in Bi-based square lattice R2O2Bi (R = rare earth +metals), accompanied by a metal-insulator transition at a shortened Bi-Bi +distance of 3.95 Å in Sm2O2Bi[36] and a subsequent SC at 3.87 Å in Y2O2Bi[37]. +As our simulations are all based on the fully occupation of Li in LiSn2As2, the +influence of Li vacancies is carefully checked. To this end, we create a 3×3×1 +superlattice and randomly removed 7 out of 18 Li atoms in the supercell +corresponding to Li0.61Sn2As2. Figure S15 shows the partial charge density at +30 GPa near EF (-0.2~0.2 eV) of the model. It again shows a px/py conducting +character, indicating that the Li vacancy does not influence our main result. +The electronic states in the pressurized NaSnAs resemble those of +Li0.6Sn2As2. As shown in Fig. S16, the EF of both β and γ phases are dominated +by px+py orbitals. The 2D feature of the electron dispersion becomes more +prominent in the γ phase (Fig. S17). It is the charge transfer from pz to px+py +that facilitates the direct hopping of electrons. Given the similarity of charge +distribution in the SnAs4 unit, we conclude that the orbital reorientation of As’s +p orbitals in a square lattice should be responsible for the similar Tc in both +compounds. +Conclusion +In this work, a new kind of structure unit SnAs4 is realized in SnAs-based +compounds through a topotactic phase transformation in both Li0.6Sn2As2 and +NaSnAs. Thanks to the thus-formed As square lattice and the dramatic +shrinkage of the in-plane As-As distance from SnAs3 to SnAs4 transformation, +the probability of in-plane hopping through px+py lobes is much enhanced. +Theoretical calculations show a charge redistribution of the electron near the +EF from pz orbital to px+py orbitals, further facilitating the electron transportation +and the subsequent record-high Tc in all reported SnAs-based compounds. Our +findings verify the effectiveness of modulating SC through directed local +structure tailoring of flexible layered compounds. +Acknowledgements +We appreciate Prof. Xianxin Wu for fruitful discussion. This work is financially +supported by the National Key Research and Development Program of China + +8 +(No. +2018YFE0202600, +2021YFA1401800), +Beijing +Natural +Science +Foundation (Grant No. Z200005), the National Natural Science Foundation of +China (No. 51922105, 11804184, and 11974208), the Strategic Priority +Research +Program +of +the +Chinese +Academy +of +Sciences +(Grant +XDB30000000). ADXRD measurements were performed at 4W2 High Pressure +Station, Beijing Synchrotron Radiation Facility (BSRF), which is supported by +Chinese Academy of Sciences (Grant KJCX2-SW-N20, KJCX2-SW-N03). Y.Q. +would like to acknowledge the National Key R&D Program of China (Grant No. +2018YFA0704300), the National Natural Science Foundation of China (grant +nos. U1932217, 11974246, and 12004252). The authors thank the support from +CħEM +(02161943) +and +Analytical +Instrumentation +Center +(SPST- +AIC10112914), SPST, ShanghaiTech University. H. 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Theoretical prediction of the structure transition of LiSn2As2 and +NaSnAs. Top and side-views of LiSn2As2 at a) ambient pressure (R3̅m, α +phase) and b) high pressure (I4/mmm, β phase) predicted by the CALYPSO +package. c) Bonding environment of SnAs-layered compounds and the +illustration of the zigzag-propagation of electrons within the ab plane. The lone- +pair electrons are denoted by light blue droplets. d) Relative enthalpy of +LiSn2As2 as a function of pressure. The insets depict the trigonal (SnAs3) and +tetragonal (SnAs4) pyramids. e) Relative enthalpy of P63mc (α phase), I4/mmm +(β phase) and P4/mmm (γ phase) for NaSnAs as a function of pressure. The +enthalpy of the α phase is taken as a reference. + + + + + + + + + + + + + + + + + + + +a +b +C +lone-pair electrons +Sn: 5s2 5p2 +As: 4s24p3 +As +LiSna +NaSnAs +d +e +150 +Enthalpy (meV/atom) +Enthalpy (meV/atom) +200 +AS +100 +Sn +100 +50 +SnAs4 +SnAs3 +100 +50 +200 +0 +20 +40 +0 +10 +20 +30 +40 +α phase +βphase +Pressure (GPa) +Pressure(GPa)11 + + +Figure 2. In-situ synchrotron diffractions of Li0.6Sn2As2 at pressures. a) Three +Rietveld refinements profiles of Li0.6Sn2As2 at 48.0 GPa, 32.8 GPa and 1.88 +GPa. b) Color contour plot of the pressure-dependent diffraction peaks from +1.88 to 48.0 GPa. c) Pressure-mediated volume fraction of the α phase and β +phase extracted from refinement. + + + + + + + + + + + + + + + + + + + + + + + + + + +a +48.0GPa +R=0.84% +40 +30 +(GPa +20 +0 +207 +32.8GPa +R.=1.15% +Intensity (a.u.) +Rwp=5.43% +10 +:aphase +8101214161820222426 +I:Bphase +2(degree) +C +100 +100 +1.88 GPa +Observed +80 +80 +(%) +Calculated +R,=1.23% +phase +60 +60 +aαphase +βphase +R +=5.43% +40 +eβphase +40 +WD +20 +20 +11111 +0 +0 +810121416182022242628 +0 +10 +20 +30 +40 +50 +29(degree) +P (GPa)12 + + +Figure 3. Pressure-dependent SC in Li0.6Sn2As2. a) Temperature-dependent +resistivity of Li0.6Sn2As2 under pressure. Insets are the enlargement of resistivity +at higher temperatures. Blue and red arrows indicate the onset Tc of α and β +phases. b) Superconducting phase diagram of Li0.6Sn2As2 under pressure. An +intermediate pressure region mixes two transition temperatures. c) Resistivity +of the β phase (75.3 GPa) from 1.5 K-8 K at different magnetic fields. d) Fitting +of the upper critical fields for the ambient (black squares) and high pressure +(red circles) phases. + + + + + + + + + + + + + + + + + + + + + + +a +b +9 +C +75.3 +2.5 T +aαphase +mixture +βphase +8 +0.2 +cm) +66.5 +7 +53.1 +6 +44.7 +OT +(a.u.) +5 +0.0 +378 +2 +4 +6 +8 +T (K) +d +3 +2.2 +O 75.3 GPa +3 +Ambient +7.7 +E +2 +pressure +Ambientpressure +20.8 +Run1 +15.5 +1 +Run2 +0GPa +Run3 +2 +4 +6 +8 +1012 +10 +20 +30 +40 +50 +60 +7080 +4 +6 +8 +T (K) +P (GPa) +T。 (K)13 + +Figure 4. Charge redistribution and orbital reorientation from SnAs3-SnAs4 +transformation in LiSn2As2. a) Projected density of states (PDOS) and b) - +COHP for α phase at 0 GPa (upper panel) and β phase at 30 GPa (lower panel), +respectively. Insets in b) show the interaction of As-As and Sn-As. Positive +values indicate bonding states, and negative ones are antibonding states. c-e) +Partial charge density around the EF (-0.2 eV ~0.2 eV) with iso-value 0.0025 +e/Bohr3, section projections, and illustration of bond connections of the α phase +(left) and β phase (right), respectively. f-g) Molecular orbital diagram of the Sn- +As bonding state in the α phase and out-of-plane As-As antibonding state in the +β phase. Note that extra electrons in the πxy* band come from the donation of +Li and Sn. + +a +αphase +βphase +f +As-py+Px +Sn-Py+Px +aphase +PDOS (states/eV) +As-Pz +Sn-Pz +Sn-5p2 +0 +[110] +As-4 +d +e/bohr3 +0.005 +axy +0 +-1.0 +0.5 +0.0 +0.5 +1.0 +Energy(eV) +C +b +Sn-As +[110] +βphase +As-As +0 +e +Li, Sn +Sn-As +HP +As-As +e +CO +Sn-As +As-As +AS +As-4p3 +0 +TTxy +a +Aspx +4 +-2 +0 +2 +4 +Aspy +Energy(eV) +5pz(Sn)+4pz(As) \ No newline at end of file diff --git a/nNE1T4oBgHgl3EQfhQT4/content/tmp_files/load_file.txt b/nNE1T4oBgHgl3EQfhQT4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1e1a2641a7e7132849b240b6f48f5bf7e837ed0 --- /dev/null +++ b/nNE1T4oBgHgl3EQfhQT4/content/tmp_files/load_file.txt @@ -0,0 +1,898 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf,len=897 +page_content='1 Superconductivity in an Orbital-reoriented SnAs Square Lattice: a Case Study of Li0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Du, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Ying, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Guo Institute of Physics and University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China E-mail: ying@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' jgguo@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='cn [b] Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Pei, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Qi School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China Shanghai Key Laboratory of High-resolution Electron Microscopy, ShanghaiTech University, Shanghai 201210, China E-mail: qiyp@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='cn [c] Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Yu, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dai Gusu Laboratory of Materials, Jiangsu 215123, China Suzhou Laboratory, Jiangsu 215123, China [d] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Wan, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Li, State Key Laboratory of Surface Physics and Department of Physics, Department of Physics, Fudan University, Shanghai 200438, China [e] Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Yang, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Li Beijing Synchrotron Radiation Facility and Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049, China [f] Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Iimura, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Hosono National Institute for Materials Science (NIMS), Tsukuba, Ibaraki, 305- 0047, Japan Materials Research Center for Element Strategy, Tokyo Institute of Technology, Yokohama, 226-8503, Japan [g] Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Guo Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China † These authors contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2 Abstract: Searching for functional square lattices in layered superconductor systems offers an explicit clue to modify the electron behavior and find exotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The trigonal SnAs3 structural units in SnAs-based systems are relatively conformable to distortion, which provides the possibility to achieve structurally topological transformation and higher superconducting transition temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' In the present work, the functional As square lattice was realized and activated in Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 and NaSnAs through a topotactic structural transformation of trigonal SnAs3 to square SnAs4 under pressure, resulting in a record-high Tc among all synthesized SnAs-based compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Meanwhile, the conductive channel transfers from the out-of-plane pz orbital to the in-plane px+py orbitals, facilitating electron hopping within the square 2D lattice and boosting the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The reorientation of p-orbital following a directed local structure transformation provides an effective strategy to modify layered superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Introduction Attributing the emergent properties of compounds to structural units is naturally a reductionism standpoint, but is highly desirable and easily operable to understand, manipulate, and predict the performance of the systems under various physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' In this aspect, the well-defined square lattice plane in high transition temperature (Tc) superconductors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=', cuprates[1],[2] and iron- based[3],[4] systems is deemed as an indispensable ingredient and plays a unique role in penetrating the puzzle of underlying pairing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The cuprates share the square Cu-O plane where the electrons propagate through the Cu d-orbitals with the super-exchange interaction of oxygen p-orbitals[5],[6],[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Adding an apical oxygen to enhance the out-of-plane dispersion of the electron will localize the in-plane electronic wave function, and ultimately lower the maximum Tc[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Similarly, the key ingredient of iron-based superconductors is the spatial repetition of FeAs/FeSe tetrahedron units to form a layered structure with alternating square lattices of iron and As/Se planes[9],[10],[11],[12],[13], in which the distortion of the FeSe from planar-tetragonal to 3D-trigonal connections under high pressure is accompanied by the complete loss of superconductivity (SC)[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' From a stereochemistry perspective, the square tiling can make full use of the orthogonal d- and p-orbitals, enabling easy charge carrier hopping between adjacent lobes[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Thus, searching novel layered systems with square motifs is a promising strategy for exploring high-Tc superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' SnAs-based compounds are such auspicious candidates, where the Tcs of layered compounds are surprisingly lower than their 3D counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Layered SnAs-based compounds have diverse combinations with alkali[16],[17],[18],[19], alkaline-earth[20], rare-earth metals[21], and even molecule clusters[22], resulting in highly tunable carrier concentrations and stacking sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Despite the structural flexibility, however, they are often semiconducting or barely superconducting at low temperatures of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='1~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 K[16],[17],[18],[19],[20],[21],[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The 3 binary SnAs with a rock-salt structure (Fm-3m), in contrast, has a relatively higher Tc of 4 K[23],[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' A naive interpretation of the lowered Tc in layered-SnAs compounds is related to the trigonal pyramid SnAs3 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' It is the steric effect of the misaligned p-orbitals that somehow impedes electron flow within the superconducting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' High pressure can create atypical materials by repopulating atomic orbitals and reconstructing functional units without altering the chemical composition[25],[26],[27],[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Herein, we explore the possibility of obtaining higher- Tc superconductor by constructing square-lattice configurations under pressure, employing layered metallic Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 and semiconducting NaSnAs as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Theoretical calculations predict a topotactic transformation of the functional units from SnAs3 to SnAs4, accompanied by a trigonal-tetragonal phase transition, which is verified by our in-situ synchrotron diffraction measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Prominently, the tetragonal Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 shows the highest Tc of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 K among all the synthesized SnAs-based compounds that is six times higher than its ambient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Pressurized NaSnAs exhibits a similar SnAs3- SnAs4 transformation and enhanced SC, but with a more complex three-stage phase transformation from semiconductor to two successive superconducting phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Theoretical analyses of their orbital components reveal an intriguing charge redistribution from out-of-plane to in-plane, which enhances electron hopping and contributes to the record high Tc in the SnAs square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Results and Discussion Ambient structures (α phase) of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 (R3̅m, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1a) and NaSnAs (P63mc, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S1) share the similar trigonal pyramid of SnAs3 building blocks, with the intercalation of alkali metals in between one or two SnAs layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1c, the electrons from 5p of Sn, 4p of As and one extra electron from the Li+/Na+ construct the bonding state of SnAs3, leaving a lone pair from 5s2 dangling along the c axis, away from the SnAs3 pyramid to create the canonical lone-pair electrons[26],[29],[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Due to the fully occupied Sn-As bonds and the repulsive interaction of lone-pair electrons, NaSnAs turns out to be a semiconductor with a band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='31 eV[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' It is therefore easy to understand the metallic nature of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 e/SnAs) and NaSn2As2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 e/SnAs) with reduced electron doping, which further become superconducting with similar Tc of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 K (Note that Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 is a newly discovered superconductor, and its properties are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S2 and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Given that the electrons move within the SnAs plane in a zigzag manner (illustrated as dashed curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1c), it is possible to alter the SnAs3 configuration by pressure, which may facilitate the flow of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Therefore, we begin with determining the structure under pressure through theoretical structure search package CALYPSO[32],[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' During the structure searching, the size of unit cells is limited from 2 up to 4 formulas for each stoichiometry (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Both the generation sizes and the number of generations were set to 30 for getting a converged result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4 Our structural searches up to 30 GPa reveal that a new phase with a space group of I4/mmm (β phase) is the most stable one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We superimposed the β phase on the α phase as open circles to illustrate the structural transformation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The SnAs3 building blocks only need to bear a mild distortion under this transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This is reasonable if considering that the coordination numbers generally increase under high pressure to homogenize the electron distribution [34],[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We check the pressure-dependent enthalpy of the β phase with respect to that of the α phase from 0 to 50 GPa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Beyond a critical pressure of around 25 GPa, the β phase overwhelms the α phase in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Figure S6 shows the phonon dispersion of the β phase at 30 GPa, which does not have any imaginary frequencies, indicating the stability of the high-pressure phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' It is worth noting that the tetragonal pyramid configuration of SnAs4 has not been previously reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We also did the structural search of NaSnAs under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' At low pressure, NaSnAs undergoes a structural transformation from the α phase to the β phase at 15 GPa, with the lowest enthalpy in the Na-Sn-As system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' However, calculations at 30 GPa predict that another tetragonal phase with the space group P4/mmm (γ phase) is the most stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Figure S7 depicts the successive structural transformation of the three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The γ phase can be viewed as a change in the stacking sequence of the Na spacer layer from the β phase, requiring every other Na layer transmits through its adjacent SnAs layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The β phase is predicted to be stable in a pressure window of 15-20 GPa, as indicated by the pressure-dependent formation enthalpy shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Obviously, the interlayer transmission of Na from the β phase to γ phase should be more energetically costly than the intralayer SnAs3 to SnAs4 transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Nonetheless, both β phase and γ phase share the identical SnAs4 building block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' To verify the SnAs3-SnAs4 transformation, we performed in-situ synchrotron diffraction experiments on Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 at various pressures ranging from ambient pressure to 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' All diffraction peaks show a systematic shift below 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa and can be indexed into the α phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' An additional peak at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5° gradually arises as pressure above 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa, signaling a pressure-driven phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' When the pressure is increased to 40 GPa, the initial peaks corresponding to the α phase become almost indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We carried out Rietveld refinements on each synchrotron x-ray diffraction pattern, and put three typical profiles of 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='8 GPa, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='88 GPa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We refined the pattern of 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa based on the theoretically predicted β phase and obtained the agreement factors of Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='84% and Rwp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3%, which are comparable to the ambient refinement values of Rp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='21% and Rwp = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='43%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' A two-phase refinement against the pattern of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='8 GPa produces the best fitting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We note that the diffraction pattern depressurized phase is almost identical to the initial one except mild peak broadening (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S9), indicating the reversibility of the SnAs3-SnAs4 transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 5 The contour plot of the diffraction patterns reveals more details of the structural transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The critical pressure of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa is clearly with the emergence of new diffraction peaks at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5° and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' An interesting observation is that the diffraction peaks corresponding to the α phase (102, 105, 110, 025, and 207 in particular) exhibit a consistent kink at 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa, indicating substantial lattice distortion in the α phase as an intermediate transition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' A qualitative analysis of the volume fractions of α and β phase extracted from Rietveld refinements are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Another notable thing is the dramatic change in bond lengths under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Figure S10 labels the Sn-As and As-As distances at 0 and 30 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The out-of- plane As-As distance is reduced from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='04 Å to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='58 Å and the in-plane As-As distance shrinks abruptly from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='01 Å to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='51 Å along with the SnAs3-SnAs4 transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' However, the Sn-As bond is increased to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='97 Å, which is even larger than the ambient value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='72 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The elongation of the Sn-As bond suggests that the prior conduction channel through As-Sn-As is greatly impeded, whereas direct hopping between As-As anions within the square lattice should be much favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The increased coordination numbers, elongated Sn-As bonds, and collapsed As-As distance induced by the SnAs3-SnAs4 transformation significantly affect the electronic transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The α-Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 exhibits only one superconducting transition below 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='7 GPa, with its Tc gradually increasing from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 K to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 K (blue arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Noticeably, a drop in resistivity at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 K is observed at 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='7 GPa (enlarged rectangle shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 3a), which is consistent with the emergence of the β phase around 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' To the best of our knowledge, this is the highest Tc experimentally realized in the SnAs-based compounds, six times higher than the ambient value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' As pressure increasing, the decline in resistivity gets more pronounced, yet its transition temperature remains nearly constant (red arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' All the raw data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We confirm that this newly discovered kink in resistivity is superconductive according to the gradual suppression of the Tc under magnetic fields (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The superconducting drop corresponding to the α phase is seen at 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='1 GPa, indicating the survival of the trace α phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This is also in line with our estimation of the volume fraction shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Figure 3b reveals the Tc evolutions of the α phase and β phase in response to pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The Tc in the β phase remains nearly constant at moderate pressure (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0~45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa) and then decreases somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Their Tcs become to be one transition above 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Given that the structure at high pressure is dominated by the β phase, the remaining superconducting transition should be attributed to the SnAs4 arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Note that the maximum Tc is achieved in the mixture phase rather than the pure β phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' A plausible reason for this could be that the Tc-pressure diagram of the β phase has a dome-like shape, within which the maximum Tc occurring ~35 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Figure 3d shows the Hc2(0) of α (0 GPa) and β (73 GPa) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Linear extrapolation shows a higher Hc2(0) in the β phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 6 The ensuing transition from α to β to γ phases in NaSnAs is investigated by transport measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' NaSnAs undergoes a semiconductor-metal transition under lower pressure, then becomes SC at ~10 GPa (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Its Tc slowly increases to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6 K at 35 GPa, then suddenly jumps to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' While the onset pressure of the first superconducting phase at ~10 GPa is qualitatively consistent with the theoretical pressure of α-β phase transition, the pressure of Tc’s jump is higher than our theoretical prediction of 20 GPa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This is understandable because the predicted γ phase is a thermally equivalent ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Since our in-situ high-pressure measurements are carried out in a DAC cell at room temperature, the energy barrier for the inter-plane migration (β-γ) should be much higher than the intra-plane ion displacement (α-β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Despite their different carrier doping levels, the charge-balanced NaSnAs achieves a comparable Tc to that of the β-Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2, indicating the highest Tc in the SnAs system is determined the common building block of SnAs4 in terms of structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The structural transformation of SnAs3-SnAs4 can lead to charge rearrangements of conducting electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S13, the Fermi level (EF) in LiSn2As2 is mostly composed of As’s 4p and Sn’s 5p orbitals, with the 4s and 5s orbitals lying far below the EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We plot the orbital projected density of states (PDOS) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' At ambient pressure, the PDOS of As’s pz component at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='98 states/eV is higher than the sum of its px and py orbital (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='76 states/eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This feature is more prominent for Sn because the intensity ratio of pz/(px+py) reaches 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6 at the EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Specifically, the spectra weights of pz and px+py are reversed for both As and Sn at high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Specifically, the enhancement of the PDOS of As’s px+py is 4 times higher than that of the pz component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The SnAs3-SnAs4 transformation induces a prominent charge redistribution from the out-of-plane (pz) orbital to the in-plane (px, py) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We perform Crystal Orbital Hamilton Population (COHP) calculations for the α and β phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4b, the α phase is dominated by Sn-As bonding states below -1 eV, with a small component at the EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' In the β phase, however, the out-of-plane As-As bonding component of inter SnAs4 units overwhelms that of Sn-As bonding of intra unit, and dominates the EF by As-As anti-bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Moreover, the integrals of COHP (ICOHP) up to the EF for Sn-As and As-As are -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='16 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='08 eV/pair in the α phase, and change to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='48 and -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='89 eV/pair in the β phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This suggests that the interaction of Sn-As dominates in the α phase, while that of As-As prevails in the β phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We plot the partial electron distribution near the EF (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2 eV) to see the charge distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' For the α-LiSn2As2, the valence electrons mainly gather around Sn and As atoms and are oriented along the c axis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4c, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Thus the electrons can mainly migrate along As-Sn-As via zigzagging through the shared 5pz-4pz orbitals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4d left panel and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 1c right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Once the SnAs3 is transformed into SnAs4, the charge redistributions from pz to px+py are clearly seen in the right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4c and 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' We examined the partial electron distribution of the β-LiSn2As2 in each energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' With the electron 7 filling from the lowest occupied σ bond to the σ* bond, their respective electron dispersion in real space is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Summarizing all the analyses above, we provide a simple molecular orbital diagram to understand the evolution of bond connections in LiSn2As2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4f and 4g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The out-of-plane As-As interaction drives the EF sitting on the πxy* band with the extra electron donation from Li and Sn, which well explains the calculated dominating px+py PDOS (lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4a) and the As-As anti- bonding (lower panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Considering the orthogonal configuration of px and py orbitals, the electron is more likely to migrate within this As-square lattice than in the triangle ones (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 4e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The abrupt shrinkage in the As-As distance from 4 Å to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 Å, on the other hand, reinforces in-plane hopping via overlapping of the orthogonal px+py lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This reminds us of the highly tunable Bi-Bi distance from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='85 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='08 Å in Bi-based square lattice R2O2Bi (R = rare earth metals), accompanied by a metal-insulator transition at a shortened Bi-Bi distance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='95 Å in Sm2O2Bi[36] and a subsequent SC at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='87 Å in Y2O2Bi[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' As our simulations are all based on the fully occupation of Li in LiSn2As2, the influence of Li vacancies is carefully checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' To this end, we create a 3×3×1 superlattice and randomly removed 7 out of 18 Li atoms in the supercell corresponding to Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='61Sn2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Figure S15 shows the partial charge density at 30 GPa near EF (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2 eV) of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' It again shows a px/py conducting character, indicating that the Li vacancy does not influence our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The electronic states in the pressurized NaSnAs resemble those of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S16, the EF of both β and γ phases are dominated by px+py orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The 2D feature of the electron dispersion becomes more prominent in the γ phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' S17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' It is the charge transfer from pz to px+py that facilitates the direct hopping of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Given the similarity of charge distribution in the SnAs4 unit, we conclude that the orbital reorientation of As’s p orbitals in a square lattice should be responsible for the similar Tc in both compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Conclusion In this work, a new kind of structure unit SnAs4 is realized in SnAs-based compounds through a topotactic phase transformation in both Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 and NaSnAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Thanks to the thus-formed As square lattice and the dramatic shrinkage of the in-plane As-As distance from SnAs3 to SnAs4 transformation, the probability of in-plane hopping through px+py lobes is much enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Theoretical calculations show a charge redistribution of the electron near the EF from pz orbital to px+py orbitals, further facilitating the electron transportation and the subsequent record-high Tc in all reported SnAs-based compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Our findings verify the effectiveness of modulating SC through directed local structure tailoring of flexible layered compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Acknowledgements We appreciate Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Xianxin Wu for fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' This work is financially supported by the National Key Research and Development Program of China 8 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2018YFE0202600, 2021YFA1401800), Beijing Natural Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Z200005), the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 51922105, 11804184, and 11974208), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant XDB30000000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' ADXRD measurements were performed at 4W2 High Pressure Station, Beijing Synchrotron Radiation Facility (BSRF), which is supported by Chinese Academy of Sciences (Grant KJCX2-SW-N20, KJCX2-SW-N03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' would like to acknowledge the National Key R&D Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2018YFA0704300), the National Natural Science Foundation of China (grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' U1932217, 11974246, and 12004252).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='C Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Hosono, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2012, 10, 2063-2070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Prewitt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Downs in Ultrahigh Pressure Mineralogy (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Hemley), De Gruyter,1998, PP, 283-318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' [35] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Grochala, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Hoffmann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Feng, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Ashcroft, The chemical imagination at work in very tight places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2007, 46, 3620-3642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Mizoguchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Hosono, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2011, 8, 2394-2397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' [37] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Sei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Kitani, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Fukumura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Kawaji,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Hasegawa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 2016, 35, 11085-11088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' 10 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Theoretical prediction of the structure transition of LiSn2As2 and NaSnAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Top and side-views of LiSn2As2 at a) ambient pressure (R3̅m, α phase) and b) high pressure (I4/mmm, β phase) predicted by the CALYPSO package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' c) Bonding environment of SnAs-layered compounds and the illustration of the zigzag-propagation of electrons within the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The lone- pair electrons are denoted by light blue droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' d) Relative enthalpy of LiSn2As2 as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The insets depict the trigonal (SnAs3) and tetragonal (SnAs4) pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' e) Relative enthalpy of P63mc (α phase), I4/mmm (β phase) and P4/mmm (γ phase) for NaSnAs as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' The enthalpy of the α phase is taken as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a b C lone pair electrons Sn: 5s2 5p2 As: 4s24p3 As LiSna NaSnAs d e 150 Enthalpy (meV/atom) Enthalpy (meV/atom) 200 AS 100 Sn 100 50 SnAs4 SnAs3 100 50 200 0 20 40 0 10 20 30 40 α phase βphase Pressure (GPa) Pressure(GPa)11 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' In-situ synchrotron diffractions of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 at pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a) Three Rietveld refinements profiles of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 at 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='8 GPa and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='88 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' b) Color contour plot of the pressure-dependent diffraction peaks from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='88 to 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' c) Pressure-mediated volume fraction of the α phase and β phase extracted from refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0GPa R=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='84% 40 30 (GPa 20 0 207 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='8GPa R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='15% Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=') Rwp=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='43% 10 :aphase 8101214161820222426 I:Bphase 2(degree) C 100 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='88 GPa Observed 80 80 (%) Calculated R,=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='23% phase 60 60 aαphase βphase R =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='43% 40 eβphase 40 WD 20 20 11111 0 0 810121416182022242628 0 10 20 30 40 50 29(degree) P (GPa)12 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Pressure-dependent SC in Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a) Temperature-dependent resistivity of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Insets are the enlargement of resistivity at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Blue and red arrows indicate the onset Tc of α and β phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' b) Superconducting phase diagram of Li0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='6Sn2As2 under pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' An intermediate pressure region mixes two transition temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' c) Resistivity of the β phase (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 GPa) from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 K-8 K at different magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' d) Fitting of the upper critical fields for the ambient (black squares) and high pressure (red circles) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a b 9 C 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 T aαphase mixture βphase 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2 cm) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='1 6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='7 OT (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=') 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 378 2 4 6 8 T (K) d 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2 O 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='3 GPa 3 Ambient 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='7 E 2 pressure Ambientpressure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='8 Run1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 1 Run2 0GPa Run3 2 4 6 8 1012 10 20 30 40 50 60 7080 4 6 8 T (K) P (GPa) T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' (K)13 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Charge redistribution and orbital reorientation from SnAs3-SnAs4 transformation in LiSn2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a) Projected density of states (PDOS) and b) - COHP for α phase at 0 GPa (upper panel) and β phase at 30 GPa (lower panel), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Insets in b) show the interaction of As-As and Sn-As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Positive values indicate bonding states, and negative ones are antibonding states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' c-e) Partial charge density around the EF (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2 eV ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='2 eV) with iso-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0025 e/Bohr3, section projections, and illustration of bond connections of the α phase (left) and β phase (right), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' f-g) Molecular orbital diagram of the Sn- As bonding state in the α phase and out-of-plane As-As antibonding state in the β phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' Note that extra electrons in the πxy* band come from the donation of Li and Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content=' a αphase βphase f As py+Px Sn Py+Px aphase PDOS (states/eV) As Pz Sn Pz Sn 5p2 0 [110] As 4 d e/bohr3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='005 axy 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} +page_content='0 Energy(eV) C b Sn As [110] βphase As As 0 e Li, Sn Sn As HP As As e CO Sn As As As AS As 4p3 0 TTxy a Aspx 4 2 0 2 4 Aspy Energy(eV) 5pz(Sn)+4pz(As)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE1T4oBgHgl3EQfhQT4/content/2301.03240v1.pdf'} diff --git a/nNE3T4oBgHgl3EQfKwlD/content/tmp_files/2301.04356v1.pdf.txt b/nNE3T4oBgHgl3EQfKwlD/content/tmp_files/2301.04356v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6816f72569f37543d6aae882fbae9bcf6e3fd93 --- /dev/null +++ b/nNE3T4oBgHgl3EQfKwlD/content/tmp_files/2301.04356v1.pdf.txt @@ -0,0 +1,692 @@ +Threshold Voltage Control in Dual-Gate Organic +Electrochemical Transistors +Hsin Tseng∗1, Anton Weissbach1, Juzef Kucinski1, Ali Solgi1, +Rakesh Nair1, Lukas M Bongartz1, Giuseppe Ciccone1, Matteo Cucchi1,†, +Karl Leo1, & Hans Kleemann∗1 +1Dresden Integrated Center for Applied Physics and +Photonic Materials (IAPP) and Institute for Applied Physics, +Technische Universit¨at Dresden, N¨othnitzer Str. 61, +01187 Dresden, Germany +† Present address: Laboratory for Soft Bioelectronic Interfaces Neuro-X Institute, +Ecole Polytechnique F´ed´erale de Lausanne (EPFL), +Geneva, Switzerland +January 12, 2023 +Abstract +Organic electrochemical transistors (OECTs) based on Poly(3,4-ethylenedioxythiophene):poly(styrene +sulfonic acid) (PEDOT:PSS) are a benchmark system in organic bioelectronics. +In particular, the superior mechanical properties and the ionic-electronic trans- +duction yield excellent potential for the field of implantable or wearable sens- +ing technology. However, depletion-mode operation PEDOT:PSS-based OECTs +cause high static power dissipation in electronic circuits, limiting their applica- +tion in electronic systems. Hence, having control over the threshold voltage is +of utmost technological importance. Here we demonstrate PEDOT:PSS-based +dual-gate OECTs with solid-state electrolyte where the threshold voltage is +seamlessly adjustable during operation. We show that the degree of threshold +voltage tuning linearly depends on the gate capacitance, which is a straight- +forward approach for circuit designers to adjust the threshold voltage only by +the device dimensions. The PEDOT:PSS-based dual-gate OECTs show excel- +lent device performance and can be pushed to accumulation-mode operation, +resulting in a simplified and relaxed design of complementary inverters. +1 +arXiv:2301.04356v1 [physics.app-ph] 11 Jan 2023 + +1 +Introduction +Organic electrochemical transistors(OECTs) have recently been in the spotlight +of research because of their use in bioelectronics [1–3], neuromorphic computing +[4–7], and biological or chemical sensors [8–11]. All these application scenar- +ios are based on the conduction mechanism in organic mixed ionic-electronic +conductors (OMIECs) [12], enabling efficient translation of ionic or chemical +signals into electronic signals and vice versa. In addition, OMIECs offer easy +production by printing and excellent mechanical properties [13, 14]. With the +development of solid-state electrolytes [15–18], OECTs might be integrated into +wearable or even implantable systems with intelligent sensor function. +In a common OECT, ions from the electrolyte penetrate the OMIEC polymer +matrix and the distribution of ions in the OMIEC can be manipulated applying +a bias voltage to the gate electrode. As the concentration of mobile holes / +electrons in the OMIEC is regulated by the ion concentration via an electro- +chemical redox reaction, the gate bias can be used to control the conductance +of the transistor channel and hence switch the transistor on and off [19]. +One important device parameter of OECTs for sensing, computing, and cir- +cuitry is the threshold voltage (Vth). It is the point when the gate bias switches +the transistors between high current accumulation regime and low current de- +pletion regime. +It has a great technological relevance for example for logic +gates where it determines the trip-point of digital inverters. Furthermore, the +threshold voltage is also of great importance for sensing applications as in the +sub-threshold regime, the current through the transistor exponentially depends +on the gate bias offering the highest possible sensitivity of the system (comes +with the drawback of losing linearity of the sensor). +Hence, having control +over the threshold voltage is of utmost technological importance and several +strategies have been proposed to tune the threshold voltage by adjusting device +parameters or materials. For example, changing the gate electrode material af- +fects the potential drop at the gate electrode and changes the OECT operation +regime from capacitive to Faradaic [20], leading to different threshold voltages. +Alternatively, modifying the gate electrode with redox-active species or dopants +controls the gate’s work function [21, 22], also serving the same purpose. In +addition, the design of the channel material, such as its chemical structure, in- +fluences the interactions with electrolytes and can, therefore, regulate OECTs +operation and performance [23, 24]. However, even without manipulating the +material system, the threshold voltage can be tuned solely by the device geom- +etry. For example, the gating efficiency of the same gate electrode material can +be improved as the gate capacitance is increased [25], and thus reducing the +threshold voltage. +The issue of threshold control is particularly relevant for the most often used +OECT material: Poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PE- +DOT:PSS) is a working horse OECTs material because it is easy to process, com- +2 + +mercially available, and shows high transconductance. PEDOT:PSS is, however, +highly doped, resulting in depletion-mode (normally-on) transistor behavior. A +depletion-mode device is unfavorable for circuitry because of high power dissipa- +tion. In particular, for the design of inverter circuits, which are a basic element +of digital electronics, the depletion-mode behavior is very disadvantageous as it +complicates the inverter design and reduces the noise margin and gain of the +circuits. Unfortunately, chemically modifying the gate electrode does not give +an accumulation-mode device, although it helps tuning the threshold voltage +[21, 22]. On top of that, using this approach, the threshold voltage cannot be +adjusted anymore once the device has been manufactured. Although the chem- +ical modification of either the gate electrode or channel material offers control +over the threshold voltage, controlling it via the device design is technological +preferable as the use of different materials, e.g., for the gate and the channel or +additional chemical doping significantly increases the complexity of device and +circuit fabrication. +Using a dual-gate architecture is an alternative strategy to control the threshold +voltage by the device design. This approach has been put forth for conventional +organic thin-film transistors and precise control over the threshold voltage as +well as tunability during operation have been demonstrated [26–28]. Two gate +insulators are used to isolate the semiconductor channel from the top and bot- +tom gate electrodes in these architectures. Thereby, two channels are formed at +opposing interfaces of the semiconductor layer, which are used to regulate the +conductance of the transistor. Having separated electrolytes, this design prin- +ciple could also be adopted to OECTs using gate electrodes at the bottom and +top. However, OECTs are usually fabricated in a side-gate configuration due to +the conductive nature of the electrolyte. This geometry is advantageous because +of its easy processing and high production yield. In contrast to a typical dual- +gate thin-film transistor [26], the two gates in a dual-gate OECT in side-gate +configuration would be in a shared electrolyte and it is not clear whether the two +gates can be used to control the threshold voltage independently or they simul- +taneously influence the device performance. Recently, Ji et al. demonstrated +a dual-liquid-gated OECT using electropolymerization to modify the gate ca- +pacitance with PEDOT:PSS [29] and showed that the transconductance can be +tuned to some extent. However, accurate control over the threshold voltage was +not possible with their approach, and most importantly, they could not make +PEDOT:PSS-based OECTs operate as accumulation-mode transistors. +Here, we demonstrate PEDOT:PSS-based dual-gate OECTs with solid-state +electrolyte where the threshold voltage can be continuously tuned during op- +eration. We show that the degree of tuning the threshold voltage linearly de- +pends on the gate capacitance which is a straight-forward approach for cir- +cuit designers to adjust the threshold voltage only by the device dimensions. +The PEDOT:PSS-based dual-gate OECTs, which can be densely integrated us- +ing conventional photolithography or printing techniques, show excellent device +performance and can be pushed to accumulation-mode operation, leading to +3 + +simplified processing, relaxed design requirements, and improved performance +of complementary inverters. +2 +Results and Discussion +Figure 1(a, b) show the schematic layout of a dual-gate OECT. The transistor +consists of two in-plane gate electrodes, a sweeping gate (Gate 1) and a control- +ling gate (Gate 2), the semiconductor channel with source and drain electrodes, +and the solid-state electrolyte. PEDOT:PSS is used as the semiconductor chan- +nel material as well as for both gate electrodes in order to increase the capaci- +tance of the gate. Using the same material for the gate and the channel reduces +the fabrication complexity, and the volumetric capacitance of a PEDOT:PSS- +based gate strongly reduces the voltage loss at the gate/electrolyte interface [25]. +The capacitance of the PEDOT:PSS-based gate can be scaled by film area [30– +32] on the condition that this film is formed at the same spin-coating process +as the channel material, giving the same thickness of 100 nm. To evaluate the +effect of the capacitance of the control gate (Gate 2) on the threshold voltage, +we only vary the area of Gate 2, from 9200 µm2, 14700 µm2, 44100 µm2, to +60900 µm2, leaving the area of both channel and Gate 1 and the distance of 30 +µm between the gate and the channel fixed (cf. Figure 1). The solid-state elec- +trolyte is inkjet-printed on top of the channel and the gate electrodes, followed +by UV-light induced cross-linking [16]. More details on the fabrication process +of these integrated OECTs are given in the Experimental Section. +Figure 2 presents the electrical characterization of a dual-gate OECT with +AGate2 = AGate1 = 44100 µm2. +The transfer characteristics are measured at +a drain-source bias VDS of -0.1 V for Gate 2 bias ranging from 0 V to +1 V in +steps of 0.2 V. The transfer curves systematically shift with the applied VGS2. +For a controlling bias VGS2 of 0 V (grounded), the transfer curve at the far right +in Figure 2 (a, b), the effect of the Gate 2 is negligible. The dual-gate OECT +behaves like a single-gate OECT and only Gate 1 redistributes ionic charges in +the electrolyte. +The effect of the VGS2 > 0 V on the dual-gate OECT is shown in the curves on +the left in Figure 2 (a, b). Polarons in PEDOT:PSS are neutralized by cations +driven by VGS2. To compensate for VGS2 and achieve the original drain current +at VGS2 of 0 V, the sweeping gate VGS1 has to be increased to more negative +values. Therefore, the transfer curve and hence the threshold voltage is shifted +to the left. The interplay between the bias on the two gate electrodes deter- +mines the current in the dual-gate OECT at a given drain-source bias. The gate +current in Figure 2 (b) is significantly lower than the drain current because of +the precise patterning technology of the solid-state electrolyte. +As shown in Figure 2(c), output curves are measured at VGS2 =0 V. The drain +4 + +current is only affected by VGS1. Further, in Figure 2(d), the drain current is +simultaneously influenced by VGS1 and VGS2; at the same scale of x-axis and +y-axis, the drain current is close to 0 A, proving that the PEDOT-PSS-based +OECT in fact operates as an accumulation-mode transistor (in agreement with +the transfer curve shown in Figure 2(a)). It is worth to mention that the effect +of inhomogenous dedoping becomes relevant at high VDS where the curve is +supposed to saturate [33]. In this study, we report on the change of threshold +voltage for small VDS=-0.1 V where inhomoegenous dedoping can be ignored +and the threshold voltage is well defined. If inhomoegenous dedoping comes +into play, the threshold voltage becomes a function of the drain-source voltage. +Still, the VGS2 can be used to manipulate the transconductance of the transistor. +We postulate that the dual-gate OECT device behavior can be modelled as two +parallel gate capacitor connected in series to the channel capacitor, as shown in +Figure 1(c). This is because the electrolyte resistance (200kΩ) is negligible com- +pared to the shunt resistance of the electrochemical double layers formed at the +semiconductor-electrolyte interface (typical leakage current in the range of 10 +nA at 1 V as shown in Figure 2 (b)). The function of the solid-state electrolyte +does not differ from that of a liquid electrolyte such as NaCl(aq). The solid +polymer structure of the solid-state electrolyte forms a matrix for the move- +ment of the ionic liquid. As water has been used as a solvent for the solid-state +electrolyte, the PEDOT:PSS layer is always in a swollen state, which allows ions +from the ionic liquid to move in and out of the PEDOT:PSS layer and thus dopes +and dedopes the semiconductor [16]. Accordingly, the voltage only drops across +the gate capacitance/ channel capacitance and the solid-state electrolyte can +be treated as an equipotential surface. The total effective gate-source voltage +influencing the channel is then determined by the total capacitance of CGate1 +and CGate2 and the voltages applied. +We extract the threshold voltage (Vth) from the transfer characteristics in Fig- +ure 2(a). The Vth is defined by plotting the drain current against the gate-source +voltage, linearly fitting this curve, and intercepting the value on the x-axis of +VGS1. +The VDS of -0.1 V is chosen to extract Vth is to avoid the effect of +non-uniform dedoping in our system [33]. Figure 3(a) presents the threshold +voltage as a function of the controlling gate VGS2. The data is in a mean value +for 5 devices for each geometry and clearly shows the shift in threshold voltage. +It should be noted that these solid-state electrolyte OECTs show a significant +hysteresis in the transfer curve [16], which makes it technically speaking impos- +sible to derive a single threshold voltage value. However, using the dual-gate +configuration, we observe that the transfer curve including hysteresis is homoge- +neously shifted. Hence, for a sake of simplicity, we only plot the transfer curve +for switching the device from on- to off-state thereby ignoring the hysteresis. +When the area of Gate 2 is enlarged, the slope in Figure 3(a) increases, which al- +lows us to turn these PEDOT:PSS-based OECTs from depletion- to accumulation- +mode operation. Figure 3(b) shows that the degree of controlling the threshold +5 + +voltage in dual-gate OECTs linearly scales with the ratio of the gate area (be- +ing equal to the ratio of capacitances). Using the equivalent circuit proposed +in Figure 1(c), we can predict the scaling of the threshold voltage shift as a +function of the gate area ratio by the following expression: +Vth = V0 +th(AGate1 + AGate2 + AChannel) +AGate1 +− VGS2AGate2 +AGate1 +(1) +where the constant V0 +th represents the threshold voltage without the presence +of Gate 2. As shown in Figure 3(b), the experimental data fit very well to the +model predictions and even without over-sizing Gate 2 significantly, a strong +tunability of Vth is achieved. In fact, accumulation-mode operation can be al- +ready achieved if the area of Gate 2 is only 1.38-times larger than the area of +Gate 1 (Figure 3(a)). +We demonstrate the advantage of this dual-gate OECT technology for logic +circuits. As an example, a complementary inverter, combining a p-type and an +n-type OECT, is chosen here as the most simple logic circuit. It works as a +digital amplifier and is often combined with OECT-based sensors to increase +the biosignal sensitivity for bioelectronics [9]. The most common used n-type +semiconductor material for OECTs is poly(benzimidazobenzophenanthroline) +(BBL) [23, 34–36], with which an OECT operates in an accumulation-mode. +Due to the depletion-mode operation of PEDOT:PSS-based OECTs and the +large transconductance compared to BBL-based devices, good inverter charac- +teristics can only be achieved if the BBL-based OECT is significantly larger +than the PEDOT:PSS-based device. The channel width to length ratio of the +BBL-based devices is typically chosen to be at least thousand times larger than +the ratio of the PEDOT:PSS-based device (e.g., 16000-times larger in Ref.[9]). +A complementary inverter layout with a PEDOT:PSS-based dual-gate OECT +(p-type) and a BBL-based OECT (n-type) is shown in Figure 4(a). The input +voltage (Vin) is applied to the common gate, i.e., the gate of the BBL device and +the Gate 1 of the PEDOT:PSS-based dual-gate OECT and is swept from 0 V +to 0.8 V. The supply voltage VDD is set to 0.8 V. A constant voltage at Gate 2 +(VGS2 =0.25 V, 0.5 V, 1 V) is applied during the inverter measurement to con- +trol the threshold voltage of the p-channel device. The output voltage (Vout) +is measured to determine the transfer curve of the inverter which is shown in +Figure 4(b). As VGS2 increases, the inverter transfer curve shifts to the left. +In particular, the trip point of the inverter is seamlessly adjusted by VGS2 as +it controls the threshold voltage of the dual-gate devices. The dual-gate design +of OECTs offers a more robust design and operation of circuits, which can be +used to improve the sensitivity of any bioelectronics system and thus contributes +greatly to the field of bioelectronics. +6 + +Figure 1: (a) Top view and cross section configuration of a PEDOT:PSS-based +dual-gate OECT, including source and drain electrodes, PEDOT:PSS channel, +and two PEDOT:PSS gates symmetric with respect to the channel. All the +PEDOT:PSS films have the same thickness of 100 nm. +(b) Optical microscopic image of a dual-gate OECT. (c) Equivalent circuit +model of the dual-gate OECT. +7 + +(a) +Top View +(b) +Gate 2 (control) +210 x 210 um² +AGate2 +Solid +Electrolyte +W=150 um +=30 μm +Source +PEDOT:PSS +Drain +PEDOT:PSS +100 μm +210 x 210 μm² +AGatel +Gate 1 (sweep) +(c) +Gate 1 +Gate 2 +Cross section +Gatel +PEDOT:PSS +Solid +Electrolyte +Channel +Source +Drain +Channel +Gate 1 (sweep) +Gate 2 (control) +gap = 30 umFigure 2: Electrical characterization of the PEDOT:PSS-based dual-gate OECT +with AGate2 = AGate1 = 44100 µm2. (a) Transfer characteristic in linear scale +at VDS = -0.1 V. VGS2 is fixed for the loop of VGS1 sweeping from -1.5 V to +1 +V. (b) Transfer characteristic in logarithmic scale, including the drain current +(solid line) and the gate current (dotted line). +The same color of the drain +current and the gate current means they are under the same VGS2. (c) Output +characteristic of the dual-gate OECT at VGS2 = 0 V. (d) Output characteristic +at VGS2 = 0.8 V. +8 + +(b) +(a) +0.4 +10-3 +Ds = -0.1 V +10-4 +0.3 +10-5[ +mA +GS2 = 0 V +GS20 V +A +0.2 +10-6 +D +I +10-7 +0.1 +10-8 +VGS2 +10-9 +0 +=1 V +-1.5 +-0.5 +-1 +0 +0.5 +1 +-1.5 +-1 +-0.5 +0 +0.5 +1 +VGsil V +(c) +(d) +1e-4 +1e-4 +-2.5 +-2.5 +VGs2 = 0 V +VGs2 = 0.8 V +-2 +-2 +A-1.5 +A +-1.5 +LD +-1 +-1 +-0.5 +-0.5 +0 +0 +VGSI +Ves1 +-0.2 +0 +-0.1 +-0.3 +-0.4 +-0.5 +0 +-0.1 +-0.2 +-0.5 +-0.3 +-0.4 +VDsI V +VDs I VFigure 3: (a) Threshold voltage of a PEDOT:PSS-based dual-gate OECT as a +function of VGS2 for different gate area ratios. The larger area of Gate 2 (red) +gives a steeper slope, namely a larger degree of tuning. Each data point is a +mean value for 5 devices and each device is measured 3 times. (b) The degree +of threshold voltage tuning increases with the ratio of the gate area. +9 + +(b) +(a) +0.4 +1.4 +gate area ratio +Linear Fit +0.3 +1.2 +1 +V +0.2 +Gate 2 +iate +0.8 +A +0.1 +A +0.6 +AGate 2= 1.38 +0.4 +0 +AGate 1 +0.2 +AGate2= 1 +-0.1 +AGate! +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +△V th +△VGS2Figure 4: (a) A complementary inverter layout of a PEDOT:PSS-based dual- +gate OECT with ( W +L )p = 5 and a BBL OECT with ( W +L )n = 2000. The input +voltage Vin applies to the Gate 1 of the PEDOT:PSS-based dual-gate OECT +and to the gate of the BBL OECT. VGS2 is constantly applied at the Gate 2 +of the PEDOT:PSS-based dual-gate OECT. The device configuration of both +types of OECTs; the gate electrode of BBL OECT is Ag/AgCl immersed in +the ionic liquid. Further device dimension can be found in the Experimental +section. +(b) Inverter transfer characteristic: as the threshold voltage of the +PEDOT:PSS-based dual-gate OECT is tuned by increasing VGS2 (from 0.25 V, +0.5 V to 1 V), the transfer curve (solid line) shifts to the left, and the inverter +gain (dashed line) increases. +10 + +(a) +(b) +-10 +DD +0.8 +8-- +0.6 +VGS2 +LS1 +JS2 + 0.4 +0.25 V +V +In +out +0.5 V +0.2 +1V +:0 +0 +0 +0.2 +0.4 +0.6 +0.8 +),= 2000 +Vin\V +BBL +ionic liquid3 +Conclusion +In conclusion, we demonstrate continuous tuning of the threshold voltage in +PEDOT:PSS-based dual-gate OECTs. These dual-gate structures are easy to +fabricate, employing the often used side-gate architecture. The threshold volt- +age scales linearly with the voltage at the control gate (Gate 2), and the degree +of tuning linearly increases with the gate area ratio. Furthermore, we mod- +eled the device behavior with an equivalent circuit, and the experimental data +fit very well with the model predictions. +The PEDOT:PSS-based dual-gate +OECTs, which can be densely integrated using conventional photolithography +or printing techniques, show excellent device performance, and they can be +pushed to accumulation-mode operation, leading to improved performance and +relaxed design requirements of complementary OECT inverters. +4 +Experimental +Device fabrication: The process of structuring the electrode and channel pat- +tern of a dual-gate OECT follows ref.[8]. Source, drain, and gate electrodes were +patterned on a glass substrate with 50 nm Au and 3 nm Cr by photolithography +and wet-etching using Standard Gold Etchant and Standard Chromium Etchant. +PEDOT:PSS-based solution was prepared with 95 wt.% of PEDOT:PSS (Hear- +aeus Clevios PH 1000, 1.1 wt.% solids in water, 1:2.5) and 5 wt.% of ethylene +glycol. This solution was spin-coated at 3000 rpm on the electrodes and the 100 +nm-PEDOT:PSS thin film was patterned by fluorine-based photolithography +[37] and dry etching [38] using O2 and Ar. The solid-state electrolyte precur- +sor solution containing 1 mL deionized water, 750 mg N-isopropylacrylamide, +20 mg N,N’-methylenebisacrylamide, 200 mg 2-hydroxy-4’-(2-hydroxyethoxy)- +2-methylpropiophenone, and 1.5 mL 1-ethyl-3-methylimidazolium ethyl sulfate +[16] was inkjet printed on top of the active area followed by 2 minutes UV cross- +linking. The PEDOT:PSS-based dual-gate OECT was stored in the glovebox +overnight and then was encapsulated with glass for further measurement. +BBL solution was prepared by dissolving 5 mg BBL (Sigma Aldrich) in 1 +mL methanesulfonic acid and stirring at 70 ◦C overnight. The BBL solution +was then spin-coated at 1000 rpm on a gold substrate with W=10 mm, L=5 +µm. 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Organic Elec- +tronics, 76:105357, 2020. +Acknowledgements +We thank the European Social Fund (Project OrgaNanoMorph, project num- +ber: +100382168) and the Hector Fellow Academy (30000619) for providing +the financial support. Furthermore, the authors thank the Bundesministerium +f¨ur Bildung und Forschung (BMBF) for funding from the project BAYOEN +(01IS21089). The EMERGE project has received funding from the European +Union’s Horizon 2020 research and Innovation programme under grant agree- +ment Nº 101008701. +15 + +Data Availability +The data that support the findings of this study are available from the corre- +sponding author upon reasonable request. +16 + diff --git a/nNE3T4oBgHgl3EQfKwlD/content/tmp_files/load_file.txt b/nNE3T4oBgHgl3EQfKwlD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9aafa2b763bcc1f88b0498c3d4a5663a7ae91c8 --- /dev/null +++ b/nNE3T4oBgHgl3EQfKwlD/content/tmp_files/load_file.txt @@ -0,0 +1,362 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf,len=361 +page_content='Threshold Voltage Control in Dual-Gate Organic Electrochemical Transistors Hsin Tseng∗1, Anton Weissbach1, Juzef Kucinski1, Ali Solgi1, Rakesh Nair1, Lukas M Bongartz1, Giuseppe Ciccone1, Matteo Cucchi1,†, Karl Leo1, & Hans Kleemann∗1 1Dresden Integrated Center for Applied Physics and Photonic Materials (IAPP) and Institute for Applied Physics, Technische Universit¨at Dresden, N¨othnitzer Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 61, 01187 Dresden, Germany † Present address: Laboratory for Soft Bioelectronic Interfaces Neuro-X Institute, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Geneva, Switzerland January 12, 2023 Abstract Organic electrochemical transistors (OECTs) based on Poly(3,4-ethylenedioxythiophene):poly(styrene sulfonic acid) (PEDOT:PSS) are a benchmark system in organic bioelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In particular, the superior mechanical properties and the ionic-electronic trans- duction yield excellent potential for the field of implantable or wearable sens- ing technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' However, depletion-mode operation PEDOT:PSS-based OECTs cause high static power dissipation in electronic circuits, limiting their applica- tion in electronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Hence, having control over the threshold voltage is of utmost technological importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Here we demonstrate PEDOT:PSS-based dual-gate OECTs with solid-state electrolyte where the threshold voltage is seamlessly adjustable during operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' We show that the degree of threshold voltage tuning linearly depends on the gate capacitance, which is a straight- forward approach for circuit designers to adjust the threshold voltage only by the device dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The PEDOT:PSS-based dual-gate OECTs show excel- lent device performance and can be pushed to accumulation-mode operation, resulting in a simplified and relaxed design of complementary inverters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='04356v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='app-ph] 11 Jan 2023 1 Introduction Organic electrochemical transistors(OECTs) have recently been in the spotlight of research because of their use in bioelectronics [1–3], neuromorphic computing [4–7], and biological or chemical sensors [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' All these application scenar- ios are based on the conduction mechanism in organic mixed ionic-electronic conductors (OMIECs) [12], enabling efficient translation of ionic or chemical signals into electronic signals and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In addition, OMIECs offer easy production by printing and excellent mechanical properties [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' With the development of solid-state electrolytes [15–18], OECTs might be integrated into wearable or even implantable systems with intelligent sensor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In a common OECT, ions from the electrolyte penetrate the OMIEC polymer matrix and the distribution of ions in the OMIEC can be manipulated applying a bias voltage to the gate electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' As the concentration of mobile holes / electrons in the OMIEC is regulated by the ion concentration via an electro- chemical redox reaction, the gate bias can be used to control the conductance of the transistor channel and hence switch the transistor on and off [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' One important device parameter of OECTs for sensing, computing, and cir- cuitry is the threshold voltage (Vth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' It is the point when the gate bias switches the transistors between high current accumulation regime and low current de- pletion regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' It has a great technological relevance for example for logic gates where it determines the trip-point of digital inverters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Furthermore, the threshold voltage is also of great importance for sensing applications as in the sub-threshold regime, the current through the transistor exponentially depends on the gate bias offering the highest possible sensitivity of the system (comes with the drawback of losing linearity of the sensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Hence, having control over the threshold voltage is of utmost technological importance and several strategies have been proposed to tune the threshold voltage by adjusting device parameters or materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' For example, changing the gate electrode material af- fects the potential drop at the gate electrode and changes the OECT operation regime from capacitive to Faradaic [20], leading to different threshold voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Alternatively, modifying the gate electrode with redox-active species or dopants controls the gate’s work function [21, 22], also serving the same purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In addition, the design of the channel material, such as its chemical structure, in- fluences the interactions with electrolytes and can, therefore, regulate OECTs operation and performance [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' However, even without manipulating the material system, the threshold voltage can be tuned solely by the device geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' For example, the gating efficiency of the same gate electrode material can be improved as the gate capacitance is increased [25], and thus reducing the threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The issue of threshold control is particularly relevant for the most often used OECT material: Poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PE- DOT:PSS) is a working horse OECTs material because it is easy to process, com- 2 mercially available, and shows high transconductance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' PEDOT:PSS is, however, highly doped, resulting in depletion-mode (normally-on) transistor behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' A depletion-mode device is unfavorable for circuitry because of high power dissipa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In particular, for the design of inverter circuits, which are a basic element of digital electronics, the depletion-mode behavior is very disadvantageous as it complicates the inverter design and reduces the noise margin and gain of the circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Unfortunately, chemically modifying the gate electrode does not give an accumulation-mode device, although it helps tuning the threshold voltage [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' On top of that, using this approach, the threshold voltage cannot be adjusted anymore once the device has been manufactured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Although the chem- ical modification of either the gate electrode or channel material offers control over the threshold voltage, controlling it via the device design is technological preferable as the use of different materials, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=', for the gate and the channel or additional chemical doping significantly increases the complexity of device and circuit fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Using a dual-gate architecture is an alternative strategy to control the threshold voltage by the device design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' This approach has been put forth for conventional organic thin-film transistors and precise control over the threshold voltage as well as tunability during operation have been demonstrated [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Two gate insulators are used to isolate the semiconductor channel from the top and bot- tom gate electrodes in these architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Thereby, two channels are formed at opposing interfaces of the semiconductor layer, which are used to regulate the conductance of the transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Having separated electrolytes, this design prin- ciple could also be adopted to OECTs using gate electrodes at the bottom and top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' However, OECTs are usually fabricated in a side-gate configuration due to the conductive nature of the electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' This geometry is advantageous because of its easy processing and high production yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In contrast to a typical dual- gate thin-film transistor [26], the two gates in a dual-gate OECT in side-gate configuration would be in a shared electrolyte and it is not clear whether the two gates can be used to control the threshold voltage independently or they simul- taneously influence the device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Recently, Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' demonstrated a dual-liquid-gated OECT using electropolymerization to modify the gate ca- pacitance with PEDOT:PSS [29] and showed that the transconductance can be tuned to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' However, accurate control over the threshold voltage was not possible with their approach, and most importantly, they could not make PEDOT:PSS-based OECTs operate as accumulation-mode transistors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Here, we demonstrate PEDOT:PSS-based dual-gate OECTs with solid-state electrolyte where the threshold voltage can be continuously tuned during op- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' We show that the degree of tuning the threshold voltage linearly de- pends on the gate capacitance which is a straight-forward approach for cir- cuit designers to adjust the threshold voltage only by the device dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The PEDOT:PSS-based dual-gate OECTs, which can be densely integrated us- ing conventional photolithography or printing techniques, show excellent device performance and can be pushed to accumulation-mode operation, leading to 3 simplified processing, relaxed design requirements, and improved performance of complementary inverters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 2 Results and Discussion Figure 1(a, b) show the schematic layout of a dual-gate OECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The transistor consists of two in-plane gate electrodes, a sweeping gate (Gate 1) and a control- ling gate (Gate 2), the semiconductor channel with source and drain electrodes, and the solid-state electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' PEDOT:PSS is used as the semiconductor chan- nel material as well as for both gate electrodes in order to increase the capaci- tance of the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Using the same material for the gate and the channel reduces the fabrication complexity, and the volumetric capacitance of a PEDOT:PSS- based gate strongly reduces the voltage loss at the gate/electrolyte interface [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The capacitance of the PEDOT:PSS-based gate can be scaled by film area [30– 32] on the condition that this film is formed at the same spin-coating process as the channel material, giving the same thickness of 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' To evaluate the effect of the capacitance of the control gate (Gate 2) on the threshold voltage, we only vary the area of Gate 2, from 9200 µm2, 14700 µm2, 44100 µm2, to 60900 µm2, leaving the area of both channel and Gate 1 and the distance of 30 µm between the gate and the channel fixed (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The solid-state elec- trolyte is inkjet-printed on top of the channel and the gate electrodes, followed by UV-light induced cross-linking [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' More details on the fabrication process of these integrated OECTs are given in the Experimental Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Figure 2 presents the electrical characterization of a dual-gate OECT with AGate2 = AGate1 = 44100 µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The transfer characteristics are measured at a drain-source bias VDS of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 V for Gate 2 bias ranging from 0 V to +1 V in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The transfer curves systematically shift with the applied VGS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' For a controlling bias VGS2 of 0 V (grounded), the transfer curve at the far right in Figure 2 (a, b), the effect of the Gate 2 is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The dual-gate OECT behaves like a single-gate OECT and only Gate 1 redistributes ionic charges in the electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The effect of the VGS2 > 0 V on the dual-gate OECT is shown in the curves on the left in Figure 2 (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Polarons in PEDOT:PSS are neutralized by cations driven by VGS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' To compensate for VGS2 and achieve the original drain current at VGS2 of 0 V, the sweeping gate VGS1 has to be increased to more negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Therefore, the transfer curve and hence the threshold voltage is shifted to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The interplay between the bias on the two gate electrodes deter- mines the current in the dual-gate OECT at a given drain-source bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The gate current in Figure 2 (b) is significantly lower than the drain current because of the precise patterning technology of the solid-state electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' As shown in Figure 2(c), output curves are measured at VGS2 =0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The drain 4 current is only affected by VGS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Further, in Figure 2(d), the drain current is simultaneously influenced by VGS1 and VGS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' at the same scale of x-axis and y-axis, the drain current is close to 0 A, proving that the PEDOT-PSS-based OECT in fact operates as an accumulation-mode transistor (in agreement with the transfer curve shown in Figure 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' It is worth to mention that the effect of inhomogenous dedoping becomes relevant at high VDS where the curve is supposed to saturate [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In this study, we report on the change of threshold voltage for small VDS=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 V where inhomoegenous dedoping can be ignored and the threshold voltage is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' If inhomoegenous dedoping comes into play, the threshold voltage becomes a function of the drain-source voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Still, the VGS2 can be used to manipulate the transconductance of the transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' We postulate that the dual-gate OECT device behavior can be modelled as two parallel gate capacitor connected in series to the channel capacitor, as shown in Figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' This is because the electrolyte resistance (200kΩ) is negligible com- pared to the shunt resistance of the electrochemical double layers formed at the semiconductor-electrolyte interface (typical leakage current in the range of 10 nA at 1 V as shown in Figure 2 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The function of the solid-state electrolyte does not differ from that of a liquid electrolyte such as NaCl(aq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The solid polymer structure of the solid-state electrolyte forms a matrix for the move- ment of the ionic liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' As water has been used as a solvent for the solid-state electrolyte, the PEDOT:PSS layer is always in a swollen state, which allows ions from the ionic liquid to move in and out of the PEDOT:PSS layer and thus dopes and dedopes the semiconductor [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Accordingly, the voltage only drops across the gate capacitance/ channel capacitance and the solid-state electrolyte can be treated as an equipotential surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The total effective gate-source voltage influencing the channel is then determined by the total capacitance of CGate1 and CGate2 and the voltages applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' We extract the threshold voltage (Vth) from the transfer characteristics in Fig- ure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The Vth is defined by plotting the drain current against the gate-source voltage, linearly fitting this curve, and intercepting the value on the x-axis of VGS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The VDS of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 V is chosen to extract Vth is to avoid the effect of non-uniform dedoping in our system [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Figure 3(a) presents the threshold voltage as a function of the controlling gate VGS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The data is in a mean value for 5 devices for each geometry and clearly shows the shift in threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' It should be noted that these solid-state electrolyte OECTs show a significant hysteresis in the transfer curve [16], which makes it technically speaking impos- sible to derive a single threshold voltage value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' However, using the dual-gate configuration, we observe that the transfer curve including hysteresis is homoge- neously shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Hence, for a sake of simplicity, we only plot the transfer curve for switching the device from on- to off-state thereby ignoring the hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' When the area of Gate 2 is enlarged, the slope in Figure 3(a) increases, which al- lows us to turn these PEDOT:PSS-based OECTs from depletion- to accumulation- mode operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Figure 3(b) shows that the degree of controlling the threshold 5 voltage in dual-gate OECTs linearly scales with the ratio of the gate area (be- ing equal to the ratio of capacitances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Using the equivalent circuit proposed in Figure 1(c), we can predict the scaling of the threshold voltage shift as a function of the gate area ratio by the following expression: Vth = V0 th(AGate1 + AGate2 + AChannel) AGate1 − VGS2AGate2 AGate1 (1) where the constant V0 th represents the threshold voltage without the presence of Gate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' As shown in Figure 3(b), the experimental data fit very well to the model predictions and even without over-sizing Gate 2 significantly, a strong tunability of Vth is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In fact, accumulation-mode operation can be al- ready achieved if the area of Gate 2 is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='38-times larger than the area of Gate 1 (Figure 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' We demonstrate the advantage of this dual-gate OECT technology for logic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' As an example, a complementary inverter, combining a p-type and an n-type OECT, is chosen here as the most simple logic circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' It works as a digital amplifier and is often combined with OECT-based sensors to increase the biosignal sensitivity for bioelectronics [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The most common used n-type semiconductor material for OECTs is poly(benzimidazobenzophenanthroline) (BBL) [23, 34–36], with which an OECT operates in an accumulation-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Due to the depletion-mode operation of PEDOT:PSS-based OECTs and the large transconductance compared to BBL-based devices, good inverter charac- teristics can only be achieved if the BBL-based OECT is significantly larger than the PEDOT:PSS-based device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The channel width to length ratio of the BBL-based devices is typically chosen to be at least thousand times larger than the ratio of the PEDOT:PSS-based device (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=', 16000-times larger in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' A complementary inverter layout with a PEDOT:PSS-based dual-gate OECT (p-type) and a BBL-based OECT (n-type) is shown in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The input voltage (Vin) is applied to the common gate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=', the gate of the BBL device and the Gate 1 of the PEDOT:PSS-based dual-gate OECT and is swept from 0 V to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The supply voltage VDD is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' A constant voltage at Gate 2 (VGS2 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='25 V, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 V, 1 V) is applied during the inverter measurement to con- trol the threshold voltage of the p-channel device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The output voltage (Vout) is measured to determine the transfer curve of the inverter which is shown in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' As VGS2 increases, the inverter transfer curve shifts to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' In particular, the trip point of the inverter is seamlessly adjusted by VGS2 as it controls the threshold voltage of the dual-gate devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The dual-gate design of OECTs offers a more robust design and operation of circuits, which can be used to improve the sensitivity of any bioelectronics system and thus contributes greatly to the field of bioelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 6 Figure 1: (a) Top view and cross section configuration of a PEDOT:PSS-based dual-gate OECT, including source and drain electrodes, PEDOT:PSS channel, and two PEDOT:PSS gates symmetric with respect to the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' All the PEDOT:PSS films have the same thickness of 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (b) Optical microscopic image of a dual-gate OECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (c) Equivalent circuit model of the dual-gate OECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 7 (a) Top View (b) Gate 2 (control) 210 x 210 um² AGate2 Solid Electrolyte W=150 um =30 μm Source PEDOT:PSS Drain PEDOT:PSS 100 μm 210 x 210 μm² AGatel Gate 1 (sweep) (c) Gate 1 Gate 2 Cross section Gatel PEDOT:PSS Solid Electrolyte Channel Source Drain Channel Gate 1 (sweep) Gate 2 (control) gap = 30 umFigure 2: Electrical characterization of the PEDOT:PSS-based dual-gate OECT with AGate2 = AGate1 = 44100 µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (a) Transfer characteristic in linear scale at VDS = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' VGS2 is fixed for the loop of VGS1 sweeping from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 V to +1 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (b) Transfer characteristic in logarithmic scale, including the drain current (solid line) and the gate current (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The same color of the drain current and the gate current means they are under the same VGS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (c) Output characteristic of the dual-gate OECT at VGS2 = 0 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (d) Output characteristic at VGS2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 8 (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 10-3 Ds = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 V 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='3 10-5[ mA GS2 = 0 V GS20 V A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 10-6 D I 10-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 10-8 VGS2 10-9 0 =1 V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 1 VGsil V (c) (d) 1e-4 1e-4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 VGs2 = 0 V VGs2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 V 2 2 A-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 LD 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0 0 VGSI Ves1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 VDsI V VDs I VFigure 3: (a) Threshold voltage of a PEDOT:PSS-based dual-gate OECT as a function of VGS2 for different gate area ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The larger area of Gate 2 (red) gives a steeper slope, namely a larger degree of tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Each data point is a mean value for 5 devices and each device is measured 3 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (b) The degree of threshold voltage tuning increases with the ratio of the gate area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 9 (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 gate area ratio Linear Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 1 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 Gate 2 iate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='6 AGate 2= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 0 AGate 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 AGate2= 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 AGate!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='7 △V th △VGS2Figure 4: (a) A complementary inverter layout of a PEDOT:PSS-based dual- gate OECT with ( W L )p = 5 and a BBL OECT with ( W L )n = 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The input voltage Vin applies to the Gate 1 of the PEDOT:PSS-based dual-gate OECT and to the gate of the BBL OECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' VGS2 is constantly applied at the Gate 2 of the PEDOT:PSS-based dual-gate OECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The device configuration of both types of OECTs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' the gate electrode of BBL OECT is Ag/AgCl immersed in the ionic liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Further device dimension can be found in the Experimental section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' (b) Inverter transfer characteristic: as the threshold voltage of the PEDOT:PSS-based dual-gate OECT is tuned by increasing VGS2 (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='25 V, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 V to 1 V), the transfer curve (solid line) shifts to the left, and the inverter gain (dashed line) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 10 (a) (b) 10 DD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 8-- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='6 VGS2 LS1 JS2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='25 V V In out 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 1V :0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='8 ),= 2000 Vin\\V BBL ionic liquid3 Conclusion In conclusion, we demonstrate continuous tuning of the threshold voltage in PEDOT:PSS-based dual-gate OECTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' These dual-gate structures are easy to fabricate, employing the often used side-gate architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The threshold volt- age scales linearly with the voltage at the control gate (Gate 2), and the degree of tuning linearly increases with the gate area ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Furthermore, we mod- eled the device behavior with an equivalent circuit, and the experimental data fit very well with the model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The PEDOT:PSS-based dual-gate OECTs, which can be densely integrated using conventional photolithography or printing techniques, show excellent device performance, and they can be pushed to accumulation-mode operation, leading to improved performance and relaxed design requirements of complementary OECT inverters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 4 Experimental Device fabrication: The process of structuring the electrode and channel pat- tern of a dual-gate OECT follows ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Source, drain, and gate electrodes were patterned on a glass substrate with 50 nm Au and 3 nm Cr by photolithography and wet-etching using Standard Gold Etchant and Standard Chromium Etchant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' PEDOT:PSS-based solution was prepared with 95 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='% of PEDOT:PSS (Hear- aeus Clevios PH 1000, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='% solids in water, 1:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5) and 5 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='% of ethylene glycol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' This solution was spin-coated at 3000 rpm on the electrodes and the 100 nm-PEDOT:PSS thin film was patterned by fluorine-based photolithography [37] and dry etching [38] using O2 and Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The solid-state electrolyte precur- sor solution containing 1 mL deionized water, 750 mg N-isopropylacrylamide, 20 mg N,N’-methylenebisacrylamide, 200 mg 2-hydroxy-4’-(2-hydroxyethoxy)- 2-methylpropiophenone, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content='5 mL 1-ethyl-3-methylimidazolium ethyl sulfate [16] was inkjet printed on top of the active area followed by 2 minutes UV cross- linking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The PEDOT:PSS-based dual-gate OECT was stored in the glovebox overnight and then was encapsulated with glass for further measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' BBL solution was prepared by dissolving 5 mg BBL (Sigma Aldrich) in 1 mL methanesulfonic acid and stirring at 70 ◦C overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The BBL solution was then spin-coated at 1000 rpm on a gold substrate with W=10 mm, L=5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Afterwards, the BBL film was soaked into ethanol for 1 minute and then dried on a hot plate at 150 oC for 5 minutes, and the resulting BBL film is 70 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Electrical characteristics: Transfer and output characterizations were done with Keithley SMUs controlled by the software SweepMe!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' BBL OECTs were mea- sured with a Ag/AgCl gate and ionic liquid 1-ethyl-3-methylimidazolium ethyl sulfate in 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' n-type organic electrochemical transistors: materials and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Jour- nal of Materials Chemistry C, 6(44):11778–11784, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' [36] Hanyu Jia and Ting Lei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Emerging research directions for n-type conjugated polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Journal of Materials Chemistry C, 7(41):12809–12821, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' [37] Hans Kleemann, Alex A Zakhidov, Merve Anderson, Torben Menke, Karl Leo, and Bj¨orn L¨ussem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Direct structuring of c60 thin film transis- tors by photo-lithography under ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Organic Electronics, 13(3):506–513, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' [38] Marco H¨oppner, David Kneppe, Hans Kleemann, and Karl Leo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Precise patterning of organic semiconductors by reactive ion etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Organic Elec- tronics, 76:105357, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Acknowledgements We thank the European Social Fund (Project OrgaNanoMorph, project num- ber: 100382168) and the Hector Fellow Academy (30000619) for providing the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' Furthermore, the authors thank the Bundesministerium f¨ur Bildung und Forschung (BMBF) for funding from the project BAYOEN (01IS21089).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' The EMERGE project has received funding from the European Union’s Horizon 2020 research and Innovation programme under grant agree- ment Nº 101008701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 15 Data Availability The data that support the findings of this study are available from the corre- sponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE3T4oBgHgl3EQfKwlD/content/2301.04356v1.pdf'} diff --git a/o9AyT4oBgHgl3EQfzPm_/content/2301.00699v1.pdf b/o9AyT4oBgHgl3EQfzPm_/content/2301.00699v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..317e1f9c1b910bd4c0953779f59acdc4988d2155 --- /dev/null +++ b/o9AyT4oBgHgl3EQfzPm_/content/2301.00699v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Ramaswamy1, Sing Yu Lin1, Dora Zhao2*, Aaron B. Adcock3, +Laurens van der Maaten3, Deepti Ghadiyaram, Olga Russakovsky1 +1Princeton University +2Sony AI +3Meta AI +*Work done as a graduate student at Princeton University +Abstract +Current dataset collection methods typically scrape +large amounts of data from the web. While this technique +is extremely scalable, data collected in this way tends to re- +inforce stereotypical biases, can contain personally identifi- +able information, and typically originates from Europe and +North America. In this work, we rethink the dataset col- +lection paradigm and introduce GeoDE , a geographically +diverse dataset with 61,940 images from 40 classes and +6 world regions, and no personally identifiable informa- +tion, collected through crowd-sourcing. We analyse GeoDE +to understand differences in images collected in this man- +ner compared to web-scraping. Despite the smaller size +of this dataset, we demonstrate its use as both an evalu- +ation and training dataset, highlight shortcomings in cur- +rent models, as well as show improved performances when +even small amounts of GeoDE (1000 - 2000 images per re- +gion) are added to a training dataset. We release the full +dataset and code at https://geodiverse-data- +collection.cs.princeton.edu/ +1. Introduction +The creation of large-scale image datasets has enabled +advances in the performance of computer vision models. +Although previously limited by manual collection and an- +notation efforts [11, 12, 14], recently the size of these +datasets has rapidly grown. This growth has been empow- +ered by a new data collection framework: scraping web im- +ages at scale. These images are either human-labelled (e.g., +ImageNet [8,23]), use tags (e.g., CLIP-400M [21]) or used +for self-supervised learning (e.g., PASS [2]). +However, these web-scraped datasets come with their +downsides. First, the datasets can contain pernicious gen- +der and racial biases by underrepresenting certain demo- +graphic groups and using stereotypical depictions of these +groups [4, 29, 35]. Second, significant geographic bias is +present in these datasets with most images coming from +GeoYFCC distribution [9] +0 +10 +20 +30 +40 +Percentage of Images from Region +GeoDE distribution (ours) +0 +5 +10 +15 +Percentage of Images from Region +Figure 1. We construct a geographically diverse dataset GeoDE +that is approximately balanced across 6 world regions. We visual- +ize the images per region, and compare our distribution (bottom) +to that of a previously created diverse dataset GeoYFCC [9] (top). +North America or Western Europe [24]. +As de Vries et +al. [7] show, this lack of geo-diversity propagates into +downstream recognition tasks–resulting in difficulty in rec- +ognizing common household objects from non-Western re- +gions. Third, copyright and consent may pose challenges +for web-scraped data. Dataset creators sometimes do not +obtain full permission of the content creators and of the +people featured in the content.1 Finally, while annotators +are compensated for their time, content creators and image +subjects are rarely compensated for their contributions to +the dataset [3]. Though there have been efforts to balance +datasets [9], clean datasets [31], and protect privacy of indi- +viduals by blurring [34], methods that rely on web-scraping +1While images used are sometimes under the most permissive Creative +Commons license, it is unclear if creators know the full impacts of their +images being used in the training of large scale models. +1 +arXiv:2301.02560v1 [cs.CV] 5 Jan 2023 + +cannot fully eliminate the aforementioned issues [3,17]. +Rather than improving web-scraped datasets, we rethink +the entire paradigm: we explore a data collection approach +where we do not scrape images, but rather collect images +via crowd-sourcing. We commission photos of different ob- +jects from people across the world. This approach naturally +resolves copyright concerns, and enables much tighter con- +trol of image distribution which may reduce biases. We +partnered with a company called Appen (www.appen. +com) and crowd-sourced the Geographically Diverse Eval- +uation (GeoDE) dataset. GeoDE contains 61,940 images +roughly balanced across 40 object categories and six ge- +ographic regions. +Despite the small size of GeoDE, the +dataset has several key advantages. +First, the object recognition problem becomes surpris- +ingly challenging since the images represent the diverse +appearance of common objects across six global regions: +Africa, the Americas, East Asia, Europe, Southeast Asia, +and West Asia. Similar to de Vries et al. [7], we show that +modern object recognition models perform poorly on rec- +ognizing objects from Africa, East Asia, and SE Asia. Aug- +menting current training datasets (like ImageNet [8, 23]) +with images from GeoDE yields an improvement of 12.0% +on DollarStreet [22] and 20.9% on a test split of GeoDE . +Second, requesting images containing specific object +classes removes selection bias: objects present in images +that are web-scraped are uploaded by creators with different +incentives, e.g. to make exciting/unique content or to gen- +erate engagement [25], and disincentives showing mundane +everyday content. We show that the distribution of images +in GeoDE is different to that in other datasets, even when +controlling for world region and object class. +Third, we own the copyright to all of the images in +GeoDE , have explicit permission from content creators +to use these images for machine learning applications, and +ensured fair compensation to the content creators. +We acknowledge that a main drawback of this method is +the cost: this approach does not scale as well as scraping +images from the web and is partially the result of ensuring +fair compensation for content creators and curators. How- +ever, we demonstrate that even small amounts of data col- +lected in this way can be beneficial in partially remedying +some of the concerns with large-scale web-scraped datasets. +Data and code can be found at https://geodiverse- +data-collection.cs.princeton.edu/ +2. Related Work +There are three key research directions that inspired this +work. The first is the call to increase geographic diversity in +visual datasets [7,24]. In response, there have been attempts +to construct geographically diverse datasets [1,9,22], sum- +marized in Tab. 1. However, these datasets are still geo- +graphically concentrated (in Europe for GeoYFCC [9], In- +dia for OpenImages Extended [1]), and/or are still relatively +small scale (DollarStreet [22]), prompting our work. +Second, in using crowd-sourcing to generate visual con- +tent rather than using web-scraped images, we follow re- +cent video datasets Charades [25], Epic Kitchens [6] and +Ego4d [13]. However, we differ in that our key goal is to +ensure a geographically balanced dataset. This poses chal- +lenges in recruitment and dataset scope (more in Sec. 3). +Finally, in our data collection efforts we take into ac- +count the extensive literature around selection bias in com- +puter vision datasets [3, 4, 10, 26, 27, 29, 32, 35] and ensure +that our dataset is collected responsibly, with attention to +privacy, consent, copyright and worker compensation [3]. +3. Collecting GeoDE +We describe our data collection process, including our +selection of object classes and world regions. +Selecting the object classes. We focus on object classes +that are likely to be visually distinct in different parts of the +world. Selecting such objects is a chicken-and-egg prob- +lem: without a geographically diverse dataset at our dis- +posal, it is unclear which objects are diverse. We adopt a +number of heuristics using existing datasets to find a plau- +sible set. The full process is detailed in the appendix, but +briefly, we use simple computer vision techniques (linear +models and visual clustering, using features extracted from +PASS-pretrained models [2]) along with manual examina- +tion to identify a set of candidate tags from DollarStreet [22] +and GeoYFCC [9] (e.g., “chili,” “footstool,” “stove”). To +prune these tags, we remove those that are not objects (e.g., +“arctic”, “descent”), removing wild animals not found in all +regions (e.g., “gnu”, “camel”) and group variants of objects +(e.g., “stupa”, “temple”, “church”, “mosque” and “chapel”). +The final set of objects is in Tab. 2. +Selecting diverse geographic regions. We chose six re- +gions across the world: Africa, Central and South Amer- +ica (“Americas”), East Asia, Europe, Southeast Asia and +West Asia. Within each region, we targeted 3-4 countries +(Tab. 3). These were chosen due to the lack of available im- +ages from these regions in most public datasets [7, 24, 30]; +the countries were chosen based on the presence of partic- +ipants within Appen’s database. We obtain a roughly even +distribution of images across each class and region pair. +Image collection and worker demographics. Work- +ers were asked to upload images for a given object class, +following the instructions in Fig 2. There were more than +4,500 workers, representing a range of genders, ages and +races (Fig. 3). All images submitted were manually checked +by Appen’s quality assurance (QA) team. +2 + +Dataset +Size; +distribution +Collection process; +annotation process +Geographic coverage +Personally +Identifi- +able Info (PII) +ImageNet +[8,23] +14.2M images; +mostly balanced +across classes +Scraped images from the web +based on the class label; crowd- +sourced annotations +Mostly North America +and Western Europe [24] +Contains +people, +some +images +have +faces blurred [33] +OpenImages +[18] +9.1M images; +long-tailed class +distribution +Flickr images with CC-BY li- +censes; +automatic labels with +some human verification +Mostly North America +and Western Europe [24] +Contains people +OpenImages +Extended +[1] +478K images; +long-tailed class +distribution +Crowd-sourced gamified app to +collect images; automatic labels +and manual descriptions +More than 80% of im- +ages are from India +People are blurred +GeoYFCC +[9] +330K images; +long-tailed class +distribution +Flickr images subsampled to be +geodiverse; noisy tags +Geographically +diverse +(62 +countries), +but +concentrated in Europe +Contains people +PASS +[2] +1.4M images; +N/A (no labels) +Random images from Flickr; no +annotations +Collected from Flickr, +thus mostly North Amer- +ica and Western Europe +No people +DollarStreet +[22] +38,479 +images; +mostly balanced +across topics +Images by professional and vol- +unteer photographers; manual la- +bels including household income +63 countries in four re- +gions (Africa, America, +Asia and Europe) +Yes, with permission +GeoDE +(ours) +61,940 images; +balanced across +classes®ions +Crowd-sourced collection using +paid workers; manual annotation +Even distribution over +six geographical regions +(Tab. 3) +No identifiable peo- +ple and no other PII +Table 1. We compare recent approaches to dataset collection. Although GeoDE is smaller than recent datasets, we ensure that the images +are sourced with permission of the creator, contain no identifiable people, and are balanced across both regions and object classes. +Indoor +common +bag, chair, dustbin, hairbrush/comb, hand soap, +hat, light fixture, light switch, toothbrush, +toothpaste/toothpowder +Indoor +rare +candle, cleaning equipment, cooking pot, jug, +lighter, medicine, plate of food, spices, stove, toy +Outdoor +common +backyard, car, fence, front door, house, road sign, +streetlight/lantern, tree, truck, waste container +Outdoor +rare +bicycle, boat, bus, dog, flag, monument, religious +building, stall, storefront, wheelbarrow +Table 2. GeoDE consists of 40 object classes, loosely organized. +West Asia +Saudi Arabia, United Arab Emirates, Turkey +Africa +Egypt, Nigeria, South Africa +East Asia +China, Japan, South Korea +SE Asia +Indonesia, Philippines, Thailand +Americas +Argentina, Colombia, Mexico +Europe +Italy, Romania, Spain, United Kingdom +Table 3. GeoDE consists of images from six world regions. Within +each region, there are 3-4 countries contributing to most of the im- +ages, chosen to balance the diversity of the images against practi- +cal data collection considerations. Participants from outside these +countries but within the same region were still accepted. +General Instructions +In this task, you will submit up to 3 photos of the same type of object +(e.g., upload 3 photos of 3 different bags; please do not upload 3 +photos of the same bag from different angles). +1. Please make sure the location function is enabled for the camera. +2. The photo resolution should be at least 640 x 480. +3. All images should be new photos captured with +Appen Mobile. +4. Please make sure it’s a single object per image. +5. Please make sure it’s a well-lit environment and the object is +clearly visible in the photos. +6. Please make the object occupy at least 25% of the image. +7. Objects captured are foregrounded and not occluded. +8. Objects should not be blurred, e.g., motion blur. +9. No effects or filters added (cropping is acceptable). +10. Please try to avoid capturing people in the images (it’s OK if +people are blurry in the background and far from the camera). +11. Please try to avoid capturing vehicle license plates in images. +Figure 2. Image collection instructions given to GeoDE workers. +4. Lessons learned from collecting GeoDE +A key part of this study was to understand if crowd- +sourcing images is a viable alternative to web-scraping. In +this section we detail the challenges faced and the lessons +learned in constructing the GeoDE dataset. +Getting sufficient images of all object classes. While +some object classes expectedly proved more difficult than +3 + +Male +Female +Gender +White +African +East +Asian +Latino or + Hispanic +South + Asian +Middle +Eastern +Other +Race +< 20 +20-29 +30-39 +40-49 +> 50 +Age +Figure 3. Demographics of workers contributing GeoDE images. +Figure 4. Some issues we encountered within GeoDE. (Left) Three +images submitted for “hat” by the same worker; these are near- +identical except for the color. (Right) Especially for outdoor im- +ages, the target class may be ambiguous: two of these images were +submitted for “fence”, and one for “tree” +others (e.g., instances of “monument” or “flag” were simply +hard for workers to find), others surprised us. For example, +“stove” was originally underrepresented until the definition +of “stove” was clarified to “any cooking surface either elec- +tric, gas, induction.” Workers did not always consider their +cooking appliance to be a “stove,” highlighting a vocabulary +challenge unique to geographically diverse data collection. +Multiple copies of images. Two most common cate- +gories of error were incorrect images (i.e, having an class +other than the one selected) and multiple copies of an im- +age. The QA team found that participants often submitted +multiple copies of the same object instance from different +angles despite clear instructions not to do so. Workers also +sometimes submitted very similar objects (see Fig. 4 (left), +where there are three hats submitted by the same worker +with slightly varying colors). We filtered out such images. +Multiple objects per image. For some of the (especially +larger and outdoor) object categories, it was difficult to en- +sure that additional objects were not present. For example, +we found that images of “fences” often have “trees” present, +and it was not always possible to discern between objects +in the foreground and background (see Fig. 4 (right)). We +found that this was primarily an issue for the “tree” class, +and thus requested that images that had a significant portion +of the image covered by trees are tagged. These additional +annotations can be used to filter and remove such images +and/or to analyze errors made by a model. +Other. Beyond these, the rest of the data collection went +smoothly. Following instructions, only 0.78% images con- +tained identifiable information. Some images contain non- +identifiable incidental people in the background (especially +for larger object classes, like “monument”). All such im- +ages are tagged in GeoDE. We were also able to ensure that +the number of images per region is roughly equal, although +unfortunately it was harder to obtain an even number of im- +ages per country within each region. +Cost. Collecting images in this way is expensive: each +image costed $1.08 (not including researcher time). This +allowed us to fairly compensate workers for their labour as +well as the QA team to ensure the quality of these images. +5. Comparing GeoDE to current datasets +We compare GeoDE with two datasets: the canonical +ImageNet [8] and the geographically diverse GeoYFCC [9]. +Qualitative. In Fig. 5, we show a subset of 60 GeoDE +images of “stoves” and “houses” (more in the appendix). +Compared to images from ImageNet, we see a larger variety +of stoves: e.g., induction coils, single and two burner stoves. +The stoves also appear more used than those in ImageNet. +Similarly, for “house,” we see a larger range in terms of +materials used and size. In the Secs. 6 and 7 we examine +the impact of this diversity on visual recognition models. +Statistics. We compare with GeoYFCC, a dataset sam- +pled from YFCC100m to be more geo-diverse. Fig. 6 (left) +shows that most images in GeoYFCC come from Europe; +e.g., images from Africa and West Asia comprise less than +10% of the dataset. We also consider tags in GeoYFCC that +correspond to classes within GeoDE, and show the per-class +distribution of images in Fig. 6 (right). We see selection +bias in the objects people choose to upload, with “monu- +ments” accounting for more than 25% of these GeoYFCC +images. +In contrast, GeoDE is approximately balanced +across both regions and objects (details in the appendix). +Object appearance. Finally, we take a stab at quanti- +fying the differences in the appearance of images collected +through crowd-sourcing and web-scraping, by comparing +the images in GeoDE with those in GeoYFCC [9]. We ex- +tract features for each dataset using a ResNet50 model [16] +trained with self-supervised learning SwAV [5] on the PASS +dataset [2]. We first train a linear classifier to predict which +dataset an image was drawn from. The classifier achieves +an accuracy of 96.3%. To understand how the dataset distri- +butions are differ beyond just the class/region frequencies +we obtain low-dimensional TSNE embeddings [28]. The +TSNE plots are in Fig. 7. Even within an object class and +region, the image features are very different, suggesting that +different image collection methods lead to different distri- +bution of images.2 +6. GeoDE as an evaluation dataset +We now analyze the use of GeoDE as an evaluation +dataset, by using it to evaluate two canonical models: the +recent CLIP [21] and an ImageNet [8]-trained model. +Implementation details. +For the CLIP model, we use +the weights provided for the ViT-B/32 model. We use text +2We note that GeoYFCC just has tags, not labels, hence, these might +be noisy. We also visualize these plots for Imagenet in the appendix. +4 + +Figure 5. Sample images of two object classes in different regions within GeoDE (and ImageNet in the bottom row, for comparison). +Product labels on images have been blurred. +West Asia +Africa +East Asia +SE Asia +Americas +Europe +0 +20 +40 +% of images +GeoYFCC +GeoDE +Uniform +Indoor common +Indoor rare +Outdoor common +Outdoor rare +Dataset +bag +chair +dustbin +hat +light fixture +clean. equip. +cooking pot +lighter +plate of food +spices +stove +toy +car +fence +front door +house +streetlight +tree +truck +waste cont. +bicycle +boat +bus +flag +monument +religious bld +stall +storefront +wheelbarrow +GeoYFCC 1 1 0 3 +2 +0 0 0 10 1 0 2 2 1 2 11 0 2 1 0 5 4 2 1 29 16 0 4 0 +GeoDE +5 5 4 4 +3 +4 3 3 +4 +4 4 4 4 4 3 3 +3 4 3 3 3 2 3 3 +3 +3 3 3 2 +Figure 6. Distribution (in %) of object counts, across the 6 regions (left) and for the 30 object classes (right) that are present in both +GeoYFCC [9] and GeoDE. GeoYFCC has a long tail in both cases, with images from Europe comprising over 40% of the dataset and with +monuments, religious buildings, houses and plates of food comprising more than 66% of the images, while objects like dustbin appear in +less than 0.5% of these images (denoted as 0). In contrast, our GeoDE dataset is balanced across these regions and object classes. +prompts for all 40 object categories as described in the zero- +shot recognition setup of [21]. To train a model on Ima- +geNet [8], we first match the classes of GeoDE and Ima- +geNet. We find the corresponding synset for each GeoDE +class in WordNet [19], and include all images of that synset +and its children. For two object categories (“backyard” and +“toothpaste/toothpowder”) we do not find any matching cat- +egories, and so we ignore these categories in the quantita- +tive analysis. We split our filtered ImageNet [8] dataset into +train (38,353 images), validation (12,794 images), and test +(12,795 images) datasets. As in Sec. 5 we extract features +using a ResNet50 model [16] trained with self-supervised +learning SwAV [5] on PASS [2], and retrain the final layer. +Results. Tab. 4 shows the accuracy across different re- +gions on these two models. Compared to CLIP, the over- +all performance on GeoDE by an ImageNet trained model +is considerably lower. However, both models perform the +best on images from Europe and the worst on images from +5 + +Stove +House +West Asia +Africa +East Asia +Southeast +Asia +Americas +Europe +ImageNetWest Asia +Africa +East Asia +Southeast Asia +Americas +Europe +Plate of food +20 +10 +0 +10 +20 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +20 +10 +0 +10 +20 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +30 +20 +10 +0 +10 +20 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +20 +10 +0 +10 +20 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +20 +10 +0 +10 +20 +30 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +30 +20 +10 +0 +10 +20 +30 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +Storefront +20 +15 +10 +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +GeoYFCC +GeoDE +20 +15 +10 +5 +0 +5 +10 +15 +20 +15 +10 +5 +0 +5 +10 +15 +GeoYFCC +GeoDE +20 +10 +0 +10 +20 +10 +5 +0 +5 +10 +GeoYFCC +GeoDE +20 +10 +0 +10 +20 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +15 +10 +5 +0 +5 +10 +15 +15 +10 +5 +0 +5 +10 +15 +GeoYFCC +GeoDE +20 +15 +10 +5 +0 +5 +10 +15 +20 +20 +10 +0 +10 +20 +GeoYFCC +GeoDE +Figure 7. We visualize the TSNE plots for several of object classes per region for GeoYFCC and GeoDE . While the features do overlap +slightly, on the whole, they are very different for dataset distributions, even within each (region, object) tuple. +Model +WAsia Africa EAsia SEAsia Americas Europe +ImageNet +69.4 +62.7 +63.3 +67.3 +68.6 +69.9 +CLIP +84.0 +78.7 +79.9 +81.9 +84.4 +85.8 +Table 4. Accuracies (in %) on GeoDE of a model trained on a +subset of Imagenet [8] and of CLIP [21]. The models perform +best on images from Europe, and worst on images from Africa. +Africa (difference of more than 7% in both cases). +Tab. 5 breaks out the per-object accuracy for CLIP. +While the average accuracy is 82.8%, classes like “dust- +bin” (37.3%), “medicine” (54.1%), “cleaning equipment” +(59.0%), “spices” (63.2%) and “house” (63.3%) perform +poorly. Fig. 8 shows some example errors. +When investigating the different classes with the CLIP +model, we find geographic disparities. Classes like “fence”, +“stove” and “spices” have different accuracies in different +regions: “fence” is over 88% for images from Europe, but +only 60% and 59% for images from Africa and Southeast +Asia respectively. Similarly, “stove” has accuracy of 95% +in the Americas but only 67% in East Asia. We visualize +this using the TSNE plots of the features for these classes +in Fig. 9. We find that these classes have objects that are re- +gion specific. For example, “religious buildings” from East +and Southeast Asia can include buildings like monasteries +and temples. Similarly, single- and two-burner “stoves” are +primarily from countries in Africa and Southeast Asia. +7. Impact of training with GeoDE +Finally, we attempt to answer how training with GeoDE +data can improve the performance of these models. We in- +vestigate training a model where we combine images from +GeoDE with images from ImageNet. We find the combina- +tion of the two can improve results across regions. +7.1. Training a model with GeoDE +We would like to understand how training a model with +data from GeoDE affects the object recognition capabilities +of the dataset. Fixing the total number of images used for +training, we train a linear model, using a pre-trained feature +extractor, on a dataset comprised entirely of ImageNet im- +ages and a dataset comprised of both ImageNet and GeoDE +images. The feature extractor is a ResNet50 [15] model +trained on PASS [2] using SwAV [5] 3. +Implementation details. We split the GeoDE dataset +into a train (4,970 images per region), validation (between +1657 and 2188 images per region), and test (between 1657 +and 2189 images per region) datasets. We use the validation +dataset to select training hyperparameters. We consider the +training set for our ImageNet only model as the same 38,353 +image training set constructed in Sec. 6, only considering +tags in ImageNet that correspond to classes within GeoDE,. +To construct the training set of our ImageNet and all regions +in GeoDE model, we start with the training set for ImageNet +and add in the training sets for all 6 regions in GeoDE while +removing proportionately per class the same number of im- +ages from ImageNet. This procedure gives a training set +of 29,820 GeoDE images and 8,533 ImageNet [8] images. +The final models are trained using an SGD optimizer, with a +learning rate of 0.1, and momentum of 0.9, for 500 epochs +with cross entropy loss. We use models with the highest +accuracy on the validation set. Unless specified otherwise, +results are reported on the test set. +Results. We first report results on the GeoDE test set, +and notice a significant improvement in accuracy across all +regions, as a result of training with both GeoDE and Im- +ageNet (Tab. 6). However, this improvement could come +from the ImageNet + GeoDE dataset matching the domain +of the GeoDE evaluation set and may not generalize to other +datasets. To address this, we also test these models on a dif- +ferent dataset: the DollarStreet dataset [22]. This dataset +has been used before as an evaluation benchmark [7], to un- +derstand if current object recognition models can perform +well on objects from a diverse set of regions. Tab. 7 lists the +per class accuracies for the object categories that overlap be- +tween GeoDE and DollarStreet. We see an increase in per- +formance across most categories, suggesting that GeoDE is +more geo-diverse than ImageNet and that there is an advan- +tage to using geo-diverse data in the training set. +3We also try training a ResNet50 [15] model from scratch as well +as finetuning the model trained on ImageNet, and do not find significant +changes to our results. Results are provided in the appendix +6 + +lightswitch +bus +chair +bag +dog +monument +car +hairbrush +boat +cooking pot +hat +road sign +bicycle +religious bld +flag +toothbrush +toothpaste +storefront +wheelbarrow +light fixture +truck +plate of food +hand soap +front door +jug +stove +lighter +stall +streetlight +fence +backyard +toy +candle +waste cont. +tree +house +spices +clean. equip. +medicine +dustbin +98 97 97 96 96 96 96 95 95 95 93 93 92 92 91 90 90 89 89 88 88 88 86 85 85 78 77 76 76 75 74 73 71 69 68 63 63 59 54 37 +Table 5. Per-class accuracy (in %; decreasing order) of CLIP [21] on GeoDE. Object in bold are poorly recognized by the model. +Figure 8. Example errors that the CLIP [21] model makes on GeoDE images (the ground truth label on the left, CLIP prediction at the +bottom). There are some systematic errors, e.g., classifying “house” as a “religious building”, particularly on images from regions in Asia. +(product labels on images have been blurred. ) +Model +WAsia Africa EAsia SEAsia Americas Europe Avg +ImageNet +69.4 +62.7 +63.3 +67.3 +68.6 +69.9 +66.9 ++GeoDE +88.2 +86.7 +86.4 +86.5 +89.1 +90.0 +87.8 +Table 6. +We notice significant performance improvements on +GeoDE after augmenting ImageNet with images from GeoDE +across all regions. +bicycle +car +cooking pot +front door +hand soap +house +light fixture +light switch +plate of food +spices +toothbrush +toy +waste cont. +Average +Imagenet 89 90 53 77 28 55 56 80 62 30 45 41 6 55 ++ GeoDE 92 89 67 82 45 57 89 78 85 52 55 52 24 67 +Table 7. +We compare the per class accuracies of the Dol- +larStreet [22] dataset for a model trained on only ImageNet [8] +and a model trained on both ImageNet and GeoDE . We find that +even adding such a small amount of diverse data into the training +pipeline can improve the overall performance of the model. +7.2. Cost-vs-Diversity tradeoffs +The main drawback of GeoDE is the cost of this dataset: +images collected in this way cost more than the standard +pipeline of web-scraping and crowd-sourcing annotations. +Thus, it is important to identify which classes and regions +contribute most to the overall model. To investigate this, +we start with the filtered ImageNet dataset described above, +vary the amount of GeoDE data from a particular region, +and analyze the change in overall regional performance and +regional performance for specific objects. +Implementation Details. We start with a dataset fully +comprised of the 38,353 filtered ImageNet images. We then +add a region of GeoDE’s data back into the dataset and re- +move the same number of ImageNet images to keep both +the number of images and class balance the same for each +model we train. The other training details remain the same +as in Sec. 7.1. +Evaluation. As we are evaluating on the GeoDE test set, +there are two possible sources of performance gain. The +first is that the model is able to take advantage of the ad- +ditional regional information from the GeoDE data. The +second is that the GeoDE images were collected using the +same collection method as the test set and from Sec. 5, we +saw that there is a difference in the feature space that can be +attributed to the collection method itself (crowd-sourcing +rather than web-scraping). In order to distinguish between +7 + +West Asia +Africa +East Asia +Southeast Asia +Americas +Europe +Cleaning +Equipment +Toothbrush +Toothbrush +Toothbrush +Hairbrush +Hairbrush +Hand soap +Toothbrush +Dustbin +Hand soap +Toothbrush +Stall +Hand soap +Dustbin +Waste +Cooking pot +Cooking pot +Cooking pot +Waste +Waste +Stall +Waste +Waste +Cooking pot +Waste +Waste +container +container +container +container +container +container +container +鱼 +House +Religious +Religious +Cooking pot +Car +Front door +Wheel- +Religious +Backyard +Religious +Storefront +Religious +building +Storefront +building +barrow +building +building +building +Medicine +Hand soap +Toothpaste/ +Light fixture +Hand soap +Toothpaste/ +Toothpaste/ +Toothpaste/ +Toothpaste/ +Hand soap +Toothpaste/ +Toothpaste/ +Toothpaste/ +toothpowder +toothpowder +toothpowder +toothpowder +toothpowder +toothpowder +toothpowder +toothpowder +Spices +Waste +Hand soap +Waste +Stove +Candle +Toothpaste/ +Medicine +Toothpaste/ +Candle +Bag +Jug +Cooking pot +container +container +toothpowder +toothpowderreligious building +spices +stove +Figure 9. We show the TSNE plots of classes which have large regional disparities in accuracy from the CLIP trained model and show +images from different parts of the plots. For “religious buildings”, we see that GeoDE contains a cluster of monasteries and temples, mostly +from East and Southeast Asia. For “spices”, we see a separation based on the spice container which can be region dependent. +0 +50 +100 +% of images used from GeoDE +60 +70 +80 +Accuracy (%) +Trained on Africa, + tested on Africa +Trained on Africa, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on SE Asia, + tested on SE Asia +Trained on SE Asia, + tested on Europe +Figure 10. We show the effect of incrementally adding regional +GeoDE data for Africa and SE Asia on both the specific region and +on Europe. We find that while adding any GeoDE regional images +increases the performance of the model on European images, it has +a larger effect on the region the images were drawn from. +these two sources, we measure the accuracy on both the re- +gion in the train set and accuracy on the images from Eu- +rope4. We also measure the increase in average precision +for specific objects to better understand which objects ben- +efit the most from GeoDE data. +Results. In Figure 10 we visualize the increase in ac- +curacy as we incrementally add in images from Africa and +Southeast Asia (all other regions are presented in the ap- +pendix). We can see that the performance both within the +specific region and in Europe increase with the additional +GeoDE data. The relative increase in performance for the +specific region (i.e. Africa or Southeast Asia) is larger than +the increase for Europe, showing the value of data for each +region. We also note that the improvements have not sat- +urated, suggesting that obtaining more regional data could +lead to further gains. +We also examine the classes that have the largest increase +in average precision (AP) as the regional GeoDE images are +added to the dataset in Figure 11. We present the object +classes that see the most improvement in Tab. 8. In general, +we see a large overlap in which objects benefit the most +from regional GeoDE data. In particular, the performance +4We use Europe as this region had the best performance when using a +model trained on just ImageNet. +Figure 11. We measure the relative improvement in average preci- +sion per object when GeoDE images from that region are included +in training. Each vertical line represents an object, and we sort +them by the region where that object saw the largest improvement. +We see that Africa and East Asia see the largest improvement for +the most classes. +Region +Classes with largest percent increase in AP +Africa +waste cont., spices, dustbin, clean. equip., hand soap +Americas +dustbin, spices, clean. equip., medicine, waste cont. +E. Asia +relig. blg., spices, dustbin, waste cont., clean. equip. +SE. Asia +waste cont., spices, medicine, clean. equip., dustbin +W. Asia +dustbin, hand soap, clean. equip., spices, jug +Table 8. We highlight the classes with the largest increases in the +average precision when adding in training images from GeoDE. +Classes like cleaning equipment, spices and dustbin and waste +container are among the classes most improved across all regions. +on “spices”, “waste container” and “cleaning equipment” +see large improvements in AP across all regions. +8. Conclusions +We have introduced a new dataset, GeoDE , which uses +crowd-sourcing for image collection, a significant depar- +ture from the popular computer vision dataset collection +paradigm of web-scraping for image collection. +Using +crowd-sourcing for image collection allows us to ensure +that our dataset does not contain personally identifiable in- +formation, that we own the rights to the images, that the +image creators were compensated for their work, and we +are able to control for geographic diversity and object dis- +tribution in the dataset. In addition, we showed that this +collection method results in significantly different images +and image features than standard web-scraped dataset, even +8 + +60 +40 +20 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +5040 +20 +0 +-20 +-40 +-60 +-40 +-20 +0 +20 +40 +6040 +20 +-20 +-40 +-40 +-20 +0 +20 +40W. Asia +East Asia () +Europe +Africa () +(+) +(★) Americas +大 +(x) +SE Asiawhen controlling for the types of objects. We also show that +even small amounts of geodiverse data are useful for eval- +uating and highlighting shortcomings in common models +(e.g. CLIP) and can improve performance when added to +the training dataset. The GeoDE dataset shows that crowd- +sourcing is a viable image collection technique for creating +diverse and responsible image datasets. Our work also sug- +gests avenues for scaling crowd-sourcing through the use +of targeting, i.e. focusing on collecting the most valuable +images for improving model training. +Acknowledgements. This material is based upon work +partially supported by the National Science Foundation un- +der Grant No. 2145198. Any opinions, findings, and con- +clusions or recommendations expressed in this material are +those of the author(s) and do not necessarily reflect the +views of the National Science Foundation. +We also ac- +knowledge support from Meta AI and the Princeton SEAS +Howard B. Wentz, Jr. Junior Faculty Award to OR. 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In +EMNLP, 2017. 1, 2 +Appendix +Here, we provide some more details about our experi- +ments. +• In Sec A, we describe our heuristic to select object cat- +egories in more detail. +• In Sec B, we compare the GeoDE feature space to that +of ImageNet [8] +• In Sec. C, we provide results when finetuning pre- +trained models rather than just training the final layer +of a ResNet. +• In Sec. D, we give more details about GeoDE , in- +cluding the counts of images of different regions and +categories, as well as more sample images from this +dataset. +A. Selecting object categories +In this section, we provide more details about the ob- +ject selection heuristic we employed. We used 2 different +datasets that were collected to be geodiverse: GeoYFCC [9] +and DollarStreet [22]. +Implementation details. Features for GeoYFCC were +extracted using a ResNet50 [15] pretrained on Ima- +geNet [8]. We used Logistic regression, Linear SVM and +KMeans clustering implementations from the sklearn li- +brary [20]. We used continents as regions. GeoYFCC [9] +contains over 1200 tags, we ignored all tags with counts in +the bottom 20th percentile, giving us a total of 745 tags. +First, we apply each of these methods to GeoYFCC to +identify candidate tags. +• For each region R, we train a linear model using a fea- +ture extractor and images from all regions except R and a +linear model trained on all images from all regions, to pre- +dict the presence or absence of each tag. We then applied +both models to images from R. The difference in perfor- +mance between these models allows us to measure the dif- +ference in appearance of the tag. We selected tags where +in the weighted average precision on the region was less +than 0.8* the performance on other regions. This gave us +a set of 277 tags such as “footstool”, “chili”, “case”, etc. +• For each tag T, we train a linear SVM to predict the re- +gion given the features of images containing tag T. If this +model has high accuracy, this suggests that this tag is vi- +sually different across regions. We selected tags that had +an accuracy of over 50%. 223 tags were identified in this +10 + +manner. “Cork”, “bowler hat” and “mountain bike” are +examples of tags found in this way. +• We clustered features of images containing tag T. We +then computed the Gini impurity of each world region, +and selected tags that had a median Gini value of at least +0.5. This gave us 75 tags in total. Examples of tags found +in this way were “chili”, “footstool” and “stove”. +After identifying these tags, we first pruned them by re- +moving tags that did not appear to correspond to an object. +Examples of this include “arctic”, “descent”, etc. Second, +we removed tags corresponding to wild animals, since these +would not be found in all regions. Examples of tags re- +moved in this manner were “gnu”, “camel”, etc. This gave +us a list of 265 tags. Third, these tags were sometimes vari- +ants of objects, for example, we had tags like “stupa”, “tem- +ple”, “church”, “mosque” and “chapel”. Thus we grouped +tags based on meaning. Other examples included “break- +fast”, “dinner” and “dessert” as “plate of food”, “stool”, +“footstool” and “bench” as “chair”, etc. +This gave us a +list of objects we could collect, e.g. “religious buildings”, +“plate of food”, “toy”. +B. Comparison with ImageNet +We note that the comparison to GeoYFCC [9] in Sec. 5 +in the main text required us to use tags which are noisy. +Here, we compare GeoDE to ImageNet, checking how +much the feature spaces differ. +We find a subset of ImageNet21k as outlined in Sec. 6, +and extract features using a PASS [2] trained ResNet50 [15] +model. Other implementation details remain the same as in +Sec 5 in the main paper. +We first use a Logistic regression model to predict the +dataset that the features are taken from and this has an ac- +curacy of 96.0%, showing that the feature space is very dif- +ferent. We also visualize TSNE plots of different objects in +figure 12. +C. More results when training with GeoDE +In this section, we provide results for the incremental +training with GeoDE for different regions that were not +presented in Sec 7, and provide results when fine-tuning a +ResNet50 model, rather than freezing the layer weights. +C.1. Results from incrementally adding additional +regions +We again visualize the improvement in the accuracy as +we incrementally add in images from West Asia, East Asia, +Americas and Europe. We can see that the performance +both within the specific region and in Europe (compared to +Americas when considering Europe) increase with the ad- +ditional GeoDE data. Similar to before, we see that the +− 40 +− 20 +0 +20 +40 +− 60 +− 40 +− 20 +0 +20 +40 +60 +Boat +ImageNet +GeoDE +− 60 +− 40 +− 20 +0 +20 +40 +60 +− 60 +− 40 +− 20 +0 +20 +40 +60 +80 +Car +ImageNet +GeoDE +− 60 +− 40 +− 20 +0 +20 +40 +60 +− 80 +− 60 +− 40 +− 20 +0 +20 +40 +60 +Hand soap +ImageNet +GeoDE +− 60 +− 40 +− 20 +0 +20 +40 +60 +80 +− 80 +− 60 +− 40 +− 20 +0 +20 +40 +60 +80 +Spices +ImageNet +GeoDE +Figure 12. +We visualize the TSNE plots for several of object +classes using ImageNet and GeoDE . While the features do overlap +slightly, on the whole, they are very different for dataset distribu- +tions, even within each category. +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on W. Asia, + tested on W. Asia +Trained on W. Asia, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on E. Asia, + tested on E. Asia +Trained on E. Asia, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on Americas, + tested on Americas +Trained on Americas, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on Europe, + tested on Europe +Trained on Europe, + tested on Americas +Figure 13. We visualize the increase in accuracy as images are in- +crementally added in from a region. Similar to Fig. 10 in the main +text, we see that adding in GeoDE images increases performance +more in the region than a control. +increase within the region is larger than that of the con- +trol, showing that these images are from different domains. +(Fig. 13) +C.2. Results from finetuning a ResNet50 model +Implementation details. We use a ResNet50 [15] model +pretrained on Imagenet and fine tune the weights using dif- +ferent fractions of the ImageNet and GeoDE datasets as +mentioned in Sec. 7 in the main paper. We train the model +with an SGD optimizer, learning rate = 0.1, and momen- +tum=0.9. Other implementation details remain the same as +before. +Results. While the overall trend of the results are the +same, we see that these results are slightly noisier, poten- +tially because the model overfits to the small training set +(Fig. 14). +11 + +Leave one out training +curler, fan, footstool, chili, coconut, toilet, canoe, motorboat, mountain-bike, stupa, villa, backpack, baseball- +glove, basin, basket, bat, bathtub, battery, beer-mug, belt, blade, bowl, bowl, broom, bucket, carryall, case, cash- +machine, cassette, cleaver, cologne, cooler, counter, dinner-dress, dinner-jacket, dish, gown, grille, hammer, jacket, +kettle, microphone, parka, porch, rack, remote-control, sandal, scale, shelf, shot-glass, stereo, stocking, stool, +sweater, tape, teddy, timer, tripod, trouser, turntable, wardrobe, weight, wok, woodcarving, hot-pot, chewing-gum, +cucumber, lime, fig, pineapple, jackfruit, kiwi, mango, basil, garlic, sage, lager, ale, porter, stout, champagne, rum, +tequila, vodka, whiskey, mocha +Linear SVM for region +mountain-bike, bicycle, raft, ferry, ship, kayak, streetcar, bus, impala, car, footstool, bench, chair, mushroom, +breakfast, vegetable, dessert, dinner, door, bowler-hat, house, building, chandelier, lamp, light, castle, acropolis, +fortress, tower, palace, dome, architecture, memorial, statue, sculpture, gravestone, arch, temple, stupa, monastery, +church, cathedral, chapel, mosque, signboard, grocery-store, shop, kitchen, lantern, doll, coati, cork, primate, alp, +shore, curler, cologne, seashore, gnu, hog, giraffe, arctic, ice-rink, ski, elephant, guinness, makeup, circuit, geyser, +skyscraper, hippopotamus, basketball, paintball, sword, hijab, fortification, craft, clock, stage, tractor, dagger, +defile, bikini, swing, windmill, motor, brick, snowboard, course, volleyball, display, opera, railing, playground, +veranda, wind-instrument, city-hall, ruin, portfolio, newspaper, airbus, bridge, airfield, global-positioning-system, +brake-drum, kid, mangrove, motor-scooter, crane, intersection, plain, column, wardrobe, interface, guitar, cos- +tume, grand-piano, aircraft, factory, seaside, ball, sweet, gravy-boat, spotlight, american-bison, sail, beer, pier, +road, tulip, grass, miniskirt, willow, flood, street, roof, slide, cliff, track, train, vehicle, boot, world, patio, window, +rainbow, beacon, sidewalk, organ-pipe, tank, cable-car, grey, hall, map, cattle, airport, school, mountain, promon- +tory, monkey, motorcycle, bubble, black, mirror, golf-club, skateboard, computer, university, denim, sky, rock, +earphone, descent, garden, hill, library, tea, blush-wine, radio, bill, sunglass, ballpark, apparel, web, field-glass, +reef, fountain, downhill, pen, cable, step, graffito, conveyance, fabric, hovel, umbrella, iron, cloud, strand, toilet, +walker, valley, airplane, cup, base, wire, camel, pizza, bathroom, lounge, dock, van, circuit-board, bell, sheep, +book, fish, canyon, fire, array, rangefinder, coca-cola +Clustering +acropolis, cork, coati, footstool, stupa, impala, chili, primate, cologne, gnu, guinness, alp, hog, shore, boater, +walker, plain, hippopotamus, raft, chandelier, curler, giraffe, arctic, bowler-hat, castle, geyser, boot, streetcar, +rum, hijab, ski, temple, windmill, dagger, fortification, snowboard, coffee, ice-rink, display, cathedral, bench, +bikini, lantern, slope, elephant, strand, sword, paintball, gravestone, tulip, golf-club, downhill, swing, volleyball, +mushroom, monastery, american-bison, stage, cup, church, wardrobe, wind-instrument, skyscraper, sweet, course, +tower, opera, sketch, circuit, chapel, col, motor, clock, railing, mangrove +Table 9. Prospective tags identified from GeoYFCC [9]. Tags in red seemed hard to picture. Tags in blue are of animals that might be hard +to crowdsource +. +D. More details about the dataset +In this supplementary section, we provide counts of the +objects per region in GeoDE as well as more examples of +images from this dataset. . +As mentioned before, GeoDE is mostly balanced across +both region and object: for most part, we were able to get +atleast 150 images per region per object, with a few excep- +tions (“wheelbarrow” in 2 regions; “monument”, “boat” and +“flag” in 1 region). See Tab. 10 for full counts. +We also provide more examples of the images from +GeoDE in the Figures Figs. 15 to 28. +12 + +W.Asia +Africa +E.Asia +SE.Asia +Americas +Europe +backyard +213 +670 +191 +218 +216 +232 +bag +267 +388 +379 +593 +298 +437 +bicycle +234 +252 +303 +235 +228 +244 +boat +159 +227 +174 +237 +84 +225 +bus +203 +234 +228 +214 +217 +227 +candle +232 +243 +219 +239 +188 +272 +car +242 +323 +284 +235 +273 +363 +chair +279 +353 +338 +512 +344 +349 +clean. equip. +257 +275 +316 +305 +270 +363 +cooking pot +216 +261 +237 +202 +213 +304 +dog +216 +188 +191 +244 +206 +196 +dustbin +218 +417 +267 +203 +271 +301 +fence +258 +315 +251 +302 +226 +283 +flag +206 +258 +148 +223 +209 +272 +front door +210 +248 +222 +224 +200 +235 +hairbrush +269 +250 +312 +300 +290 +431 +hand soap +222 +204 +281 +191 +245 +362 +hat +209 +294 +340 +316 +294 +336 +house +198 +434 +212 +192 +277 +197 +jug +217 +202 +195 +249 +236 +194 +light fixture +231 +337 +251 +209 +191 +307 +lightswitch +215 +237 +249 +273 +273 +234 +lighter +219 +309 +225 +237 +217 +273 +medicine +240 +275 +321 +330 +328 +302 +monument +161 +189 +188 +183 +254 +245 +plate of food +206 +479 +294 +364 +241 +310 +relig. blg. +221 +223 +211 +226 +197 +230 +road sign +224 +407 +264 +270 +235 +289 +spices +243 +242 +339 +216 +290 +300 +stall +139 +216 +201 +227 +197 +226 +storefront +209 +309 +191 +240 +246 +204 +stove +199 +537 +201 +263 +206 +287 +streetlight +202 +343 +214 +196 +208 +227 +toothbrush +264 +252 +336 +361 +337 +270 +toothpaste +209 +286 +271 +230 +245 +315 +toy +224 +220 +281 +292 +323 +287 +tree +223 +296 +257 +357 +300 +331 +truck +205 +239 +214 +231 +212 +225 +waste cont. +225 +210 +212 +213 +214 +259 +wheelbarrow +122 +256 +141 +197 +152 +243 +Table 10. We show the counts of objects per region in GeoDE. Bolded are the ones categories for which we were not able to get 150 images +per region. +13 + +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on West Asia, + tested on West Asia +Trained on West Asia, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on Africa, + tested on Africa +Trained on Africa, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on East Asia, + tested on East Asia +Trained on East Asia, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on SE Asia, + tested on SE Asia +Trained on SE Asia, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on Americas, + tested on Americas +Trained on Americas, + tested on Europe +0 +50 +100 +% of images used from GeoDE +65 +75 +85 +Trained on Europe, + tested on Europe +Trained on Europe, + tested on Americas +Figure 14. We visualize the increase in accuracy as images are +incrementally added in from a region when finetuning a ResNet50 +model. Similar to Fig. 10 in the main text, we see that adding in +GeoDE images increases performance more in the region than a +control. +14 + +Backyard +West Asia +Africa +East Asia +Figure 15. Randomly chosen images for “backyard” for 3 regions. We notice that some of these are backyards made of concrete (West +Asia: r2c1, r3c4, etc., Africa: r3c1 ,r4c9, r5c3, etc.,) or do not contain lawns (West Asia: r3c1, r2c5, etc., Africa:r5c1-5, etc., East Asia: +r2c4, r5c1, r5c4-6, etc.) +15 + +Backyard +Southeast Asia +Americas +Europe +Figure 16. Randomly chosen images for “backyard” for the 3 other regions. Again, we see that regions tend to have backyards made of +concrete or paved (Southeast Asia: r1c8, r2c5, r3c3 as examples, Americas: r1c9, r1c10, r2c10, etc., Europe: r3c2, r4c5, etc ), or do not +contain lawns (Southeast Asia: r1c1, r1c9, etc., Americas:r3c2, r3c3, etc., Europe: r3c2, r5c9, etc.) +16 + +Bicycle +West Asia +Africa +East Asia +Figure 17. Randomly chosen images for “bicycle” for 3 regions. While most images are of standard bicycles, we notice a couple of +interesting images: tricycle (West Asia: r4c9), rickshaws (Africa: r2c8, r2c10, r4c8, r4c10), and motorized cycles (West Asia: r1c8). There +are also a lot of children’s bicycles. +17 + +Bicycle +Southeast Asia +Americas +Europe +Figure 18. Randomly chosen images for “bicycle” for the 3 other regions. We see more motorized cycles (Southeast Asia: r5c6) as well as +several children’s bicycles. +18 + +8Boat +West Asia +Africa +East Asia +Figure 19. Randomly chosen images for “boat” for 3 regions. We see a variety of boats including larger ships in West Asia (r1c1, r1c2, +r1c5, r4c1,r5c1), smaller kayaks and canoes in Africa (r1c8-9, r2c1-4, r4c2 , etc ), and a mix in East Asia. +19 + +Boat +Southeast Asia +Americas +Europe +Figure 20. Randomly chosen images for “boat” for the 3 other regions. We again see a variety of boats ranging from motor boats in Europe +and the Americas to smaller boats in Southeast Asia. +20 + +Cleaning equipment +West Asia +Africa +East Asia +Figure 21. Randomly chosen images for “cleaning equipment” for 3 regions. This appears to be a diverse category within all regions +containing images of mops, buckets, products, brooms, etc. +21 + +Cleaning equipment +Southeast Asia +Americas +Europe +Figure 22. Randomly chosen images for “cleaning equipment” for the 3 other regions. This appears to be a diverse category within all +regions containing images of mops, buckets, products, brooms, etc. +22 + +PFSpices +West Asia +Africa +East Asia +Figure 23. Randomly chosen images for “spices” for 3 regions. We see a wide range of containers, ranging from packets (mostly in Africa), +glass jars (in West Asia) to some bottles (all regions). +23 + +05CoalingHETACO,Spices +Southeast Asia +Americas +Europe +Figure 24. Randomly chosen images for “spices” for 3 regions. We see a wide range of containers, ranging from packets (some in Southeast +Asia and Americas) to bottles (some in Southeast Asia) +24 + +DBONUS,DIZLDARYLiaStove +West Asia +Africa +East Asia +Figure 25. Randomly chosen images for “stove” for 3 regions. We see that Africa and East Asia contain one-burner and two burner stoves +(along with 4 burner stoves). We also see a variety of stoves in terms of induction, gas, ovens, etc. +25 + +Stove +Southeast Asia +Americas +Europe +Figure 26. Randomly chosen images for “stove” for 3 regions. We see that Southeast Asia contains one-burner and two burner stoves +(along with 4 burner stoves). We also see a variety of stoves in terms of induction, coils, gas, ovens, etc. +26 + +OWaste container +West Asia +Africa +East Asia +Figure 27. Randomly chosen images for “waste container” for 3 regions. We see that different regions have containers of varying sizes +(Africa seems to be smaller than West Asia or East Asia), and have different closing mechanisms (see West Asia r5c6 as an interesting +example.) East Asia also tends to have segregated waste containers. +27 + +4Waste containers +Southeast Asia +Americas +Europe +Figure 28. Randomly chosen images for “waste container” for 3 regions. We see that different regions have containers of varying sizes +(Europe seems to have containers of very different sizes) and have different closing mechanisms (see Southeast Asia r2c6 as an interesting +example.) +28 + +PAPER2238 \ No newline at end of file diff --git a/qtE0T4oBgHgl3EQfrAFn/content/tmp_files/load_file.txt b/qtE0T4oBgHgl3EQfrAFn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..438f26d6d1dd77d421bef8b902f8e67ceb226c3f --- /dev/null +++ b/qtE0T4oBgHgl3EQfrAFn/content/tmp_files/load_file.txt @@ -0,0 +1,1640 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf,len=1639 +page_content='Beyond web-scraping: Crowd-sourcing a geographically diverse image dataset Vikram V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Ramaswamy1, Sing Yu Lin1, Dora Zhao2*, Aaron B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Adcock3, Laurens van der Maaten3, Deepti Ghadiyaram, Olga Russakovsky1 1Princeton University 2Sony AI 3Meta AI Work done as a graduate student at Princeton University Abstract Current dataset collection methods typically scrape large amounts of data from the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While this technique is extremely scalable, data collected in this way tends to re- inforce stereotypical biases, can contain personally identifi- able information, and typically originates from Europe and North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In this work, we rethink the dataset col- lection paradigm and introduce GeoDE , a geographically diverse dataset with 61,940 images from 40 classes and 6 world regions, and no personally identifiable informa- tion, collected through crowd-sourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We analyse GeoDE to understand differences in images collected in this man- ner compared to web-scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Despite the smaller size of this dataset, we demonstrate its use as both an evalu- ation and training dataset, highlight shortcomings in cur- rent models, as well as show improved performances when even small amounts of GeoDE (1000 - 2000 images per re- gion) are added to a training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We release the full dataset and code at https://geodiverse-data- collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='edu/ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Introduction The creation of large-scale image datasets has enabled advances in the performance of computer vision models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Although previously limited by manual collection and an- notation efforts [11, 12, 14], recently the size of these datasets has rapidly grown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This growth has been empow- ered by a new data collection framework: scraping web im- ages at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' These images are either human-labelled (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', ImageNet [8,23]), use tags (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', CLIP-400M [21]) or used for self-supervised learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', PASS [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' However, these web-scraped datasets come with their downsides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' First, the datasets can contain pernicious gen- der and racial biases by underrepresenting certain demo- graphic groups and using stereotypical depictions of these groups [4, 29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Second, significant geographic bias is present in these datasets with most images coming from GeoYFCC distribution [9] 0 10 20 30 40 Percentage of Images from Region GeoDE distribution (ours) 0 5 10 15 Percentage of Images from Region Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We construct a geographically diverse dataset GeoDE that is approximately balanced across 6 world regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We visual- ize the images per region, and compare our distribution (bottom) to that of a previously created diverse dataset GeoYFCC [9] (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' North America or Western Europe [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' As de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' [7] show, this lack of geo-diversity propagates into downstream recognition tasks–resulting in difficulty in rec- ognizing common household objects from non-Western re- gions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Third, copyright and consent may pose challenges for web-scraped data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Dataset creators sometimes do not obtain full permission of the content creators and of the people featured in the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1 Finally, while annotators are compensated for their time, content creators and image subjects are rarely compensated for their contributions to the dataset [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Though there have been efforts to balance datasets [9], clean datasets [31], and protect privacy of indi- viduals by blurring [34], methods that rely on web-scraping 1While images used are sometimes under the most permissive Creative Commons license, it is unclear if creators know the full impacts of their images being used in the training of large scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='02560v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='CV] 5 Jan 2023 cannot fully eliminate the aforementioned issues [3,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Rather than improving web-scraped datasets, we rethink the entire paradigm: we explore a data collection approach where we do not scrape images, but rather collect images via crowd-sourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We commission photos of different ob- jects from people across the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This approach naturally resolves copyright concerns, and enables much tighter con- trol of image distribution which may reduce biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We partnered with a company called Appen (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='appen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' com) and crowd-sourced the Geographically Diverse Eval- uation (GeoDE) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' GeoDE contains 61,940 images roughly balanced across 40 object categories and six ge- ographic regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Despite the small size of GeoDE, the dataset has several key advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' First, the object recognition problem becomes surpris- ingly challenging since the images represent the diverse appearance of common objects across six global regions: Africa, the Americas, East Asia, Europe, Southeast Asia, and West Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similar to de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' [7], we show that modern object recognition models perform poorly on rec- ognizing objects from Africa, East Asia, and SE Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Aug- menting current training datasets (like ImageNet [8, 23]) with images from GeoDE yields an improvement of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0% on DollarStreet [22] and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9% on a test split of GeoDE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Second, requesting images containing specific object classes removes selection bias: objects present in images that are web-scraped are uploaded by creators with different incentives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' to make exciting/unique content or to gen- erate engagement [25], and disincentives showing mundane everyday content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We show that the distribution of images in GeoDE is different to that in other datasets, even when controlling for world region and object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Third, we own the copyright to all of the images in GeoDE , have explicit permission from content creators to use these images for machine learning applications, and ensured fair compensation to the content creators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We acknowledge that a main drawback of this method is the cost: this approach does not scale as well as scraping images from the web and is partially the result of ensuring fair compensation for content creators and curators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' How- ever, we demonstrate that even small amounts of data col- lected in this way can be beneficial in partially remedying some of the concerns with large-scale web-scraped datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Data and code can be found at https://geodiverse- data-collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='edu/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Related Work There are three key research directions that inspired this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The first is the call to increase geographic diversity in visual datasets [7,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In response, there have been attempts to construct geographically diverse datasets [1,9,22], sum- marized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' However, these datasets are still geo- graphically concentrated (in Europe for GeoYFCC [9], In- dia for OpenImages Extended [1]), and/or are still relatively small scale (DollarStreet [22]), prompting our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Second, in using crowd-sourcing to generate visual con- tent rather than using web-scraped images, we follow re- cent video datasets Charades [25], Epic Kitchens [6] and Ego4d [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' However, we differ in that our key goal is to ensure a geographically balanced dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This poses chal- lenges in recruitment and dataset scope (more in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Finally, in our data collection efforts we take into ac- count the extensive literature around selection bias in com- puter vision datasets [3, 4, 10, 26, 27, 29, 32, 35] and ensure that our dataset is collected responsibly, with attention to privacy, consent, copyright and worker compensation [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Collecting GeoDE We describe our data collection process, including our selection of object classes and world regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Selecting the object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We focus on object classes that are likely to be visually distinct in different parts of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Selecting such objects is a chicken-and-egg prob- lem: without a geographically diverse dataset at our dis- posal, it is unclear which objects are diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We adopt a number of heuristics using existing datasets to find a plau- sible set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The full process is detailed in the appendix, but briefly, we use simple computer vision techniques (linear models and visual clustering, using features extracted from PASS-pretrained models [2]) along with manual examina- tion to identify a set of candidate tags from DollarStreet [22] and GeoYFCC [9] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', “chili,” “footstool,” “stove”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' To prune these tags, we remove those that are not objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', “arctic”, “descent”), removing wild animals not found in all regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', “gnu”, “camel”) and group variants of objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', “stupa”, “temple”, “church”, “mosque” and “chapel”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The final set of objects is in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Selecting diverse geographic regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We chose six re- gions across the world: Africa, Central and South Amer- ica (“Americas”), East Asia, Europe, Southeast Asia and West Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Within each region, we targeted 3-4 countries (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' These were chosen due to the lack of available im- ages from these regions in most public datasets [7, 24, 30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' the countries were chosen based on the presence of partic- ipants within Appen’s database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We obtain a roughly even distribution of images across each class and region pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Image collection and worker demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Work- ers were asked to upload images for a given object class, following the instructions in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' There were more than 4,500 workers, representing a range of genders, ages and races (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' All images submitted were manually checked by Appen’s quality assurance (QA) team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2 Dataset Size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' distribution Collection process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' annotation process Geographic coverage Personally Identifi- able Info (PII) ImageNet [8,23] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='2M images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' mostly balanced across classes Scraped images from the web based on the class label;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' crowd- sourced annotations Mostly North America and Western Europe [24] Contains people, some images have faces blurred [33] OpenImages [18] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1M images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' long-tailed class distribution Flickr images with CC-BY li- censes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' automatic labels with some human verification Mostly North America and Western Europe [24] Contains people OpenImages Extended [1] 478K images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' long-tailed class distribution Crowd-sourced gamified app to collect images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' automatic labels and manual descriptions More than 80% of im- ages are from India People are blurred GeoYFCC [9] 330K images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' long-tailed class distribution Flickr images subsampled to be geodiverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' noisy tags Geographically diverse (62 countries), but concentrated in Europe Contains people PASS [2] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='4M images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' N/A (no labels) Random images from Flickr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' no annotations Collected from Flickr, thus mostly North Amer- ica and Western Europe No people DollarStreet [22] 38,479 images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' mostly balanced across topics Images by professional and vol- unteer photographers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' manual la- bels including household income 63 countries in four re- gions (Africa, America, Asia and Europe) Yes, with permission GeoDE (ours) 61,940 images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' balanced across classes®ions Crowd-sourced collection using paid workers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' manual annotation Even distribution over six geographical regions (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3) No identifiable peo- ple and no other PII Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We compare recent approaches to dataset collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Although GeoDE is smaller than recent datasets, we ensure that the images are sourced with permission of the creator, contain no identifiable people, and are balanced across both regions and object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Indoor common bag, chair, dustbin, hairbrush/comb, hand soap, hat, light fixture, light switch, toothbrush, toothpaste/toothpowder Indoor rare candle, cleaning equipment, cooking pot, jug, lighter, medicine, plate of food, spices, stove, toy Outdoor common backyard, car, fence, front door, house, road sign, streetlight/lantern, tree, truck, waste container Outdoor rare bicycle, boat, bus, dog, flag, monument, religious building, stall, storefront, wheelbarrow Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' GeoDE consists of 40 object classes, loosely organized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' West Asia Saudi Arabia, United Arab Emirates, Turkey Africa Egypt, Nigeria, South Africa East Asia China, Japan, South Korea SE Asia Indonesia, Philippines, Thailand Americas Argentina, Colombia, Mexico Europe Italy, Romania, Spain, United Kingdom Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' GeoDE consists of images from six world regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Within each region, there are 3-4 countries contributing to most of the im- ages, chosen to balance the diversity of the images against practi- cal data collection considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Participants from outside these countries but within the same region were still accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' General Instructions In this task, you will submit up to 3 photos of the same type of object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', upload 3 photos of 3 different bags;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' please do not upload 3 photos of the same bag from different angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Please make sure the location function is enabled for the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The photo resolution should be at least 640 x 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' All images should be new photos captured with Appen Mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Please make sure it’s a single object per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Please make sure it’s a well-lit environment and the object is clearly visible in the photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Please make the object occupy at least 25% of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Objects captured are foregrounded and not occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Objects should not be blurred, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', motion blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' No effects or filters added (cropping is acceptable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Please try to avoid capturing people in the images (it’s OK if people are blurry in the background and far from the camera).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Please try to avoid capturing vehicle license plates in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Image collection instructions given to GeoDE workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Lessons learned from collecting GeoDE A key part of this study was to understand if crowd- sourcing images is a viable alternative to web-scraping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In this section we detail the challenges faced and the lessons learned in constructing the GeoDE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Getting sufficient images of all object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While some object classes expectedly proved more difficult than 3 Male Female Gender White African East Asian Latino or Hispanic South Asian Middle Eastern Other Race < 20 20-29 30-39 40-49 > 50 Age Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Demographics of workers contributing GeoDE images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Some issues we encountered within GeoDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' (Left) Three images submitted for “hat” by the same worker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' these are near- identical except for the color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' (Right) Especially for outdoor im- ages, the target class may be ambiguous: two of these images were submitted for “fence”, and one for “tree” others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', instances of “monument” or “flag” were simply hard for workers to find), others surprised us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For example, “stove” was originally underrepresented until the definition of “stove” was clarified to “any cooking surface either elec- tric, gas, induction.” Workers did not always consider their cooking appliance to be a “stove,” highlighting a vocabulary challenge unique to geographically diverse data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Multiple copies of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Two most common cate- gories of error were incorrect images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='e, having an class other than the one selected) and multiple copies of an im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The QA team found that participants often submitted multiple copies of the same object instance from different angles despite clear instructions not to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Workers also sometimes submitted very similar objects (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4 (left), where there are three hats submitted by the same worker with slightly varying colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We filtered out such images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Multiple objects per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For some of the (especially larger and outdoor) object categories, it was difficult to en- sure that additional objects were not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For example, we found that images of “fences” often have “trees” present, and it was not always possible to discern between objects in the foreground and background (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4 (right)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We found that this was primarily an issue for the “tree” class, and thus requested that images that had a significant portion of the image covered by trees are tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' These additional annotations can be used to filter and remove such images and/or to analyze errors made by a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Beyond these, the rest of the data collection went smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Following instructions, only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='78% images con- tained identifiable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Some images contain non- identifiable incidental people in the background (especially for larger object classes, like “monument”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' All such im- ages are tagged in GeoDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We were also able to ensure that the number of images per region is roughly equal, although unfortunately it was harder to obtain an even number of im- ages per country within each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Collecting images in this way is expensive: each image costed $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='08 (not including researcher time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This allowed us to fairly compensate workers for their labour as well as the QA team to ensure the quality of these images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Comparing GeoDE to current datasets We compare GeoDE with two datasets: the canonical ImageNet [8] and the geographically diverse GeoYFCC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Qualitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5, we show a subset of 60 GeoDE images of “stoves” and “houses” (more in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Compared to images from ImageNet, we see a larger variety of stoves: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', induction coils, single and two burner stoves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The stoves also appear more used than those in ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similarly, for “house,” we see a larger range in terms of materials used and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In the Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6 and 7 we examine the impact of this diversity on visual recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We compare with GeoYFCC, a dataset sam- pled from YFCC100m to be more geo-diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6 (left) shows that most images in GeoYFCC come from Europe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', images from Africa and West Asia comprise less than 10% of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also consider tags in GeoYFCC that correspond to classes within GeoDE, and show the per-class distribution of images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see selection bias in the objects people choose to upload, with “monu- ments” accounting for more than 25% of these GeoYFCC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In contrast, GeoDE is approximately balanced across both regions and objects (details in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Object appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Finally, we take a stab at quanti- fying the differences in the appearance of images collected through crowd-sourcing and web-scraping, by comparing the images in GeoDE with those in GeoYFCC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We ex- tract features for each dataset using a ResNet50 model [16] trained with self-supervised learning SwAV [5] on the PASS dataset [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We first train a linear classifier to predict which dataset an image was drawn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The classifier achieves an accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' To understand how the dataset distri- butions are differ beyond just the class/region frequencies we obtain low-dimensional TSNE embeddings [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The TSNE plots are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Even within an object class and region, the image features are very different, suggesting that different image collection methods lead to different distri- bution of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' GeoDE as an evaluation dataset We now analyze the use of GeoDE as an evaluation dataset, by using it to evaluate two canonical models: the recent CLIP [21] and an ImageNet [8]-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For the CLIP model, we use the weights provided for the ViT-B/32 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We use text 2We note that GeoYFCC just has tags, not labels, hence, these might be noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also visualize these plots for Imagenet in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Sample images of two object classes in different regions within GeoDE (and ImageNet in the bottom row, for comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Product labels on images have been blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' West Asia Africa East Asia SE Asia Americas Europe 0 20 40 % of images GeoYFCC GeoDE Uniform Indoor common Indoor rare Outdoor common Outdoor rare Dataset bag chair dustbin hat light fixture clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' cooking pot lighter plate of food spices stove toy car fence front door house streetlight tree truck waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' bicycle boat bus flag monument religious bld stall storefront wheelbarrow GeoYFCC 1 1 0 3 2 0 0 0 10 1 0 2 2 1 2 11 0 2 1 0 5 4 2 1 29 16 0 4 0 GeoDE 5 5 4 4 3 4 3 3 4 4 4 4 4 4 3 3 3 4 3 3 3 2 3 3 3 3 3 3 2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Distribution (in %) of object counts, across the 6 regions (left) and for the 30 object classes (right) that are present in both GeoYFCC [9] and GeoDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' GeoYFCC has a long tail in both cases, with images from Europe comprising over 40% of the dataset and with monuments, religious buildings, houses and plates of food comprising more than 66% of the images, while objects like dustbin appear in less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5% of these images (denoted as 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In contrast, our GeoDE dataset is balanced across these regions and object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' prompts for all 40 object categories as described in the zero- shot recognition setup of [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' To train a model on Ima- geNet [8], we first match the classes of GeoDE and Ima- geNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We find the corresponding synset for each GeoDE class in WordNet [19], and include all images of that synset and its children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For two object categories (“backyard” and “toothpaste/toothpowder”) we do not find any matching cat- egories, and so we ignore these categories in the quantita- tive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We split our filtered ImageNet [8] dataset into train (38,353 images), validation (12,794 images), and test (12,795 images) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' As in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5 we extract features using a ResNet50 model [16] trained with self-supervised learning SwAV [5] on PASS [2], and retrain the final layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4 shows the accuracy across different re- gions on these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Compared to CLIP, the over- all performance on GeoDE by an ImageNet trained model is considerably lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' both models perform the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='best on images from Europe and the worst on images from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Stove ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='House ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='West Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Africa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='East Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Southeast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Americas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Europe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='ImageNetWest Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Africa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='East Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Southeast Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Americas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Europe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Plate of food ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='20 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='GeoYFCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='GeoDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='GeoYFCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='GeoDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='GeoYFCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='GeoDE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We visualize the TSNE plots for several of object classes per region for GeoYFCC and GeoDE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While the features do overlap slightly, on the whole, they are very different for dataset distributions, even within each (region, object) tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Model WAsia Africa EAsia SEAsia Americas Europe ImageNet 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9 CLIP 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='8 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Accuracies (in %) on GeoDE of a model trained on a subset of Imagenet [8] and of CLIP [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The models perform best on images from Europe, and worst on images from Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Africa (difference of more than 7% in both cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5 breaks out the per-object accuracy for CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While the average accuracy is 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='8%, classes like “dust- bin” (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3%), “medicine” (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1%), “cleaning equipment” (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0%), “spices” (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='2%) and “house” (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3%) perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 8 shows some example errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' When investigating the different classes with the CLIP model, we find geographic disparities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Classes like “fence”, “stove” and “spices” have different accuracies in different regions: “fence” is over 88% for images from Europe, but only 60% and 59% for images from Africa and Southeast Asia respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similarly, “stove” has accuracy of 95% in the Americas but only 67% in East Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We visualize this using the TSNE plots of the features for these classes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We find that these classes have objects that are re- gion specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For example, “religious buildings” from East and Southeast Asia can include buildings like monasteries and temples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similarly, single- and two-burner “stoves” are primarily from countries in Africa and Southeast Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Impact of training with GeoDE Finally, we attempt to answer how training with GeoDE data can improve the performance of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We in- vestigate training a model where we combine images from GeoDE with images from ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We find the combina- tion of the two can improve results across regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Training a model with GeoDE We would like to understand how training a model with data from GeoDE affects the object recognition capabilities of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Fixing the total number of images used for training, we train a linear model, using a pre-trained feature extractor, on a dataset comprised entirely of ImageNet im- ages and a dataset comprised of both ImageNet and GeoDE images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The feature extractor is a ResNet50 [15] model trained on PASS [2] using SwAV [5] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We split the GeoDE dataset into a train (4,970 images per region), validation (between 1657 and 2188 images per region), and test (between 1657 and 2189 images per region) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We use the validation dataset to select training hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We consider the training set for our ImageNet only model as the same 38,353 image training set constructed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6, only considering tags in ImageNet that correspond to classes within GeoDE,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' To construct the training set of our ImageNet and all regions in GeoDE model, we start with the training set for ImageNet and add in the training sets for all 6 regions in GeoDE while removing proportionately per class the same number of im- ages from ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This procedure gives a training set of 29,820 GeoDE images and 8,533 ImageNet [8] images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The final models are trained using an SGD optimizer, with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1, and momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9, for 500 epochs with cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We use models with the highest accuracy on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Unless specified otherwise, results are reported on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We first report results on the GeoDE test set, and notice a significant improvement in accuracy across all regions, as a result of training with both GeoDE and Im- ageNet (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' However, this improvement could come from the ImageNet + GeoDE dataset matching the domain of the GeoDE evaluation set and may not generalize to other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' To address this, we also test these models on a dif- ferent dataset: the DollarStreet dataset [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This dataset has been used before as an evaluation benchmark [7], to un- derstand if current object recognition models can perform well on objects from a diverse set of regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7 lists the per class accuracies for the object categories that overlap be- tween GeoDE and DollarStreet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see an increase in per- formance across most categories, suggesting that GeoDE is more geo-diverse than ImageNet and that there is an advan- tage to using geo-diverse data in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3We also try training a ResNet50 [15] model from scratch as well as finetuning the model trained on ImageNet, and do not find significant changes to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results are provided in the appendix 6 lightswitch bus chair bag dog monument car hairbrush boat cooking pot hat road sign bicycle religious bld flag toothbrush toothpaste storefront wheelbarrow light fixture truck plate of food hand soap front door jug stove lighter stall streetlight fence backyard toy candle waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tree house spices clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' medicine dustbin 98 97 97 96 96 96 96 95 95 95 93 93 92 92 91 90 90 89 89 88 88 88 86 85 85 78 77 76 76 75 74 73 71 69 68 63 63 59 54 37 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Per-class accuracy (in %;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' decreasing order) of CLIP [21] on GeoDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Object in bold are poorly recognized by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Example errors that the CLIP [21] model makes on GeoDE images (the ground truth label on the left, CLIP prediction at the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' There are some systematic errors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', classifying “house” as a “religious building”, particularly on images from regions in Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' (product labels on images have been blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' ) Model WAsia Africa EAsia SEAsia Americas Europe Avg ImageNet 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9 +GeoDE 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='7 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='8 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We notice significant performance improvements on GeoDE after augmenting ImageNet with images from GeoDE across all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' bicycle car cooking pot front door hand soap house light fixture light switch plate of food spices toothbrush toy waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Average Imagenet 89 90 53 77 28 55 56 80 62 30 45 41 6 55 + GeoDE 92 89 67 82 45 57 89 78 85 52 55 52 24 67 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We compare the per class accuracies of the Dol- larStreet [22] dataset for a model trained on only ImageNet [8] and a model trained on both ImageNet and GeoDE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We find that even adding such a small amount of diverse data into the training pipeline can improve the overall performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Cost-vs-Diversity tradeoffs The main drawback of GeoDE is the cost of this dataset: images collected in this way cost more than the standard pipeline of web-scraping and crowd-sourcing annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Thus, it is important to identify which classes and regions contribute most to the overall model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' To investigate this, we start with the filtered ImageNet dataset described above, vary the amount of GeoDE data from a particular region, and analyze the change in overall regional performance and regional performance for specific objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We start with a dataset fully comprised of the 38,353 filtered ImageNet images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We then add a region of GeoDE’s data back into the dataset and re- move the same number of ImageNet images to keep both the number of images and class balance the same for each model we train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The other training details remain the same as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' As we are evaluating on the GeoDE test set, there are two possible sources of performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The first is that the model is able to take advantage of the ad- ditional regional information from the GeoDE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The second is that the GeoDE images were collected using the same collection method as the test set and from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5, we saw that there is a difference in the feature space that can be attributed to the collection method itself (crowd-sourcing rather than web-scraping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In order to distinguish between ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='West Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Africa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='East Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Southeast Asia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Cooking pot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='container ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='container ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='toothpowder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='toothpowderreligious building ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='spices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='stove ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We show the TSNE plots of classes which have large regional disparities in accuracy from the CLIP trained model and show images from different parts of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For “religious buildings”, we see that GeoDE contains a cluster of monasteries and temples, mostly from East and Southeast Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For “spices”, we see a separation based on the spice container which can be region dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 0 50 100 % of images used from GeoDE 60 70 80 Accuracy (%) Trained on Africa, tested on Africa Trained on Africa, tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on SE Asia, tested on SE Asia Trained on SE Asia, tested on Europe Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We show the effect of incrementally adding regional GeoDE data for Africa and SE Asia on both the specific region and on Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We find that while adding any GeoDE regional images increases the performance of the model on European images, it has a larger effect on the region the images were drawn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' these two sources, we measure the accuracy on both the re- gion in the train set and accuracy on the images from Eu- rope4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also measure the increase in average precision for specific objects to better understand which objects ben- efit the most from GeoDE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In Figure 10 we visualize the increase in ac- curacy as we incrementally add in images from Africa and Southeast Asia (all other regions are presented in the ap- pendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We can see that the performance both within the specific region and in Europe increase with the additional GeoDE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The relative increase in performance for the specific region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Africa or Southeast Asia) is larger than the increase for Europe, showing the value of data for each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also note that the improvements have not sat- urated, suggesting that obtaining more regional data could lead to further gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also examine the classes that have the largest increase in average precision (AP) as the regional GeoDE images are added to the dataset in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We present the object classes that see the most improvement in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In general, we see a large overlap in which objects benefit the most from regional GeoDE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In particular, the performance 4We use Europe as this region had the best performance when using a model trained on just ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We measure the relative improvement in average preci- sion per object when GeoDE images from that region are included in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Each vertical line represents an object, and we sort them by the region where that object saw the largest improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see that Africa and East Asia see the largest improvement for the most classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Region Classes with largest percent increase in AP Africa waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', spices, dustbin, clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', hand soap Americas dustbin, spices, clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', medicine, waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia relig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' blg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', spices, dustbin, waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', spices, medicine, clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', dustbin W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia dustbin, hand soap, clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' equip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', spices, jug Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We highlight the classes with the largest increases in the average precision when adding in training images from GeoDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Classes like cleaning equipment, spices and dustbin and waste container are among the classes most improved across all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' on “spices”, “waste container” and “cleaning equipment” see large improvements in AP across all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Conclusions We have introduced a new dataset, GeoDE , which uses crowd-sourcing for image collection, a significant depar- ture from the popular computer vision dataset collection paradigm of web-scraping for image collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Using crowd-sourcing for image collection allows us to ensure that our dataset does not contain personally identifiable in- formation, that we own the rights to the images, that the image creators were compensated for their work, and we are able to control for geographic diversity and object dis- tribution in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In addition, we showed that this collection method results in significantly different images and image features than standard web-scraped dataset, even 8 60 40 20 40 30 20 10 0 10 20 30 40 5040 20 0 20 40 60 40 20 0 20 40 6040 20 20 40 40 20 0 20 40W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia East Asia () Europe Africa () (+) (★) Americas 大 (x) SE Asiawhen controlling for the types of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also show that even small amounts of geodiverse data are useful for eval- uating and highlighting shortcomings in common models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' CLIP) and can improve performance when added to the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The GeoDE dataset shows that crowd- sourcing is a viable image collection technique for creating diverse and responsible image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Our work also sug- gests avenues for scaling crowd-sourcing through the use of targeting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' focusing on collecting the most valuable images for improving model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This material is based upon work partially supported by the National Science Foundation un- der Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2145198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Any opinions, findings, and con- clusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also ac- knowledge support from Meta AI and the Princeton SEAS Howard B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Wentz, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Junior Faculty Award to OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We thank Dhruv Mahajan for his valuable insights during the project development phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also thank Jihoon Chung, Nicole Meister, Angelina Wang and the Princeton Visual AI Lab for their helpful comments and feedback during the writing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' References [1] Open images extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' https://research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='google/ tools / datasets / open - images - extended - crowdsourced/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Accessed: 2022-10-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2, 3 [2] Yuki M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1 [13] Kristen Grauman,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Andrew Westbury,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Eugene Byrne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Zachary Chavis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Antonino Furnari,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Rohit Girdhar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Jack- son Hamburger,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Hao Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Miao Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Xingyu Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Miguel Martin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tushar Nagarajan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Chen Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Siddhant Bansal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Dhruv Batra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Vincent Cartillier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Sean Crane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tien Do,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 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J´achym Kol´ar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Satwik Kottur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Anurag Kumar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Federico Lan- dini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Chao Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Yanghao Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Zhenqiang Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Karttikeya Man- galam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Raghava Modhugu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Jonathan Munro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tullie Mur- rell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Takumi Nishiyasu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Will Price,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 4 [29] Angelina Wang, Alexander Liu, Ryan Zhang, Anat Kleiman, Leslie Kim, Dora Zhao, Iroha Shirai, Arvind Narayanan, and Olga Russakovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' REVISE: A tool for measuring and mit- igating bias in visual datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' IJCV, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1, 2 [30] Angelina Wang, Arvind Narayanan, and Olga Russakovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' REVISE: A tool for measuring and mitigating bias in visual datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In ECCV, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2 [31] Kaiyu Yang, Klint Qinami, Li Fei-Fei, Jia Deng, and Olga Russakovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Towards fairer datasets: Filtering and balanc- ing the distribution of the people subtree in the imagenet hi- erarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' FAT*, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1 [32] Kaiyu Yang, Klint Qinami, Li Fei-Fei, Jia Deng, and Olga Russakovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Towards fairer datasets: Filtering and balanc- ing the distribution of the people subtree in the imagenet hi- erarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 2 [33] Kaiyu Yang, Jacqueline Yau, Li Fei-Fei, Jia Deng, and Olga Russakovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' A study of face obfuscation in imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' CoRR, abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='06191, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 3 [34] Kaiyu Yang, Jacqueline Yau, Li Fei-Fei, Jia Deng, and Olga Russakovsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' A study of face obfuscation in imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In ICML, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1 [35] Jieyu Zhao, Tianlu Wang, Mark Yatskar, Vicente Ordonez, and Kai-Wei Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Men also like shopping: Reducing gender bias amplification using corpus-level constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In EMNLP, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 1, 2 Appendix Here, we provide some more details about our experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In Sec A, we describe our heuristic to select object cat- egories in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In Sec B, we compare the GeoDE feature space to that of ImageNet [8] In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' C, we provide results when finetuning pre- trained models rather than just training the final layer of a ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' D, we give more details about GeoDE , in- cluding the counts of images of different regions and categories, as well as more sample images from this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Selecting object categories In this section, we provide more details about the ob- ject selection heuristic we employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We used 2 different datasets that were collected to be geodiverse: GeoYFCC [9] and DollarStreet [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Features for GeoYFCC were extracted using a ResNet50 [15] pretrained on Ima- geNet [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We used Logistic regression, Linear SVM and KMeans clustering implementations from the sklearn li- brary [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We used continents as regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' GeoYFCC [9] contains over 1200 tags, we ignored all tags with counts in the bottom 20th percentile, giving us a total of 745 tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' First, we apply each of these methods to GeoYFCC to identify candidate tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For each region R, we train a linear model using a fea- ture extractor and images from all regions except R and a linear model trained on all images from all regions, to pre- dict the presence or absence of each tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We then applied both models to images from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' The difference in perfor- mance between these models allows us to measure the dif- ference in appearance of the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We selected tags where in the weighted average precision on the region was less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='8* the performance on other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This gave us a set of 277 tags such as “footstool”, “chili”, “case”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' For each tag T, we train a linear SVM to predict the re- gion given the features of images containing tag T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' If this model has high accuracy, this suggests that this tag is vi- sually different across regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We selected tags that had an accuracy of over 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 223 tags were identified in this 10 manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' “Cork”, “bowler hat” and “mountain bike” are examples of tags found in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We clustered features of images containing tag T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We then computed the Gini impurity of each world region, and selected tags that had a median Gini value of at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This gave us 75 tags in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Examples of tags found in this way were “chili”, “footstool” and “stove”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' After identifying these tags, we first pruned them by re- moving tags that did not appear to correspond to an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Examples of this include “arctic”, “descent”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Second, we removed tags corresponding to wild animals, since these would not be found in all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Examples of tags re- moved in this manner were “gnu”, “camel”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This gave us a list of 265 tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Third, these tags were sometimes vari- ants of objects, for example, we had tags like “stupa”, “tem- ple”, “church”, “mosque” and “chapel”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Thus we grouped tags based on meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Other examples included “break- fast”, “dinner” and “dessert” as “plate of food”, “stool”, “footstool” and “bench” as “chair”, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This gave us a list of objects we could collect, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' “religious buildings”, “plate of food”, “toy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Comparison with ImageNet We note that the comparison to GeoYFCC [9] in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 5 in the main text required us to use tags which are noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Here, we compare GeoDE to ImageNet, checking how much the feature spaces differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We find a subset of ImageNet21k as outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 6, and extract features using a PASS [2] trained ResNet50 [15] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Other implementation details remain the same as in Sec 5 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We first use a Logistic regression model to predict the dataset that the features are taken from and this has an ac- curacy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='0%, showing that the feature space is very dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also visualize TSNE plots of different objects in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' More results when training with GeoDE In this section, we provide results for the incremental training with GeoDE for different regions that were not presented in Sec 7, and provide results when fine-tuning a ResNet50 model, rather than freezing the layer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results from incrementally adding additional regions We again visualize the improvement in the accuracy as we incrementally add in images from West Asia, East Asia, Americas and Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We can see that the performance both within the specific region and in Europe (compared to Americas when considering Europe) increase with the ad- ditional GeoDE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similar to before, we see that the − 40 − 20 0 20 40 − 60 − 40 − 20 0 20 40 60 Boat ImageNet GeoDE − 60 − 40 − 20 0 20 40 60 − 60 − 40 − 20 0 20 40 60 80 Car ImageNet GeoDE − 60 − 40 − 20 0 20 40 60 − 80 − 60 − 40 − 20 0 20 40 60 Hand soap ImageNet GeoDE − 60 − 40 − 20 0 20 40 60 80 − 80 − 60 − 40 − 20 0 20 40 60 80 Spices ImageNet GeoDE Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We visualize the TSNE plots for several of object classes using ImageNet and GeoDE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While the features do overlap slightly, on the whole, they are very different for dataset distribu- tions, even within each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 0 50 100 % of images used from GeoDE 65 75 85 Trained on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia, tested on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia Trained on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia, tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia, tested on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia Trained on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Asia, tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on Americas, tested on Americas Trained on Americas, tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on Europe, tested on Europe Trained on Europe, tested on Americas Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We visualize the increase in accuracy as images are in- crementally added in from a region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 10 in the main text, we see that adding in GeoDE images increases performance more in the region than a control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' increase within the region is larger than that of the con- trol, showing that these images are from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 13) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results from finetuning a ResNet50 model Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We use a ResNet50 [15] model pretrained on Imagenet and fine tune the weights using dif- ferent fractions of the ImageNet and GeoDE datasets as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 7 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We train the model with an SGD optimizer, learning rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='1, and momen- tum=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Other implementation details remain the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While the overall trend of the results are the same, we see that these results are slightly noisier, poten- tially because the model overfits to the small training set (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 11 Leave one out training curler,' metadata={'source': 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wind-instrument,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' skyscraper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' sweet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' course,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tower,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' opera,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' sketch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' circuit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' chapel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' col,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' motor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' clock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' railing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' mangrove Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Prospective tags identified from GeoYFCC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tags in red seemed hard to picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Tags in blue are of animals that might be hard to crowdsource .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' More details about the dataset In this supplementary section, we provide counts of the objects per region in GeoDE as well as more examples of images from this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' As mentioned before, GeoDE is mostly balanced across both region and object: for most part, we were able to get atleast 150 images per region per object, with a few excep- tions (“wheelbarrow” in 2 regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' “monument”, “boat” and “flag” in 1 region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 10 for full counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also provide more examples of the images from GeoDE in the Figures Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 15 to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Asia Africa E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Asia SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='Asia Americas Europe backyard 213 670 191 218 216 232 bag 267 388 379 593 298 437 bicycle 234 252 303 235 228 244 boat 159 227 174 237 84 225 bus 203 234 228 214 217 227 candle 232 243 219 239 188 272 car 242 323 284 235 273 363 chair 279 353 338 512 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='273 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='medicine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='275 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='330 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='328 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='245 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='plate of food ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='206 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='479 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='294 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='364 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='241 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='310 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content='relig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' blg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 221 223 211 226 197 230 road sign 224 407 264 270 235 289 spices 243 242 339 216 290 300 stall 139 216 201 227 197 226 storefront 209 309 191 240 246 204 stove 199 537 201 263 206 287 streetlight 202 343 214 196 208 227 toothbrush 264 252 336 361 337 270 toothpaste 209 286 271 230 245 315 toy 224 220 281 292 323 287 tree 223 296 257 357 300 331 truck 205 239 214 231 212 225 waste cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 225 210 212 213 214 259 wheelbarrow 122 256 141 197 152 243 Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We show the counts of objects per region in GeoDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Bolded are the ones categories for which we were not able to get 150 images per region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 13 0 50 100 % of images used from GeoDE 65 75 85 Trained on West Asia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on West Asia Trained on West Asia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on Africa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Africa Trained on Africa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on East Asia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on East Asia Trained on East Asia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on SE Asia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on SE Asia Trained on SE Asia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on Americas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Americas Trained on Americas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Europe 0 50 100 % of images used from GeoDE 65 75 85 Trained on Europe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Europe Trained on Europe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' tested on Americas Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We visualize the increase in accuracy as images are incrementally added in from a region when finetuning a ResNet50 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 10 in the main text, we see that adding in GeoDE images increases performance more in the region than a control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 14 Backyard West Asia Africa East Asia Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “backyard” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We notice that some of these are backyards made of concrete (West Asia: r2c1, r3c4, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', Africa: r3c1 ,r4c9, r5c3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=',) or do not contain lawns (West Asia: r3c1, r2c5, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', Africa:r5c1-5, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', East Asia: r2c4, r5c1, r5c4-6, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=') 15 Backyard Southeast Asia Americas Europe Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “backyard” for the 3 other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Again, we see that regions tend to have backyards made of concrete or paved (Southeast Asia: r1c8, r2c5, r3c3 as examples, Americas: r1c9, r1c10, r2c10, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', Europe: r3c2, r4c5, etc ), or do not contain lawns (Southeast Asia: r1c1, r1c9, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', Americas:r3c2, r3c3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=', Europe: r3c2, r5c9, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=') 16 Bicycle West Asia Africa East Asia Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “bicycle” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' While most images are of standard bicycles, we notice a couple of interesting images: tricycle (West Asia: r4c9), rickshaws (Africa: r2c8, r2c10, r4c8, r4c10), and motorized cycles (West Asia: r1c8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' There are also a lot of children’s bicycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 17 Bicycle Southeast Asia Americas Europe Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “bicycle” for the 3 other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see more motorized cycles (Southeast Asia: r5c6) as well as several children’s bicycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 18 8Boat West Asia Africa East Asia Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “boat” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see a variety of boats including larger ships in West Asia (r1c1, r1c2, r1c5, r4c1,r5c1), smaller kayaks and canoes in Africa (r1c8-9, r2c1-4, r4c2 , etc ), and a mix in East Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 19 Boat Southeast Asia Americas Europe Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “boat” for the 3 other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We again see a variety of boats ranging from motor boats in Europe and the Americas to smaller boats in Southeast Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 20 Cleaning equipment West Asia Africa East Asia Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “cleaning equipment” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This appears to be a diverse category within all regions containing images of mops, buckets, products, brooms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 21 Cleaning equipment Southeast Asia Americas Europe Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “cleaning equipment” for the 3 other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' This appears to be a diverse category within all regions containing images of mops, buckets, products, brooms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 22 PFSpices West Asia Africa East Asia Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “spices” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see a wide range of containers, ranging from packets (mostly in Africa), glass jars (in West Asia) to some bottles (all regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 23 05CoalingHETACO,Spices Southeast Asia Americas Europe Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “spices” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see a wide range of containers, ranging from packets (some in Southeast Asia and Americas) to bottles (some in Southeast Asia) 24 DBONUS,DIZLDARYLiaStove West Asia Africa East Asia Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “stove” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see that Africa and East Asia contain one-burner and two burner stoves (along with 4 burner stoves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also see a variety of stoves in terms of induction, gas, ovens, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 25 Stove Southeast Asia Americas Europe Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “stove” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see that Southeast Asia contains one-burner and two burner stoves (along with 4 burner stoves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We also see a variety of stoves in terms of induction, coils, gas, ovens, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 26 OWaste container West Asia Africa East Asia Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “waste container” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see that different regions have containers of varying sizes (Africa seems to be smaller than West Asia or East Asia), and have different closing mechanisms (see West Asia r5c6 as an interesting example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=') East Asia also tends to have segregated waste containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' 27 4Waste containers Southeast Asia Americas Europe Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' Randomly chosen images for “waste container” for 3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=' We see that different regions have containers of varying sizes (Europe seems to have containers of very different sizes) and have different closing mechanisms (see Southeast Asia r2c6 as an interesting example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} +page_content=') 28 PAPER2238' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE0T4oBgHgl3EQfrAFn/content/2301.02560v1.pdf'} diff --git a/rdE2T4oBgHgl3EQf1AiG/content/tmp_files/2301.04147v1.pdf.txt b/rdE2T4oBgHgl3EQf1AiG/content/tmp_files/2301.04147v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..179c5892a62bd2c11cb88e332e32e0d4c5f9c877 --- /dev/null +++ b/rdE2T4oBgHgl3EQf1AiG/content/tmp_files/2301.04147v1.pdf.txt @@ -0,0 +1,614 @@ +The Basis of Design Tools for Quantum Computing: +Arrays, Decision Diagrams, Tensor Networks, and ZX-Calculus +(Invited Paper) +Robert Wille1,2*, Lukas Burgholzer3*, Stefan Hillmich3*, Thomas Grurl3, Alexander Ploier3, Tom Peham3 +1 Chair for Design Automation, Technical University of Munich, Germany +2 Software Competence Center Hagenberg (SCCH) GmbH, Austria +3 Institute for Integrated Circuits, Johannes Kepler University Linz, Austria +*Corresponding Authors: robert.wille@tum.de, lukas.burgholzer@jku.at, stefan.hillmich@jku.at +https://www.cda.cit.tum.de/research/quantum/ +Abstract—Quantum computers promise to efficiently solve +important problems classical computers never will. However, in +order to capitalize on these prospects, a fully automated quantum +software stack needs to be developed. This involves a multitude of +complex tasks from the classical simulation of quantum circuits, +over their compilation to specific devices, to the verification of +the circuits to be executed as well as the obtained results. All +of these tasks are highly non-trivial and necessitate efficient +data structures to tackle the inherent complexity. Starting from +rather straight-forward arrays over decision diagrams (inspired +by the design automation community) to tensor networks and +the ZX-calculus, various complementary approaches have been +proposed. This work provides a look “under the hood” of today’s +tools and showcases how these means are utilized in them, e.g., +for simulation, compilation, and verification of quantum circuits. +I. INTRODUCTION +We are at the dawn of a new computing age in which +quantum computers will find their way into practical applica- +tions such as cryptography [1], chemistry [2], medicine [3], +physics [4], finance [5], and machine learning [6]. In many +instances, quantum computing is believed to provide efficient +solutions for problems which are out of reach for classical +computers. Besides the ongoing discovery of new potential +applications, the capabilities of currently available quantum +computers are rapidly improving as, e.g., witnessed by IBM’s +ambitious road map for scaling quantum technology to more +than 1000 qubits by 2023 [7]. +Due to an increased number of qubits with increased +coherence time as well as faster operations with higher fidelity, +increasingly large quantum circuits can reliably be executed +on actual devices. With this increase in computational power +comes the need for corresponding software solutions and tools +that aid users and developers in making best use of the available +hardware. Similar to the design of classical circuits and systems, +realizing conceptual quantum algorithms on actual devices +requires a multitude of complex design tasks. Some of the +most important tasks are: +• Classical simulation: Simulating the execution of a +quantum circuit on classical computers is an extremely +important task in the development and testing of new +applications and use cases. In addition to lower costs, +it offers detailed insights on the quantum state during +the execution of a quantum circuit that is physically +unavailable when running the circuit on an actual quantum +computer [8]–[13]. +• Compilation: Similar to classical circuits and systems, +quantum circuits are initially described at a rather high +abstraction level and need to be compiled to a repre- +sentation that adheres to all the constraints imposed by +the target device (e.g., limited gate-set and/or limited +connectivity) [14]–[18]. +• Verification: Since compilation significantly changes the +structure of quantum circuits, it is crucial to ensure that +the resulting circuits still realize the originally intended +functionality. To this end, verification (or, more precisely, +equivalence checking) methods are employed to guarantee +equivalence [17], [19]–[25]. +Either due to the inherent exponential size of the underlying +representations of quantum states and operations or the huge +amount of degrees of freedom, each of these design tasks +represents a computationally hard challenge. Consequently, +efficient data structures and methods are needed to tackle these +challenges. In this work, we provide a brief overview of various +complementary data structures that have been proposed in the +past and briefly discuss how each of them has been used to +efficiently solve the above mentioned design tasks. With this, +we hope to provide the interested reader with an intuition on +the different kinds of approaches available and the necessary +pointers to dive deeper into the wide range of possible methods +and solutions. +The rest of this work is structured as follows: Section II +reviews the basics of quantum computing and shows how +quantum states and operations are represented as one- and +two-dimensional arrays in a straight-forward, yet hardly effi- +cient fashion. Section III introduces decision diagrams which +enable representing quantum states and functionality in a more +compact fashion in many cases by exploiting redundancies +in the underlying representations. Section IV covers the +basics of tensor networks which, instead of capitalizing on +redundancies in the underlying representations, take advantage +of the topological structure of certain quantum states and +algorithms. Section V demonstrates how the ZX-calculus— +a graphical notation for quantum circuits equipped with a +powerful set of rewrite rules—enables diagrammatic reasoning +about quantum computing. Finally, Section VI concludes the +paper. +arXiv:2301.04147v1 [quant-ph] 10 Jan 2023 + +II. ARRAYS +In quantum computing, vectors and matrices are often con- +sidered to be the most intuitive data structure for representing +quantum objects. These structures can be directly realized using +arrays and can be used for design automation tasks. Here, we +introduce this data structure along with a brief introduction to +quantum computing. The interested reader can find an in-depth +introduction in [26]. +Similar to classical bits, quantum bits (qubits) can assume +the states 0 or 1. These are are called computational basis states +and—using Dirac notation—written as |0⟩ and |1⟩. Additionally, +they can also assume an (almost) arbitrary linear combination +(i.e., a superposition) of these two basis states. More precisely, +the state of a qubit |ψ⟩ is given by |ψ⟩ = α0 · |0⟩ + α1 · |1⟩, +with α0, α1 ∈ C such that |α0|2 + |α1|2 = 1. The two factors +α0 and α1 are the amplitudes and denote how much the +qubit is related to each of the two basis states. Measuring +a qubit returns 0 with probability |α0|2 and 1 with probability +|α1|2, respectively. The individual amplitudes in a qubit are +fundamentally not observable and measurements are the only +way to extract information out of a qubit. +The concepts of a single qubit can be generalized to describe +states composed of multiple qubits—commonly referred to as +quantum registers. An n qubit register can assume 2n basis +states and is described by amplitudes α0, α1, . . . α2n−1, which +must satisfy the normalization constraint � +i∈{0,1}n |αi|2 = 1. +Quantum states are often shortened to state vectors containing +only the amplitudes, e.g., [ α00 α01 α10 α11 ]T for two qubits. +Quantum states can be manipulated using quantum opera- +tions. Quantum operations are inherently reversible and are +described by unitary matrices. They are applied to quantum +states by matrix-vector multiplication. Important single-qubit +operations include the NOT = [ 0 1 +1 0 ] operation, which negates +the state of a qubit, and the Hadamard operation H = +1/ +√ +2 +� 1 +1 +1 −1 +� +, which transforms a qubit from a basis state into a +superposition. There are also multi-qubit operations. The most +prominent two-qubit operation is the controlled-NOT operation +(CNOT), which negates the state of its target qubit iff the +control qubit is in state |1⟩. +Example 1. Consider the quantum register |ψ⟩ composed of +two qubits, which is in the state 1/ +√ +2 · [ 1 0 1 0 ]T. Applying a +CNOT operation with control on the first and target on the +second qubit yields the output state |ψ′⟩ determined by +� 1 0 0 0 +0 1 0 0 +0 0 0 1 +0 0 1 0 +� +� +�� +� +CNOT +· +1 +√ +2 +� 1 +0 +1 +0 +� +� �� � +|ψ⟩ += +1 +√ +2 +� 1 +0 +0 +1 +� +� �� � +|ψ′⟩ +. +Measuring |ψ′⟩ (also known as Bell state) collapses the state +and returns |00⟩ or |11⟩, each with probability |1/ +√ +2|2 = 1/2. +The concepts reviewed above can be realized in a straight- +forward fashion: Vectors and matrices are described in terms +of 1-dimensional and 2-dimensional arrays, respectively. While +such a representation has huge potential for concurrent exe- +cution, it incurs a huge memory footprint, since the involved +arrays growth exponentially with each considered qubit. As +|00⟩ +|01⟩ +|10⟩ +|11⟩ +q1 +q0 +q0 +1 +√ +2 +0 +0 +1 +√ +2 +� +��������� +� +��������� +(a) Vector +q1 +q0 +q0 +1 +1/ +√ +2 +0 +0 +(b) Decision diagram +Fig. 1. Different representations of the Bell state +a consequence, these memory requirements limit array-based +simulation methods to rather small/moderate quantum compu- +tations (today’s practical limit is less than 50 qubits [27]). +III. DECISION DIAGRAMS +The general idea of decision diagrams [28], [29] is about +uncovering and exploiting redundancies within the involved +quantum states and operations. More precisely, consider a +quantum register composed of n qubits qn−1, . . . , q1, q0, where +qn−1 represents the most significant qubit. The first 2n−1 en- +tries of the corresponding state vector represent amplitudes for +basis states where qn−1 is |0⟩ and the remaining 2n−1 entries +represent amplitudes where qn−1 is |1⟩. This is represented in +a decision diagram by a node labeled qn−1 connected to two +successor nodes labeled qn−2, representing the zero- and one- +successor. This process is repeated recursively until sub-vectors +of size 1 (i.e., individual complex numbers) remain, which are +connected to terminal nodes. +During this decomposition process, equivalent sub-vectors +are represented by the same node—reducing the overall size of +the decision diagram. Furthermore, instead of having distinct +terminal nodes for all amplitudes, edge weights are used to +store common factors of the amplitudes. Having encoded +a state vector into a decision diagram, specific amplitudes +can be reconstructed multiplying the edge weights along the +corresponding path. To improve the readability of decision +diagrams, edge weights of 1 are typically omitted from the +visualization and nodes with an incoming edge weight of zero +are shown as 0-stubs to indicate that the whole sub-part is +zero. +Example 2. Fig. 1 depicts the quantum register |ψ′⟩ in +both, the vector and the decision diagram representation. The +annotations of the state vector in Fig. 1a indicate how the +corresponding decision diagram is constructed. In order to +reconstruct specific amplitudes from the decision diagram, the +edge weights of the corresponding path need to be multiplied. +For example, reconstructing the amplitude of the state |00⟩ +(bold line in the figure) requires multiplying the edge weight +of the root edge (1/ +√ +2) with the right edge of q1 (1) as well +as q0 (1), i.e. 1/ +√ +2 · 1 · 1 = 1/ +√ +2. +Decision diagram representation of matrices are constructed +in an analogous fashion to vectors, decomposing the matrix +recursively into quarters instead of halves. Just as the underlying +vectors and matrices, decision diagrams support multiplication +and addition, enabling their usage in different design automa- +tion tasks, such as quantum circuit simulation (e.g., [9]) or + +|0⟩ +|0⟩ +1 +√ +2 +� +1 +1 +1 +−1 +� +1 +√ +2 +� +�� +1 +0 +0 +1 +� +�� +� +�� +1 +1 +0 +1 +1 +0 +� +�� +≡ +Fig. 2. Tensor network representation of the quantum circuit to create the +Bell state +equivalence checking (e.g., [20]). A web-based visualizing +tool providing an intuition of decision diagrams is available +at https://iic.jku.at/eda/research/quantum dd/tool/ [30]. +IV. TENSOR NETWORKS +Tensor networks can help alleviate the complexity of the +array-based simulation by exploiting redundancies in the +topological structure of the quantum circuit [31], [32]. To +translate a quantum circuit into a tensor networks, each object, +be it a state or a operation, is represented by a multidimensional +array of complex numbers, a tensor, connecting to other tensors +according to the underlying quantum circuit. The extraction of +useful information from such a network then typically requires +the pairwise contraction of tensors into a single remaining +tensor. +Example 3. Let A, B, C be matrices in CN×N. Further, let +the matrix product C = AB be given by +Ci,j = �N−1 +k=0 Ai,kBk,j, +with i, j = 0, . . . , N −1. Then, this corresponds to the contrac- +tion of the rank-2 tensors A = [Ai,k] and B = [Bk,j] over the +shared index k. This is conveniently represented graphically +as: +C +A +B +i +j +i +j +k += +The order in which all the tensors are contracted is called +contraction plan. The main goal of such a plan is to keep the +intermediate tensors and their dimension of contracted indices +(also referred to as bond dimension) during the computation in +check—a task proven to be NP-hard [33]. Therefore, a plethora +of methods have been developed to efficiently determine +suitable contraction plans [34]. +Example 4. Consider again the Bell state from Fig. 1a. Fig. 2 +shows how this translates to a tensor network. Each individual +tensor is illustrated by a “bubble” containing the actual data +of the tensor. This representation only requires a linear amount +of memory with regard to the total number of qubits and gates +(in contrast to the exponential representation in the array-based +method). The final state vector, on the other hand, still is of +size 2n, where n denotes the number of qubits in the system. +As shown by the example, the computation of the complete +output state vector with tensor networks is generally infeasible. +Different specialized types of tensor networks have been +proposed to alleviate that complexity by imposing certain +structures by decomposing the whole state into smaller tensors +(see [35] and the references therein). +This is used, e.g., in classical quantum circuit simulation, +where it is desirable to determine a single scalar quantity, such +as the expected value of some observable or an individual +(a) Bell circuit +(b) Bell state +(c) Graph-like diagram +Fig. 3. ZX-diagrams for the Bell state +amplitude. Methods based on tensor networks accomplish this +by fixing the output indices of the circuit’s tensor network, i.e., +adding “bubbles” at the end of the circuit. Contracting this +network results in a single rank-0 tensor—a scalar. Whenever +the size and bond dimension of intermediate tensors can be +kept in check, this can be done extremely efficient. +V. ZX-CALCULUS +The ZX-calculus [36], [37] is a graphical notation for +quantum circuits equipped with a powerful set of rewrite rules +that enable diagrammatic reasoning about quantum computing. +A ZX-diagram is made up of colored nodes (called spiders) +that are connected by wires. Each spider can either be green +(Z-spider +) or red (X-spider +) and is optionally attributed a +scalar phase. Spiders without inputs are called states, whereas +spiders with no outputs are called effects. +An important concept for ZX-diagrams is the only connec- +tivity matters paradigm, which expresses the fact that two +ZX-diagrams are considered equal if one can be transformed +into the other simply by bending wires. +Example 5. Consider the Bell circuit in Fig. 3a. It is equivalent +to the ZX-diagram +because they can be transformed +into each other by (un-)crossing the wires. Here, the Hadamard +box +is a short notation for the ZX-diagram +π +2 +π +2 +π +2 +and represents the Hadamard transformation. +To see how this circuit acts on the += |0⟩ states, we +can plug them into the ZX-diagram and simplify with the +ZX-calculus: += += += +At the end, the Bell state shown in Fig. 3b is obtained. +Any quantum circuit can be interpreted as a ZX-diagram. +ZX-diagrams are more general than quantum circuits however, +and allow for representations that do not have meaningful +interpretations as quantum circuits. It is this flexibility of being +able to leave the quantum circuit formalism that makes the +ZX-calculus a good intermediate language when working with +quantum circuits. +While the basic axiomatization of the ZX-calculus is a +powerful language for quantum information theory, it is +hard to apply directly in automated reasoning. The reason +for this is the lack of normal-forms for ZX-diagrams—an +important feature for automated rewriting. The backbone of +many automated methods using the ZX-calculus is an alternate +representation of ZX-diagrams using only Z-spiders and wires +with Hadamard boxes, called graph-like ZX-diagrams. The +graph-like ZX-diagram corresponding to the Bell circuit is +shown in Fig. 3c. Additional rewrite rules based on graph- +theoretic simplification are defined for these graph-like di- + +agrams enabling the definition of a terminating rewriting +procedure [38]. +This automated rewriting as well as the ability to efficiently +work with quantum operations such as phase polynomials has +led to many algorithms based on the ZX-calculus being used +to solve problems in the design automation of quantum circuits +such as compilation and optimization as well as simulation +and verification [39]–[42]. +VI. CONCLUSION +In this work, we briefly reviewed the basics of arrays, +decision diagrams, tensor networks, and ZX-calculus as well as +their applications in design automation for quantum computing. +Each of these complementary data structures provides a +certain trade-off between memory consumption, performance, +and conceptional complexity. Picking the most suitable data +structure for the job at hand is crucial to have an efficient +workflow in design automation for quantum computing. We +hope this work provides the interested reader with an intuition +on the data structures and necessary pointers to further explore +the wide range of possible methods and applications. +ACKNOWLEDGEMENTS +This project has received funding from the European Research +Council (ERC) under the European Union’s Horizon 2020 research +and innovation programme (grant agreement No. 101001318). It is part +of the Munich Quantum Valley, which is supported by the Bavarian +state government with funds from the Hightech Agenda Bayern Plus +and was partially supported by the BMK, BMDW, and the State of +Upper Austria in the frame of the COMET program (managed by the +FFG). +REFERENCES +[1] +P. W. Shor, “Polynomial-time algorithms for prime factorization and +discrete logarithms on a quantum computer,” SIAM J. Comput., 1997. +[2] +A. Kandala, A. Mezzacapo, et al., “Hardware-efficient variational +quantum eigensolver for small molecules and quantum magnets,” Nature, +vol. 549, no. 7671, 2017. +[3] +B. A. Cordier, N. P. D. Sawaya, G. G. Guerreschi, and S. K. McWeeney, +Biology and medicine in the landscape of quantum advantages, 2021. +arXiv: 2112.00760. +[4] +H.-S. Zhong, H. Wang, et al., “Quantum computational advantage using +photons,” Science, vol. 370, no. 6523, 2020. +[5] +D. Herman, C. 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Phys., vol. 93, 4 2021. +[36] +J. van de Wetering, ZX-calculus for the working quantum computer +scientist, 2020. arXiv: 2012.13966. +[37] +B. Coecke and A. Kissinger, “Picturing quantum processes,” in +Diagrammatic Representation and Inference, 2018. +[38] +R. Duncan, A. Kissinger, S. Perdrix, and J. van de Wetering, “Graph- +theoretic Simplification of Quantum Circuits with the ZX-calculus,” +Quantum, vol. 4, 2020. +[39] +A. Kissinger and J. van de Wetering, “Reducing T-count with the +ZX-calculus,” Phys. Rev. A, 2020. +[40] +A. Kissinger and J. van de Wetering, “Simulating quantum circuits +with zx-calculus reduced stabiliser decompositions,” Quantum Science +and Technology, 2022. +[41] +N. de Beaudrap, X. Bian, and Q. Wang, “Techniques to reduce π/4- +parity-phase circuits, motivated by the ZX calculus,” Electron. Proc. +Theor. Comput. Sci., vol. 318, 2020. +[42] +A. Cowtan, W. Simmons, and R. Duncan, A generic compilation strategy +for the unitary coupled cluster ansatz, 2020. arXiv: 2007.10515. + diff --git a/rdE2T4oBgHgl3EQf1AiG/content/tmp_files/load_file.txt b/rdE2T4oBgHgl3EQf1AiG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20c89ad2f887ef535d79579045ba22cc7ff6e04b --- /dev/null +++ b/rdE2T4oBgHgl3EQf1AiG/content/tmp_files/load_file.txt @@ -0,0 +1,438 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf,len=437 +page_content='The Basis of Design Tools for Quantum Computing: Arrays, Decision Diagrams, Tensor Networks, and ZX-Calculus (Invited Paper) Robert Wille1,2*, Lukas Burgholzer3*, Stefan Hillmich3*, Thomas Grurl3, Alexander Ploier3, Tom Peham3 1 Chair for Design Automation, Technical University of Munich, Germany 2 Software Competence Center Hagenberg (SCCH) GmbH, Austria 3 Institute for Integrated Circuits, Johannes Kepler University Linz, Austria Corresponding Authors: robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='wille@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='de, lukas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='burgholzer@jku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='at, stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='hillmich@jku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='cda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='de/research/quantum/ Abstract—Quantum computers promise to efficiently solve important problems classical computers never will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' However, in order to capitalize on these prospects, a fully automated quantum software stack needs to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This involves a multitude of complex tasks from the classical simulation of quantum circuits, over their compilation to specific devices, to the verification of the circuits to be executed as well as the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' All of these tasks are highly non-trivial and necessitate efficient data structures to tackle the inherent complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Starting from rather straight-forward arrays over decision diagrams (inspired by the design automation community) to tensor networks and the ZX-calculus, various complementary approaches have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This work provides a look “under the hood” of today’s tools and showcases how these means are utilized in them, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', for simulation, compilation, and verification of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' INTRODUCTION We are at the dawn of a new computing age in which quantum computers will find their way into practical applica- tions such as cryptography [1], chemistry [2], medicine [3], physics [4], finance [5], and machine learning [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' In many instances, quantum computing is believed to provide efficient solutions for problems which are out of reach for classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Besides the ongoing discovery of new potential applications, the capabilities of currently available quantum computers are rapidly improving as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', witnessed by IBM’s ambitious road map for scaling quantum technology to more than 1000 qubits by 2023 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Due to an increased number of qubits with increased coherence time as well as faster operations with higher fidelity, increasingly large quantum circuits can reliably be executed on actual devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' With this increase in computational power comes the need for corresponding software solutions and tools that aid users and developers in making best use of the available hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Similar to the design of classical circuits and systems, realizing conceptual quantum algorithms on actual devices requires a multitude of complex design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Some of the most important tasks are: Classical simulation: Simulating the execution of a quantum circuit on classical computers is an extremely important task in the development and testing of new applications and use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' In addition to lower costs, it offers detailed insights on the quantum state during the execution of a quantum circuit that is physically unavailable when running the circuit on an actual quantum computer [8]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Compilation: Similar to classical circuits and systems, quantum circuits are initially described at a rather high abstraction level and need to be compiled to a repre- sentation that adheres to all the constraints imposed by the target device (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', limited gate-set and/or limited connectivity) [14]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Verification: Since compilation significantly changes the structure of quantum circuits, it is crucial to ensure that the resulting circuits still realize the originally intended functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' To this end, verification (or, more precisely, equivalence checking) methods are employed to guarantee equivalence [17], [19]–[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Either due to the inherent exponential size of the underlying representations of quantum states and operations or the huge amount of degrees of freedom, each of these design tasks represents a computationally hard challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Consequently, efficient data structures and methods are needed to tackle these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' In this work, we provide a brief overview of various complementary data structures that have been proposed in the past and briefly discuss how each of them has been used to efficiently solve the above mentioned design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' With this, we hope to provide the interested reader with an intuition on the different kinds of approaches available and the necessary pointers to dive deeper into the wide range of possible methods and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The rest of this work is structured as follows: Section II reviews the basics of quantum computing and shows how quantum states and operations are represented as one- and two-dimensional arrays in a straight-forward, yet hardly effi- cient fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Section III introduces decision diagrams which enable representing quantum states and functionality in a more compact fashion in many cases by exploiting redundancies in the underlying representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Section IV covers the basics of tensor networks which, instead of capitalizing on redundancies in the underlying representations, take advantage of the topological structure of certain quantum states and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Section V demonstrates how the ZX-calculus— a graphical notation for quantum circuits equipped with a powerful set of rewrite rules—enables diagrammatic reasoning about quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Finally, Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='04147v1 [quant-ph] 10 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' ARRAYS In quantum computing, vectors and matrices are often con- sidered to be the most intuitive data structure for representing quantum objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' These structures can be directly realized using arrays and can be used for design automation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Here, we introduce this data structure along with a brief introduction to quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The interested reader can find an in-depth introduction in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Similar to classical bits, quantum bits (qubits) can assume the states 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' These are are called computational basis states and—using Dirac notation—written as |0⟩ and |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Additionally, they can also assume an (almost) arbitrary linear combination (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', a superposition) of these two basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' More precisely, the state of a qubit |ψ⟩ is given by |ψ⟩ = α0 · |0⟩ + α1 · |1⟩, with α0, α1 ∈ C such that |α0|2 + |α1|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The two factors α0 and α1 are the amplitudes and denote how much the qubit is related to each of the two basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Measuring a qubit returns 0 with probability |α0|2 and 1 with probability |α1|2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The individual amplitudes in a qubit are fundamentally not observable and measurements are the only way to extract information out of a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The concepts of a single qubit can be generalized to describe states composed of multiple qubits—commonly referred to as quantum registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' An n qubit register can assume 2n basis states and is described by amplitudes α0, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' α2n−1, which must satisfy the normalization constraint � i∈{0,1}n |αi|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Quantum states are often shortened to state vectors containing only the amplitudes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', [ α00 α01 α10 α11 ]T for two qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Quantum states can be manipulated using quantum opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Quantum operations are inherently reversible and are described by unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' They are applied to quantum states by matrix-vector multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Important single-qubit operations include the NOT = [ 0 1 1 0 ] operation, which negates the state of a qubit, and the Hadamard operation H = 1/ √ 2 � 1 1 1 −1 � , which transforms a qubit from a basis state into a superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' There are also multi-qubit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The most prominent two-qubit operation is the controlled-NOT operation (CNOT), which negates the state of its target qubit iff the control qubit is in state |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Consider the quantum register |ψ⟩ composed of two qubits, which is in the state 1/ √ 2 · [ 1 0 1 0 ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Applying a CNOT operation with control on the first and target on the second qubit yields the output state |ψ′⟩ determined by � 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 � � �� � CNOT 1 √ 2 � 1 0 1 0 � � �� � |ψ⟩ = 1 √ 2 � 1 0 0 1 � � �� � |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Measuring |ψ′⟩ (also known as Bell state) collapses the state and returns |00⟩ or |11⟩, each with probability |1/ √ 2|2 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The concepts reviewed above can be realized in a straight- forward fashion: Vectors and matrices are described in terms of 1-dimensional and 2-dimensional arrays, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' While such a representation has huge potential for concurrent exe- cution, it incurs a huge memory footprint, since the involved arrays growth exponentially with each considered qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' As |00⟩ |01⟩ |10⟩ |11⟩ q1 q0 q0 1 √ 2 0 0 1 √ 2 � ��������� � ��������� (a) Vector q1 q0 q0 1 1/ √ 2 0 0 (b) Decision diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Different representations of the Bell state a consequence, these memory requirements limit array-based simulation methods to rather small/moderate quantum compu- tations (today’s practical limit is less than 50 qubits [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' DECISION DIAGRAMS The general idea of decision diagrams [28], [29] is about uncovering and exploiting redundancies within the involved quantum states and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' More precisely, consider a quantum register composed of n qubits qn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' , q1, q0, where qn−1 represents the most significant qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The first 2n−1 en- tries of the corresponding state vector represent amplitudes for basis states where qn−1 is |0⟩ and the remaining 2n−1 entries represent amplitudes where qn−1 is |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This is represented in a decision diagram by a node labeled qn−1 connected to two successor nodes labeled qn−2, representing the zero- and one- successor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This process is repeated recursively until sub-vectors of size 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', individual complex numbers) remain, which are connected to terminal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' During this decomposition process, equivalent sub-vectors are represented by the same node—reducing the overall size of the decision diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Furthermore, instead of having distinct terminal nodes for all amplitudes, edge weights are used to store common factors of the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Having encoded a state vector into a decision diagram, specific amplitudes can be reconstructed multiplying the edge weights along the corresponding path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' To improve the readability of decision diagrams, edge weights of 1 are typically omitted from the visualization and nodes with an incoming edge weight of zero are shown as 0-stubs to indicate that the whole sub-part is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 1 depicts the quantum register |ψ′⟩ in both, the vector and the decision diagram representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The annotations of the state vector in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 1a indicate how the corresponding decision diagram is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' In order to reconstruct specific amplitudes from the decision diagram, the edge weights of the corresponding path need to be multiplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' For example, reconstructing the amplitude of the state |00⟩ (bold line in the figure) requires multiplying the edge weight of the root edge (1/ √ 2) with the right edge of q1 (1) as well as q0 (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 1/ √ 2 · 1 · 1 = 1/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Decision diagram representation of matrices are constructed in an analogous fashion to vectors, decomposing the matrix recursively into quarters instead of halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Just as the underlying vectors and matrices, decision diagrams support multiplication and addition, enabling their usage in different design automa- tion tasks, such as quantum circuit simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', [9]) or |0⟩ |0⟩ 1 √ 2 � 1 1 1 −1 � 1 √ 2 � �� 1 0 0 1 � �� � �� 1 1 0 1 1 0 � �� ≡ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Tensor network representation of the quantum circuit to create the Bell state equivalence checking (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' A web-based visualizing tool providing an intuition of decision diagrams is available at https://iic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='jku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='at/eda/research/quantum dd/tool/ [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' TENSOR NETWORKS Tensor networks can help alleviate the complexity of the array-based simulation by exploiting redundancies in the topological structure of the quantum circuit [31], [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' To translate a quantum circuit into a tensor networks, each object, be it a state or a operation, is represented by a multidimensional array of complex numbers, a tensor, connecting to other tensors according to the underlying quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The extraction of useful information from such a network then typically requires the pairwise contraction of tensors into a single remaining tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Let A, B, C be matrices in CN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Further, let the matrix product C = AB be given by Ci,j = �N−1 k=0 Ai,kBk,j, with i, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' , N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Then, this corresponds to the contrac- tion of the rank-2 tensors A = [Ai,k] and B = [Bk,j] over the shared index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This is conveniently represented graphically as: C A B i j i j k = The order in which all the tensors are contracted is called contraction plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The main goal of such a plan is to keep the intermediate tensors and their dimension of contracted indices (also referred to as bond dimension) during the computation in check—a task proven to be NP-hard [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Therefore, a plethora of methods have been developed to efficiently determine suitable contraction plans [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Consider again the Bell state from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 2 shows how this translates to a tensor network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Each individual tensor is illustrated by a “bubble” containing the actual data of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This representation only requires a linear amount of memory with regard to the total number of qubits and gates (in contrast to the exponential representation in the array-based method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The final state vector, on the other hand, still is of size 2n, where n denotes the number of qubits in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' As shown by the example, the computation of the complete output state vector with tensor networks is generally infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Different specialized types of tensor networks have been proposed to alleviate that complexity by imposing certain structures by decomposing the whole state into smaller tensors (see [35] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This is used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', in classical quantum circuit simulation, where it is desirable to determine a single scalar quantity, such as the expected value of some observable or an individual (a) Bell circuit (b) Bell state (c) Graph-like diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' ZX-diagrams for the Bell state amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Methods based on tensor networks accomplish this by fixing the output indices of the circuit’s tensor network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=', adding “bubbles” at the end of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Contracting this network results in a single rank-0 tensor—a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Whenever the size and bond dimension of intermediate tensors can be kept in check, this can be done extremely efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' ZX-CALCULUS The ZX-calculus [36], [37] is a graphical notation for quantum circuits equipped with a powerful set of rewrite rules that enable diagrammatic reasoning about quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' A ZX-diagram is made up of colored nodes (called spiders) that are connected by wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Each spider can either be green (Z-spider ) or red (X-spider ) and is optionally attributed a scalar phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Spiders without inputs are called states, whereas spiders with no outputs are called effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' An important concept for ZX-diagrams is the only connec- tivity matters paradigm, which expresses the fact that two ZX-diagrams are considered equal if one can be transformed into the other simply by bending wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Consider the Bell circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' It is equivalent to the ZX-diagram because they can be transformed into each other by (un-)crossing the wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Here, the Hadamard box is a short notation for the ZX-diagram π 2 π 2 π 2 and represents the Hadamard transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' To see how this circuit acts on the = |0⟩ states, we can plug them into the ZX-diagram and simplify with the ZX-calculus: = = = At the end, the Bell state shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 3b is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Any quantum circuit can be interpreted as a ZX-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' ZX-diagrams are more general than quantum circuits however, and allow for representations that do not have meaningful interpretations as quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' It is this flexibility of being able to leave the quantum circuit formalism that makes the ZX-calculus a good intermediate language when working with quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' While the basic axiomatization of the ZX-calculus is a powerful language for quantum information theory, it is hard to apply directly in automated reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The reason for this is the lack of normal-forms for ZX-diagrams—an important feature for automated rewriting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The backbone of many automated methods using the ZX-calculus is an alternate representation of ZX-diagrams using only Z-spiders and wires with Hadamard boxes, called graph-like ZX-diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' The graph-like ZX-diagram corresponding to the Bell circuit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Additional rewrite rules based on graph- theoretic simplification are defined for these graph-like di- agrams enabling the definition of a terminating rewriting procedure [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' This automated rewriting as well as the ability to efficiently work with quantum operations such as phase polynomials has led to many algorithms based on the ZX-calculus being used to solve problems in the design automation of quantum circuits such as compilation and optimization as well as simulation and verification [39]–[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' CONCLUSION In this work, we briefly reviewed the basics of arrays, decision diagrams, tensor networks, and ZX-calculus as well as their applications in design automation for quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Each of these complementary data structures provides a certain trade-off between memory consumption, performance, and conceptional complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' Picking the most suitable data structure for the job at hand is crucial to have an efficient workflow in design automation for quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' We hope this work provides the interested reader with an intuition on the data structures and necessary pointers to further explore the wide range of possible methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' 101001318).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' It is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus and was partially supported by the BMK, BMDW, and the State of Upper Austria in the frame of the COMET program (managed by the FFG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdE2T4oBgHgl3EQf1AiG/content/2301.04147v1.pdf'} +page_content=' REFERENCES 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accompanying axion clouds +in extreme mass ratio inspirals +Takuya Takahashi,1, ∗ Hidetoshi Omiya,1, † and Takahiro Tanaka1, 2, ‡ +1Department of Physics, Kyoto University, Kyoto 606-8502, Japan +2Center for Gravitational Physics and Qunatum Information, Yukawa +Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan +(Dated: February 1, 2023) +Superradiant instability of rotating black holes (BHs) leads to the formation of a cloud of ultralight +bosons, such as axions. +When the BH with the cloud belongs to a binary system and is in an +inspiraling orbit, the resonant transition between the axion’s bound states can occur. We study +the history of the evolution of the binary system accompanying the cloud composed of the fastest +growing mode, and its impact on the observational signatures, especially for small mass ratio cases. +In this case, the hyperfine resonance, which has a very small resonance frequency, is relevant. +Therefore, due to the long timescale, we should take into account the decaying process of axions +in the transition destination mode, the backreaction to the orbital motion and the central BH, and +gravitational emission from the cloud. We present a formulation to examine the evolution of the +system around the resonance and useful expressions for the analysis. As a result, we found the +mass of the cloud that can remain after the resonance is, at most, about 10−5 of the central BH. +The maximum remaining cloud mass is achieved when the mass ratio of the binary is q ∼ 10−3. In +addition, we show that the resonant transition hardly changes the BH mass and spin distribution, +while the associated modification of the gravitational wave frequency evolution when the binary +pass through the resonance can be a signature of the presence of the cloud. +I. +INTRODUCTION +Ultralight bosons, such as axions or axion-like parti- +cles, can cause various phenomena in the universe. Such +particles are universally predicted by string theory [1, 2] +and can be a candidate for dark matter [3–6]. They can +be weakly coupled to the Standard Model particles, but +even in such a case the gravitational interaction with +black holes (BHs) and related gravitational waves (GWs) +can provide a new avenue to explore them observation- +ally. +The existence of massive bosonic fields induces the su- +perradiant instability around rotating BHs [7, 8]. Bosons +with mass in the range 10−20 ∼ 10−10 eV have the Comp- +ton wavelength comparable to the size of astrophysical +BHs, and extract energy and angular momentum effi- +ciently to form a condensate [9, 10]. +We refer to the +condensate as an axion cloud and the composing parti- +cles simply as axions. The cloud formation makes astro- +physical observable imprints, such as a forbidden region +in the distribution of mass and spin of BHs [11–13] and +continuous GW emission [14–20]. +In this paper, we focus on the cases where BHs with +clouds belong to binary systems. GWs from the binary +inspiral can be a signature to examine the environment +around BHs including the cloud [21–25]. Axion clouds +occupy a quasi-bound state of axions, which is usually +the fastest growing mode. During the inspiral phase, the +∗ t.takahashi@tap.scphys.kyoto-u.ac.jp +† omiya@tap.scphys.kyoto-u.ac.jp +‡ t.tanaka@tap.scphys.kyoto-u.ac.jp +tidal interaction from the companion acts as an oscil- +lating tidal field. It induces the resonant transition to +another mode when the orbital frequency coincides with +the phase velocity difference between the original mode +of the cloud and the other [26, 27]. The change of the +orbital motion of the binary and the associated GW fre- +quency due to the backreaction can also be a signature of +the presence of the cloud [27–30]. To clarify the impact +on the observational signatures, it is important to un- +derstand the history of the evolution during the inspiral +phase. +If the separation of the binary is sufficiently small, the +cloud configuration is tidally disrupted [31, 32], and the +transition to unbound states occurs [31, 33]. However, +for binary systems formed with a sufficiently large sep- +aration, the resonant transition should first occur with +the smallest possible resonance frequency. The frequency +spectrum of axion eigenmodes possesses the structure of +hyperfine splittings due to the rotation of the central +BH [34], and the resonance frequency associated with +the hyperfine splitting is the smallest one. In Ref. [31], +we showed that, for nearly equal mass binaries, this hy- +perfine resonance can be neglected since the resonance +condition is not maintained long enough because of the +decrease of the angular momentum of the cloud itself. +We also showed that, before the transition caused by the +leading quadrupole moment of the tidal potential occurs, +the cloud is disrupted by the effects of higher multipole +moments, and finally the cloud is depleted as a result of +transitions to unbound states. +In contrast to nearly equal mass binaries, for small +mass ratio binaries, the hyperfine resonance should be +considered because of a large backreaction to the orbital +motion, which maintains the orbital frequency within the +arXiv:2301.13213v1 [gr-qc] 30 Jan 2023 + +2 +resonance band for a long period. It has great importance +to examine the dynamics of small mass ratio binaries, +because they are one of the main targets for future GW +observations, such as LISA [35]. +In this case, because +of the very long timescale of the binary evolution due to +the radiation reaction, some effects that can be neglected +for the transition for nearly equal mass binaries become +relevant. +First, the decay of non-superradiant transition destina- +tion modes and the backreaction to the central BH mass +and spin become relevant. Since the resonance band is +broadened corresponding to the imaginary part of the +frequencies of decaying destination modes, the transition +timescale staying within the resonance band becomes +even longer. Therefore, we should also take into account +the GW emission from the cloud during the transition. +We develop a formulation that includes all of these effects +within the adiabatic approximation. It is difficult to solve +the originally obtained set of equations throughout the +whole period across the resonance band, since the solu- +tion oscillates rapidly. +To overcome this difficulty, we +also present a method to give an approximate solution +with sufficient accuracy. +In this paper, we consider axion clouds in a non- +relativistic regime, and neglect the self-interaction of ax- +ions, for simplicity. For a relativistic regime, the energy +spectrum deviates significantly from the one obtained by +non-relativistic approximation, and the transition to be +considered can change [36, 37]. +In addition, the self- +interaction can play an important role during the for- +mation of the cloud [38–45]. Here, we leave considering +these effects as future work, to focus on the tidal effect +in binary systems. +This paper is organized as follows. In Sec. II, we review +the elements involved in the evolution of axion clouds in +binary systems. In Sec. III, we present a formulation for +examining the hyperfine resonance in small mass ratio bi- +naries. In Sec. IV, we discuss the results obtained using +our formulation. Finally, we give a summary and conclu- +sion in Sec. V. Throughout this paper, we use the unit +with c = ¯h = G = 1. +II. +ELEMENTS INVOLVED IN THE +EVOLUTION OF AXION CLOUDS +In this section, we summarize the elements involved in +describing the evolution of axion clouds, especially dur- +ing the binary inspirals. Consider a scalar field (axion) of +mass µ around a rotating BH belonging to a binary sys- +tem. We denote the central BH mass by M and angular +momentum by J = aM = χM 2. Formally, we can write +the equation of motion for axion on a spacetime with the +metric ˜gµν = gµν + hµν as +(˜gµν ˜∇µ ˜∇ν − µ2)φ = 0 , +(1) +where gµν is the Kerr metric. We consider the tidal field +from the binary companion and the decay due to the +gravitational wave emission from the cloud as contribu- +tions to the perturbation. +As we will see later, since +there is a hierarchy of frequencies between them, we can +treat them separately. We first review the features of ax- +ion clouds in the unperturbed background, and later the +effects of the tidal interaction and the GW emission. +A. +Energy spectrum and superradiance +In the non-relativistic regime, it is appropriate to in- +troduce a new complex scalar field variable ψ by +φ = +1 +√2µ +� +e−iµtψ + eiµtψ∗� +. +(2) +We assume that ψ changes slowly in time compared to +the timescale determined by µ−1. Then, we can ignore +the ∂2 +t ψ term and rewrite the background equation of +motion (1) as +i ∂ +∂tψ = H0ψ , +H0 = − 1 +2µ∇2 − α +r + O(α2) , +(3) +where we have introduced the gravitational fine struc- +ture constant α ≡ Mµ, and this approximation is well +justified for α ≪ 1. Solving this equation with the in- +going boundary condition at the BH horizon and the ex- +ponentially decaying boundary condition at infinity, we +have the quasi-bound eigenstate ϕnlm(r) that satisfies +H0ϕnlm = (ωnlm − µ)ϕnlm. +They are labeled by the +principal, azimuthal and magnetic quantum numbers like +a hydrogen atom. The eigenfrequency is approximately +given by +ωnlm = (ωR)nlm + i(ωI)nlm , +(4) +with [26, 34] +(ωR)nlm = µ +� +1 − α2 +2n2 − α4 +8n4 + (2l − 3n + 1)α4 +n4(l + 1/2) ++ +2mχα5 +n3l(l + 1/2)(l + 1) +� +, (5) +(ωI)nlm = 2(r+/M)Cnlm(a, α)(mΩH − ωnlm)α4l+5 , +(6) +where r+ = M + +√ +M 2 − a2 is the horizon radius, ΩH = +a/2Mr+ is the angular velocity of the BH horizon and +the explicit form of Cnlm(a, α) can be found in Ref. [26]1. +As one can see from Eq. (6), the eigenfrequency of a +mode satisfying ωR < mΩH has a positive imaginary +part, and the cloud grows exponentially by the superra- +diance. The mode |nlm⟩ = |211⟩ is the fastest growing +1 It was first derived in Ref. [9], and corrected by a factor of 1/2 +[46, 47]. + +3 +mode for α ≲ 0.45. The BH spin decreases as the cloud +grows until the superradiance condition is saturated. The +critical spin at which the superradiance terminates is ap- +proximately given by +χcrit = +4mα +m2 + 4α2 . +(7) +The real part of the eigenfrequency can be regarded as +eigenenergy, and its degeneracy among the modes with +only m being different is solved due to the rotation of the +BH at the order of O(α5), which is called the “hyperfine” +splitting. +B. +Tidal interaction +When a BH accompanied by an axion cloud belongs +to a binary system, the tidal field from the companion +introduces a perturbation. The general state of the cloud +can be expressed by +ψ = +� +i +ci(t)ϕi , +(8) +as a superposition of orthonormal eigenfunctions ϕi. Un- +der the same approximation taken in the preceding sub- +section, the equation of motion with the perturbation is +given by +idci +dt = +� +j +� +(ωj − µ)δij + +� +d3x ϕ∗ +i V∗ϕj +� +cj . +(9) +For simplicity, we assume that the binary orbit is quasi- +circular and on the plane perpendicular to the central BH +spin. By multipole expansion, we can write the tidal field +from the companion of mass M∗ at r(t) = (R∗(t), Θ∗(= +π/2), Φ∗(t)) as +V∗ = 1 +2µhtt +tidal += −qα +� +l∗m∗ +4π +2l∗ + 1 +rl∗ +< +rl∗+1 +> +Y ∗ +l∗m∗(Θ∗, Φ∗)Yl∗m∗(θ, φ) , +(10) +where q ≡ M∗/M is the mass ratio, r>(r<) is the larger +(smaller) of r and R∗, and Ylm are the spherical har- +monics. +The angular velocity of the binary is defined +by ˙Φ∗(t) = ±Ω(t), and the upper (lower) sign represents +the case of co-rotating (counter-rotating) orbits. Since +this interaction oscillates quasi-periodically, it works ef- +ficiently only when the orbital angular velocity is close +to the difference between the phase velocity of the two +modes. Therefore, it is sufficient to consider a two-mode +subspace [27]. Tidal field mixes two modes, and the time +evolution of particle number in each mode is, from Eq.(9), +given by +i ˙c = Hc +(11) +with +H = +� +−∆E/2 + iω(1) +I +ηei∆mΦ∗ +ηe−i∆mΦ∗ +∆E/2 + iω(2) +I +� +, +(12) +where ∆E += +ω(2) +R +− ω(1) +R , ∆m += +m2 − m1, and +η(t) = +��� +d3x ϕ∗ +2V∗ϕ1 +��. +To remove the rapidly oscil- +lating term, we perform the unitary transformation as +c → U−1c and H → U†H U − i U† ˙U with the matrix +U(t) = diag(ei∆mΦ∗/2, e−i∆mΦ∗/2). As a result, we can +describe the level transition due to the tidal field by +H = +� +± ∆m +2 (Ω − Ωres) + iω(1) +I +η +η +∓ ∆m +2 (Ω − Ωres) + iω(2) +I +� +, +(13) +where we defined the “resonance” frequency by Ωres = +±∆E/∆m. Now, we are interested in the time evolution +of the occupation number of each state, |ci(t)|2. +C. +Gravitational wave emission +After an axion cloud forms, it dissipates through the +emission of GWs. +Here, we assume that the cloud is +composed of a single mode as ψ = c1ϕ1. In this case, we +can neglect the GW emission due to the spontaneous level +transition, and GWs are sourced by the pair-annihilation +of axions. +The frequency of GWs is given by ωGW = +2ωR ∼ 2µ. The energy flux of GWs from the l = m = 1 +cloud is given by [16] +dEGW +dt += C +�Mc +M +�2 +α14 , +(14) +where C is a numerical factor. In our analysis, we adopt +C = (484 + 9π2)/23040 calculated in Ref. [12]. Here, Mc +is the mass of the cloud defined by Mc = − +� +d3x T tt, +where T tt is the t-t component of the energy momentum +tensor. According to this, the wave function ψ is normal- +ized as |c1|2(= +� +d3x|ψ|2) = Mc/µ at the leading order +in α. +When we consider only the effect of GW emission, en- +ergy conservation implies that +˙Mc = − ˙EGW. +We set +the initial mass of the cloud to Mc,0 at t = t0. Here, +we define the normalized particle number by n1(t) = +µ|c1(t)|2/Mc,0, and write Mc(t) = Mc,0n1(t). +Energy +conservation reads +dn1 +dt = − C +M +�Mc,0 +M +� +n2 +1α14 . +(15) +III. +FORMULATION +In this section, we first explain the setup of the problem +that we consider and then give a formulation to investi- +gate it. + +4 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +α +q +FIG. 1. +Parameter region where the hyperfine resonance is +relevant to dissipate the cloud. In the shaded region, the reso- +nance sustains longer because the effect of the backreaction to +the orbital motion is stronger than the effect of the reduction +of the hyperfine splitting. The initial angular momentum of +the cloud is set to Jc,0 → 0. See Ref. [31] for the detail. +A. +Setup +We focus on the fastest growing mode |nlm⟩ = |211⟩. +We consider the situation in which the cloud is initially +composed of the single mode |211⟩, and the hyperfine +level transition between |211⟩ and |21 − 1⟩ subsequently +occurs. +Note that this transition occurs only for co- +rotating orbit. In Ref. [31], we found that when the bi- +nary mass ratio q is not too small, this transition does +not significantly contribute to the dissipation of the cloud +because of the reduction of the hyperfine splitting asso- +ciated with the transfer of the angular momentum of the +cloud to the orbital motion. +However, when the mass +ratio is somewhat small, the resonant tidal interaction +at this hyperfine splitting frequency would largely affect +the dynamics of the system. We show the parameter re- +gion where we should consider the hyperfine resonance +as a process that contributes to the cloud dissipation in +Fig. 1. We investigate the latter case. +For the transition between |211⟩ and |21 − 1⟩, from +Eq.(5), the resonance frequency is given by2 +Ωres = µ +12χα5 . +(16) +This is smaller by a factor of α3 than that of the +“Bohr” transition between modes with different values +of n. When we study the Bohr transition, ωI in Eq.(13) +2 We do not include the contribution from the angular momentum +of the cloud itself, focusing on the case where it is negligible. +and GW flux are so small in the timescale for passing +through the resonance band that we can usually neglect +them3. However, for hyperfine transition, binary evolu- +tion around the resonance frequency is very slow and the +timescale for passing through the resonance band can be +large, especially for q ≪ 1. In addition, since the angu- +lar momentum of the cloud is transferred to the orbital +motion, the timescale becomes even larger. As a result, +we should take into account not only the backreaction +to the orbital motion, but also the backreaction to the +mass and spin of the central BH and the effect of the GW +emission from the cloud. We summarize the timescales +involved in the current problem in Appendix A. +In the following, we label the quantities associated with +the mode |211⟩ by 1, and those with |21 − 1⟩ by 2. For +these modes, the imaginary parts of the eigenfrequencies +are given by +ω(i) +I += 1 +24 +r+ +M +�� +1 − χ2� ++ 4r2 ++(miΩH − ωR)2� +×(miΩH − ωR)α9 , +(17) +where i is 1 or 2, and m1 = 1 and m2 = −1 represent +the magnetic quantum number. The mixing term in the +Hamiltonian (13) is given by +η = 9.0 +q +1 + q +MΩ2 +α3 +. +(18) +B. +Evolution of the system +The dynamical timescale of the cloud can be estimated +by ω−1 +R +≃ µ−1. +It is always short compared to the +growth/decay rate of the cloud, i.e., (ω(i) +I )−1 ≫ µ−1. +Thus, we describe the evolution of the cloud and the cen- +tral BH within the adiabatic approximation. The local +energy and angular momentum conservation at the BH +horizon reads +dM +dt + 2ω(1) +I M (1) +c ++ 2ω(2) +I M (2) +c += 0 , +(19) +dJ +dt + 2ω(1) +I +µ +M (1) +c +− 2ω(2) +I +µ +M (2) +c += 0 , +(20) +with M (i) +c += Mc,0ni(t). Here, we used the relation be- +tween the energy flux and the angular momentum flux +for each mode ˙J(i) +c += (mi/ω(i) +R ) ˙E(i) +c +and the approxima- +tion ωR = µ. We denote the initial mass and angular +momentum of the BH just before entering the resonance +band by M0 and J0, and accordingly α0 = M0µ. +3 When we consider a higher l mode, the transition to the mode +with smaller l is allowed by the selection rule. In that case, the +decay rate of the second mode can be large, and it would be +important [48]. + +5 +Next, we consider the evolution of the binary sys- +tem at the leading post-Newtonian order. In clean bi- +nary systems, angular momentum conservation implies +˙Jorb = −TGW, where Jorb = q(1 + q)−1/3M 5/3 +0 +Ω−1/3 is +the orbital angular momentum and TGW is the torque +caused by the radiation reaction due to the GW emis- +sion. It can be rewritten as [49, 50] +dΩ +dt = γ +� Ω +Ω0 +�11/3 +, +(21) +γ +Ω2 +0 += 96 +5 +q +(1 + q)1/3 (M0Ω0)5/3 , +(22) +where the reference frequency is chosen as Ω0 += +(µ/12)(J0/M 2 +0 )α5 +0 (which is the “initial” resonance fre- +quency). +Here, we add the cloud and the BH contri- +butions to the total angular momentum conservation as +˙Jorb + ˙J + ˙J(1) +c ++ ˙J(2) +c ++ ˙JGW = −TGW, where ˙JGW = +(1/µ) ˙EGW is the angular momentum flux of the GW from +the cloud in Eq. (14). Note that we consider GW emis- +sion only from the first mode |211⟩. (As we will see later, +the particle number occupying the second mode, which is +non-superradiant, is always tiny and does not contribute +to the GW emission.) Then, we obtain4 +dΩ +dt =γ +� Ω +Ω0 +�11/3 ++ R +� Ω +Ω0 +�4/3 Ω0 +M 2 +0 +× +� d +dt +� +J + J(1) +c ++ J(2) +c +� ++ 1 +µ +dEGW +dt +� +, +(23) +with R = 3(1 + q)1/3q−1(M0Ω0)1/3. We take Ω(t0) = +Ω0(1 + (8/3)(γ/Ω0)|t0|)−3/8 as the initial value so that +Ω = Ω0 at t = 0 when there are no clouds. +Finally, we describe the level transition between two +modes. It is described by the Schr¨odinger equation with +the Hamiltonian (13). Note that the particle number oc- +cupying the first mode decreases due to the GW emission +by pair annihilation. Since the frequency of the emitted +GW (ωGW ∼ 2µ) is much larger than that of the tidal +field (Ωres ∼ µα6), we can treat them separately. Thus, +we add the effect of the GW emission into the Schr¨odinger +equation as +idc1 +dt = +� +−(Ω − Ωres) + iω(1) +I +− iΓGW +� +c1 + ηc2 , +(24) +idc2 +dt = ηc1 + +� +(Ω − Ωres) + iω(2) +I +� +c2 , +(25) +where |ci(t)|2= Mc,0ni(t)/µ. Here, ΓGW represents the +decay rate through the GW emission, whose explicit ex- +pression does not become necessary below. +4 Strictly speaking, we should take the mass of the one paired +with the companion as M + Mc. However, since the cloud mass +is small compared to the central BH mass, we approximated it +as M0. +From the above, the variables in this problem are +{M, J, Ω, c1, c2}, and we should solve the Eqs. (19), +(20), (23), (24), and (25). However, because of the highly +oscillatory behavior of the solutions for Eqs. (24) and +(25), it is difficult to solve these equations for a long time +with sufficient accuracy. To overcome this difficulty, we +derive a set of approximate equations that can be solved +easily. +C. +Adiabatic elimination +Here, we take advantage of the fact that the decay rate +of the second mode |ω(2) +I | is large compared to the transi- +tion rate due to the mixing term η around the resonance +frequency. Indeed, their ratio is estimated as5 +|ω(2) +I | +η +∼ 8 × 102 +�10−3 +q +� �0.1 +α +� +. +(26) +In this case, we can carry out an adiabatic elimination +of the second mode and discuss with only the particle +number of the first mode. +First, we redefine the variables as +˜ci(t) = e +−i +� t dt′� +(Ω−Ωres)−iω(1) +I ++iΓGw +� +ci(t) , +(27) +for i = 1, 2. Then, we can rewrite Eqs.(24) and (25) as +id˜ci +dt = +� +j=1,2 +˜Hij˜cj , +˜H = +� +0 +η(t) +η(t) ∆(t) + iΓ(t) +� +, (28) +with +∆(t) = 2 (Ω(t) − Ωres(t)) , +(29) +Γ(t) = ω(2) +I (t) − ω(1) +I (t) + ΓGW(t) . +(30) +Redefined particle numbers |˜ci|2 are related to |ci|2 as +|˜ci(t)|2= e−2 +� t dt′(ω(1) +I +−ΓGW)|ci(t)|2 . +(31) +Now, we write +˜c2(t) = y(t)e−i +� t +−∞ dt′(∆+iΓ) . +(32) +Substituting this into Eq.(28), we have +dy +dt = −iη˜c1ei +� t +−∞ dt′(∆+iΓ) . +(33) +By integrating this, we formally obtain +y(t) = −i +� t +−∞ +dt′ η˜c1ei +� t′ +−∞ dt′′(∆+iΓ) . +(34) +5 Here, we approximate |ω(2) +I +|≃ +1 +48 µχα8 and χ = χcrit ≃ 4α. + +6 +Total +w/o backreaction +GW only +-1.0 × 106 -500000 +0 +500000 +1.0 × 106 +1.5 × 106 +10-15 +10-11 +10-7 +0.001 +2 γ t +n1(t) +Total +w/o backreaction +-1.0 × 106 +-500000 +0 +500000 +1.0 × 106 +1.5 × 106 +0.98 +0.99 +1.00 +1.01 +1.02 +2 γ t +Ω(t)/Ω0 +FIG. 2. +Evolution of the normalized particle number of the first mode n1(t) (left) and the orbital frequency Ω(t) (right) +around the resonance frequency for q = 10−4, α0 = 0.1 and Mc,0 = 10−3M0. Blue solid lines show the results of solving all +equations, and orange dashed lines show the results without taking into account the backreaction to the orbital motion and the +mass and spin of the central BH. Green dashed line, in the left panel, shows the evolution of n1(t) considering only the effect +of the GW emission. +-1.0 × 106-500000 +0 +500000 1.0 × 106 1.5 × 106 +0 +1. × 10-6 +2. × 10-6 +3. × 10-6 +4. × 10-6 +2 γ t +(M(t)-M0)/M0 +-1.0 × 106-500000 +0 +500000 1.0 × 106 1.5 × 106 +0 +1. × 10-6 +2. × 10-6 +3. × 10-6 +4. × 10-6 +5. × 10-6 +2 γ t +(J(t)-J0)/M02 +-1.0 × 106-500000 +0 +500000 1.0 × 106 1.5 × 106 +-2.5 × 10-9 +-2. × 10-9 +-1.5 × 10-9 +-1. × 10-9 +-5. × 10-10 +0 +2 γ t +χ(t)- χcrit(t) +FIG. 3. +Evolution of the central BH mass M(t) (left), the angular momentum J(t) (middle) and the deviation from the +critical spin χ(t) − χcrit (right) for q = 10−4, α0 = 0.1 and Mc,0 = 10−3M0. Due to the absorption of particles belonging to +the primary cloud, the mass and the angular momentum of the BH increase slightly, but it maintains the BH spin parameter +slightly below the threshold value of the superradiance condition. +If |∆ + iΓ|≫ η, we can assume that the change rate of +˜c1(t) is much slower than |∆+iΓ|. Then, we can carry out +repeated integration by parts of the integral in Eq. (34), +to obtain an expansion in the inverse power of |∆ + iΓ|. +At the leading order of this expansion, we have +y(t) = − +η +∆ + iΓ˜c1ei +� t +−∞ dt′(∆+iΓ) . +(35) +Then, substituting this expression for y(t) into ˜c2 in the +equation for d˜c1/dt, Eq. (28), and integrating it, we ob- +tain +˜c1(t) = exp +� +i +� t +−∞ +dt′ +η2 +∆ + iΓ +� +. +(36) +From the above expressions, we find that the change rate +of the amplitude ˜c1 is much smaller than |Γ|. Thus, from +Eq. (26), the assumed conditions are all satisfied. Finally, +we can write the redefined particle number for each mode +as +|˜c1(t)|2 = exp +� +2 +� t +−∞ +dt′ +Γη2 +∆2 + Γ2 +� +, +(37) +|˜c2(t)|2 = +η2 +∆2 + Γ2 |˜c1(t)|2 . +(38) +Under this approximation, the equations that we need +to solve are Eqs. (19), (20) , (23), and +dn1 +dt = 2ω(1) +I n1 + +2Γη2 +∆2 + Γ2 n1 − +1 +Mc,0 +dEGW +dt +, +(39) +with +n2 = +η2 +∆2 + Γ2 n1 . +(40) +The last term of Eq. (39) comes from the iΓGW in the +exponential of Eq. (31), and can be identified with the +right-hand side of Eq. (15). +In practical calculations, +ΓGW should be so small compared to |ω(2) +I | that we can +neglect it in Γ (Eq. (30)). We also neglect the time deriva- +tive of η and the higher order term of |∆ + iΓ|−1. Now, +the set of variables to be solved are {M, J, Ω, n1}, and +we can easily solve the equations numerically for a wide +range of parameters. +IV. +RESULTS +In this section, we show the evolution of the system +obtained by solving the equations we formulated in the + +7 +-30 +-20 +-10 +0 +10 +20 +30 +1. × 10-19 +1. × 10-14 +1. × 10-9 +1. × 10-4 +t/ts +Mc(t)/M0 +α0 +0.05 +0.10 +0.15 +0.20 +0.25 +FIG. 4. +Dependence of the evolution of the cloud on the +gravitational fine structure constant α0. Each line shows the +evolution of the cloud mass for q = 10−4, Mc,0 = 10−3M0 and +various α0. The cloud mass at a late epoch monotonically +increases, as α0 increases. +-30 +-20 +-10 +0 +10 +20 +30 +1. × 10-19 +1. × 10-14 +1. × 10-9 +1. × 10-4 +t/ts +Mc(t)/M0 +log10q +-6.0 +-5.5 +-5.0 +-4.5 +-4.0 +-3.5 +-3.0 +FIG. 5. +Dependence of the cloud on the mass ratio q. Each +line shows the evolution of the cloud mass for α0 = 0.1, +Mc,0 = 10−3M0 and various q. As q becomes smaller, the +timescale of the binary evolution becomes longer, and thus +the decay due to the GW emission becomes dominant. +preceding section. In addition, we discuss their implica- +tions for observable signatures. +A. +Time evolution +We first discuss the initial conditions. To form a some- +what large cloud, the BH must have a large spin when it +is formed. However, the growth timescale of the cloud is +much faster than the timescale of the binary evolution, +and hence the BH spin will be quickly reduced to the +threshold value for the superradiance of the dominant +cloud. Thus, we set the initial BH spin to the threshold +value, J0 = acritM0. Also, how to choose the initial time +is not trivial because of the decay of the cloud through +the GW emission. From the analysis of the simplified toy +model in Appendix B, we can estimate the “start time” +at which the tidal field begins to be relevant as +t ∼ − +� +1 + η2 +γ +� |ω(2) +I | +2γ +≡ −ts . +(41) +We adopt t0 = −30ts evaluated with α = α0, Ω = Ω0 +and a = acrit as the the initial time. +Now, the initial condition of this system is param- +eterized by {q, α0, Mc,0}. +First, let us discuss the re- +sults for the fiducial set of parameters: {q = 10−4, α0 = +0.1, Mc,0 = 10−3M0}. +The time evolution of the nor- +malized particle number of the primary cloud n1(t) and +that of the binary’s orbital frequency Ω(t) are shown in +Fig. 2. Before reaching the resonance frequency, the par- +ticle number decreases mainly through the GW emission. +However, since the resonance band is widened due to the +presence of rapid decay of the secondary mode, charac- +terized by ω(2) +I , the orbital frequency is slightly modified +by the effect of transition, even in this stage. +Then, when the orbital frequency gets close to the res- +onance frequency, the tidal interaction works more effi- +ciently. The particles in the first mode are transferred to +the second mode, and the number n1 decreases dramat- +ically. +With the transition, the angular momentum of +the cloud is transferred to the binary orbital motion, and +the orbital frequency stagnates around the resonance fre- +quency. Here, we should note that, because of this stag- +nation, the duration to pass through the resonance band +becomes much longer and the net transition rate is much +larger than the case when the backreaction is neglected. +After the resonance, the particle number is exponen- +tially reduced owing to the backreaction to the central +BH shown in Fig. 3. Let us explain the reason why it +can give such a large influence on the cloud decay after +the resonance. Initially, the superradiance condition of +the primary cloud is saturated, i.e., ω(1) +I += 0. However, +once even a small number of particles are transferred to +the second mode, which has an angular momentum in +the opposite direction to the central BH spin, and is ab- +sorbed by the BH, the BH spin decreases slightly. Then, +the first mode becomes a non-superradiant mode, and +the particles belonging to the primary cloud also begin +to be absorbed by the BH. Thus, the BH mass and angu- +lar momentum gradually increase maintaining the spin +parameter slightly below the threshold value until the +resonant transition becomes more efficient. +At around the peak of the resonance, the particle num- +ber of the second mode increases, and the flux to the BH +of the second mode with negative angular momentum +dominates that of the first mode with a positive spin. +After passing the resonance frequency, the flux of the +first mode dominates again, but at that time there are +not enough particles left to spin-up the BH beyond the +superradiance threshold. As a result, the BH spin settles +to a value slightly below the threshold for the first mode +to be superradiant. Although the deviation from the crit- +ical spin is tiny, |ω(1) +I | is sufficiently large to eliminate the +cloud within the timescale of the binary inspiral. +In summary, the cloud, first, dissipates through the +GW emission. +Then, the particle number of the first +mode decreases dramatically with the resonant transi- +tion, and the transferred particles to the second mode +are absorbed by the BH immediately. After that, the pri- + +8 +-30 +-20 +-10 +0 +10 +20 +1. × 10-19 +1. × 10-14 +1. × 10-9 +1. × 10-4 +t/ts +Mc(t)/M0 +α0=0.2, q=10-3 +log10Mc,0/M0 +-10 +-8 +-6 +-4 +-2 +-30 +-20 +-10 +0 +10 +20 +-1.5 × 10-6 +-1. × 10-6 +-5. × 10-7 +0 +t/ts +χ(t)- χcrit(t) +log10Mc,0/M0 +-10 +-8 +-6 +-4 +-2 +FIG. 6. +Case 1. Evolution of the cloud mass (top) and the +BH spin (below) for α0 = 0.2, q = 10−3 and various initial +cloud mass Mc,0. Black dotted line in the below panel shows +the minimum value of the spin estimated in Sec. IV C. In this +case, the particle number after the transition is too small to +spin-up the BH after the transition for a large initial cloud +mass. The small initial cloud mass such that the BH spin- +down is negligible gives the largest final cloud mass. +mary cloud decreases exponentially due to the BH spin- +down below the superradiance threshold. +We also show the parameter dependence of this sys- +tem. In Fig. 4 and Fig. 5, we show the evolution of the +particle number for the same parameter but varying α0 +and q, respectively. If we neglect the backreaction and +GW emission, and approximate the binary orbital fre- +quency evolution by a linear function of t, the survival +probability of the primary cloud is analytically evaluated +as exp(−πη2/γ) [27]6. +This means that the efficiency +of the tidal effect is determined by the product of the +amplitude of the tidal perturbation η and the timescale +passing through the resonance band η/γ. This measure of +the tidal effect η2/γ is proportional to qα−11/3 +0 +for q ≪ 1. +Thus, the cloud mass after the resonance becomes tiny, +when α0 is small and q is somewhat large. +B. +Initial and final cloud mass +In terms of observation, it is interesting to clarify how +much of the cloud can remain after the satellite passes +6 Surprisingly, this result is not changed by the presence of ω(2) +I +. +-30 +-20 +-10 +0 +10 +20 +30 +40 +1. × 10-10 +1. × 10-7 +1. × 10-4 +0.1 +t/ts +Mc(t)/M0 +α0=0.2, q=10-4 +log10Mc,0/M0 +-10 +-8 +-6 +-4 +-2 +-30 +-20 +-10 +0 +10 +20 +30 +40 +-1.5 × 10-8 +-1. × 10-8 +-5. × 10-9 +0 +t/ts +χ(t)- χcrit(t) +log10Mc,0/M0 +-10 +-8 +-6 +-4 +-2 +FIG. 7. +Case 2. The same plot as Fig. 6, but for q = 10−4. In +this case, since the transition rate is not large, there is enough +particle number left to spin-up the BH after the transition for +a large initial cloud mass. +Thus, the cloud mass does not +decrease at the late epoch, and the largest initial cloud mass +gives the largest final cloud mass. +through the resonance frequency. The evolution of this +system and the fate of the cloud also depend on the ini- +tial cloud mass. If there are no processes that prevent the +cloud’s growth and the BH has nearly extremal spin when +it forms, the cloud mass can be estimated as ∼ αM [17]. +In reality, however, there can be other dissipation pro- +cesses besides GW emission, such as dissipation due to +axion’s self-interaction [40, 43]. +Therefore, it is worth +discussing the dependence on the initial cloud mass. +We find that the initial value of the cloud mass that +maximizes the final cloud mass is mainly determined by +the value of the BH spin after the resonance. It can be +classified into two cases, which we describe below. We +show the example of the cloud mass and BH spin evolu- +tion for the initial cloud mass from 10−10M0 to 10−1M0 +in Fig. 6 (case 1) and Fig. 7 (case 2). Here, we take the +final time as τbin/4, where τbin = Ω0/γ is the timsescale +of the binary evolution (see also Appendix A). +In case 1 (Fig. 6), for large initial cloud mass, the cloud +mass decreases exponentially due to the BH spin-down. +In this case, since the transition rate is large, there are +not enough particles left to spin-up the BH after the res- +onance. On the other hand, for somewhat small initial +cloud mass, the particle number is too small to spin-down +the BH efficiently from the beginning. In this case, the +absorption to the BH can be neglected, and the final mass +is determined only by the transition due to the tidal in- + +9 +0.10 +0.15 +0.20 +0.25 +0.30 +1.×10-6 +1.×10-5 +1.×10-4 +0.001 +0.010 +α0 +q +log10Mc,fin/M0 +-14 +-13 +-12 +-11 +-10 +-9 +-8 +-7 +-6 +FIG. 8. +Possible maximum final mass of the cloud after the +hyperfine resonance. The area above the red boundary be- +longs to the case1 (e.g., Fig. 6), and the area below it belongs +to the case2 (e.g., Fig. 7). +teraction. Thus, the case with such a small initial cloud +mass gives the maximum final cloud mass, for example, +Mc,0 = 10−9M0 in Fig. 6. +In case 2 (Fig. 7), for the largest initial cloud mass +(Mc,0 = 10−1M0), the cloud mass does not decrease at +the late epoch in this timescale. This is because the tran- +sition rate is small and there are enough particles left to +spin-up the BH by almost the threshold value of the su- +perradiance after the resonance. Thus, in this case, the +largest initial cloud mass simply gives the maximum final +cloud mass. +We summarize the possible maximum final mass Mc,fin +of the cloud after the resonance in the parameter space +(α0, q) in Fig. 8. We take 10−1M0 as the largest initial +cloud mass, and contours below 10−15M0 are not shown. +The area above the red boundary belongs to the case1, +and the area below it belongs to the case2. In case1 re- +gion, the final cloud mass is mainly determined by the +transition rate, i.e., the strength of the tidal interaction +characterized by η2/γ. Thus, for small α0 and somewhat +large q, the cloud hardly remains. In case2 region, the fi- +nal cloud mass is mainly determined by the GW emission. +For small α0 and q, the timescale of the binary evolution +becomes large, and thus the cloud has small mass by the +time orbital frequency reaches around the resonance. As +a result, we find that the largest final mass of the cloud is +∼ 10−5M0, which is achieved at α0 ≳ 0.2 and q ∼ 10−3. +C. +BH spin-down +In this subsection, we discuss the impact on the statis- +tical distribution of BH spin. If axions exist, most of BHs +which experienced sufficiently large spin-up in the past +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +1.×10-6 +1.×10-5 +1.×10-4 +0.001 +0.010 +0.100 +1 +α0 +q +log10( χcrit- χmin) +-14 +-12 +-10 +-8 +-6 +-4 +-2 +FIG. 9. +Approximate minimum value of the spin parameter +of the central BH obtained by dχ/dt = 0 in Eq. (42). +It +gives an estimation of the upper limit of the deviation from +the critical spin, which can be reached by the BH spin-down. +Blue solid line shows the boundary below which the hyperfine +transition is relevant as Fig. 1. +are expected to remain at the critical spin corresponding +to the threshold for the superradiance (Eq. (7)). Such +an accumulation of the spin distribution can be an ob- +servational signature of the existence of axions [11, 12]. +However, as we saw in the preceding subsections, axions +transferred to the mode with m = −1 by the tidal inter- +action make the BH spin smaller than the critical spin. +Then, the question is, how small can the BH spin be? +To answer it, we analyze the evolution of the BH spin +parameter. From Eqs. (19) and (20), we have +dχ +dt = −2 χ +M +dM +dt + +1 +M 2 +dJ +dt += 4χMc,0 +M (ω(1) +I n1 + ω(2) +I n2) ++ 2 +α +Mc,0 +M (−ω(1) +I n1 + ω(2) +I n2) . +(42) +For χ < χcrit (, i.e. ω(1,2) +I +< 0), the first term on the +right-hand side is always negative. Near the resonance, +the flux of the second mode can be dominant, at which +point the second term is also negative. +On the other +hand, when n2 decreases and the flux of the first mode +becomes dominant, the second term becomes positive. +Thus, we can estimate the minimum value of the BH +spin parameter achieved by the reabsorption of trans- +ferred axions as χmin satisfying dχ/dt = 0 around the +resonance. +Here, we use the approximation obtained in Eq. (40) +for n2. In particular, near the resonance, we can write +n2 ≃ η2 +Γ2 n1 . +(43) + +10 +Substituting it in Eq. (42) and approximating M ≃ M0 +and α ≃ α0, we can find the root of dχ/dt = 0 numeri- +cally. In Fig. 9, we show the deviation of χmin obtained +in this way from the critical spin χcrit for the param- +eter space (α0, q). +However, it is important to stress +that the deviation obtained here is only an approximate +upper bound. +In fact, if the cloud mass is too small, +χcrit(dχ/dt)−1 can be larger than the timescale of binary +evolution as the cloud mass decreases. In that case, the +BH spin-down stops before reaching the χmin. In Fig. 6 +and Fig. 7, we show the evolution of the BH spin, with +the dotted line corresponding to χmin. When the cloud +mass is somewhat large, the BH spin can only go down +to about χmin at most. On the other hand, if the cloud +mass is too small, BH spin-down terminates before reach- +ing χmin, and the absorption to the BH is negligible. In +particular, although it seems that the deviation of χmin +from the critical spin for large α0 and small q can be +O(0.1) from Fig. 9, in that region, the timescale of the +binary evolution becomes small and there are no enough +time to spin-down the BH. Therefore, although the spin- +down due to the absorption can be sufficiently large to +deplete the cloud, it would not affect the constraints on +axions from the BH spin measurements. +D. +Modification of the orbital frequency +Next, we discuss the modification of the GW frequency +evolution at around the resonance. The GW frequency +at which resonance occurs is given by [26] +fres = Ω0 +π = 2.2 mHz +1 +1 + 4α2 +0 +� α0 +0.1 +�7 �10M⊙ +M +� +. (44) +For typical binary systems with a supermassive BH hav- +ing an extreme mass ratio companion, the resonance fre- +quency is too low to detect. However, GWs at around +the resonance frequency from an intermediate mass BH +accompanied by a stellar mass or an even smaller mass +exotic compact object could be observed by space-based +GW detectors, such as LISA. +Around the resonance frequency, the orbital frequency +stagnates due to the angular momentum transfer asso- +ciated with the transition. This backreaction effect also +causes the delay of the rapid decrease of the cloud and +enhances the transition rate. +In Fig. 10, we show the +evolution of the cloud mass and the orbital frequency +for α0 = 0.1, q = 10−5 and various initial cloud mass. +When the cloud mass is large enough, this backreaction +greatly affects the evolution. We can estimate the thresh- +old value of the cloud mass before the transition for the +backreaction works effectively from Eq. (23). For sim- +plicity, neglecting the GW emission from the cloud and +considering only the primary cloud, the orbital evolution +around the resonance is approximated as +dΩ +dt ≃ γ + R Ω0 +M 2 +0 +dJ(1) +c +dt +. +(45) +Here, from Eq. (39), the time derivative of J(1) +c +is given +by +dJ(1) +c +dt +≃ Mc +µ +2η2 +ω(2) +I +. +(46) +For the orbital frequency to stagnate, in the right-hand +side of Eq. (45), the first term γ (GW radiation reaction) +and the second term must be comparable. Thus, we can +estimate the threshold value of the cloud mass required +for the backreaction to work by equating these terms. +We denote it as Mc,float, and it is given as +Mc,float = γ|ω(2) +I |α +2RΩ0η2 M +≃ 9.5 × 10−8M0(1 + q)4/3 � α0 +0.1 +�16/3 +. +(47) +Therefore, even with a small mass of the cloud, we can +expect that this modification can be a clear signature of +the presence of an axion condensate. +Unfortunately, the timescale of the binary evolution +τbin is typically much longer than the observation time +(≲ 10 yr). At first glance, it seems difficult to resolve +the degeneracy with the uncertainties in the chirp mass +and the mass ratio by observing the time derivatives of +the GW frequency ˙f and ¨f. However, we point out that +f ¨f/ ˙f 2 can be a good indicator of deviation from clean +binaries. If the binary system is clean and the mass ratio +is sufficiently small, q ≪ 1, this non-dimensional quantity +becomes a model-independent constant, i.e., f ¨f/ ˙f 2 = +11/3 in the early stage of the inspiral. +It would be natural to assume that the cloud mass is +bounded from above by the mass where the GW emis- +sion timescale τGW equals the timescale of the binary +evolution τbin (see Appendix. A). Then, around the reso- +nance, the cloud mass, reduced only by the GW emission, +is bounded by +Mc,GW = M 2 +τbin +α−14 +C +≃ 8.9 × 10−4M0 +q +(1 + q)1/3 +� α0 +0.1 +�14/3 +. +(48) +In Fig. 11, we show the value of the indicator f ¨f/ ˙f in +the presence of a cloud with Mc,0 = Mc,GW, for α0 = +0.1 and α0 = 0.2. They show that the deviation from +clean binaries can be larger than O(1), even if the axion +cloud has only a tiny fraction of the mass of the central +BH. While q dependence on the indicator for the same +cloud mass is weak7, Mc,GW is approximately linearly +proportional to q. Thus, when the mass ratio q is too +small, the effect of the angular momentum transfer due +to the tidal interaction also becomes small. +7 In Eq. (23), the main contribution to the square brackets in the +second term of the right-hand side is ˙J(1) +c +. From Eq. (39), ˙J(1) +c +is roughly proportional to q2 around the resonance. Thus, ˙Ω ≃ +O(q). One can find that ¨Ω ≃ O(q2) by differentiating Eq. (23). + +11 +-30 +-20 +-10 +0 +10 +20 +30 +1. × 10-8 +1. × 10-5 +0.01 +t/ts +Mc(t)/M0 +α0=0.1, q=10-5 +log10Mc,0/M0 +-10 +-8 +-6 +-4 +-2 +-30 +-20 +-10 +0 +10 +20 +30 +0.990 +0.995 +1.000 +1.005 +1.010 +t/ts +Ω(t)/Ω0 +α0=0.1, q=10-5 +log10Mc,0/M0 +-10 +-8 +-6 +-4 +-2 +FIG. 10. +Evolution of the cloud mass (left) and the orbital frequency (right) for α0 = 0.1, q = 10−5 and various initial cloud +mass Mc,0. Black dotted line in the left panel shows the threshold value of the cloud mass required for the backreaction to +work effectively, obtained in Eq. (47). Black dashed line in the right panel shows the evolution of the orbital frequency in the +clean binary. +-4 +-2 +0 +2 +4 +-50 +0 +50 +100 +t/ts +f f +.. +/ f 2 +α0=0.1 +log10q +-6.0 +-5.5 +-5.0 +-4.5 +-4.0 +-3.5 +-3.0 +-4 +-2 +0 +2 +4 +-5 +0 +5 +10 +15 +20 +t/ts +f f +.. +/ f 2 +α0=0.2 +log10q +-6.0 +-5.5 +-5.0 +-4.5 +-4.0 +-3.5 +-3.0 +FIG. 11. +Indicator in GW frequency of the presence of the cloud f ¨f/ ˙f 2 around the resonance t = 0 for various q, α0 = 0.1 +(left) and α0 = 0.2 (right). The initial cloud mass is set where the timescale of the GW emission and the binary evolution +are equal, i.e., Mc,0 = Mc,GW in Eq. (48). If the binary system is clean, f ¨f/ ˙f 2 = 11/3 model-independently. Around the +resonance, this quantity can be largely changed with the level transition of the cloud. +V. +SUMMARY AND DISCUSSION +In this paper, we have investigated the evolution of in- +spiralling binary systems accompanying an axion cloud +before and after the orbital frequency crosses the hyper- +fine resonance frequency, focusing on small mass ratio +(q ≪ 1) cases. Our main interest is how the hyperfine +level transition proceeds and affects the observational sig- +natures. From the comparison of timescales, we found it +necessary to take into account the following components; +the decaying process of the axion in the destination mode +of the hyperfine transition (imaginary part of the eigen- +frequency), the GW emission from the cloud, and the +backreaction to the orbital motion and that to the mass +and spin of the central BH. We presented a formulation to +examine the evolution of the cloud, the central BH, and +the orbital motion including all these effects. In particu- +lar, carrying out the adiabatic elimination of the degree +of freedom of the amplitude of the second mode allows +us to examine a wide parameter region numerically, and +gives useful expressions for analyzing the behavior of the +system. +Our results show that the cloud mass is typically signif- +icantly reduced by the GW emission before the resonant +transition occurs. If q is sufficiently large or α is suffi- +ciently small, axions in the m = 1 fastest growing mode +are almost transferred to the m = −1 mode, which has +angular momentum in the opposite direction to the BH +spin and is easily absorbed by the BH. Then, the pri- +mary cloud becomes non-superradiant and can fall into +the BH, which results in the increase of the BH angular +momentum, counter-intuitively. +However, the increase +of the BH mass dominates to maintain the first mode +to be non-superradiant. As a result, the cloud almost +completely disappears by the absorption to the BH. On +the other hand, if q is extremely small or α is sufficiently +large, the transition rate due to tidal interaction is small. +In such cases, since there are enough particles left to spin- +up the BH again after the transition, the absorption to +the BH at the late epoch can be neglected, and the cloud +does not disappear completely. +However, it dissipates + +12 +mainly owing to the GW emission before the transition, +and the maximum mass of the cloud that can remain af- +ter the resonance is ∼ 10−5M0 at most. How much of +axion clouds can remain after the resonance might have +an implication to the survey of the cloud as an environ- +ment around the BH, such as [22, 23, 25]. +We also discussed the implication to the observational +signatures. First, we confirmed that the time variation of +the BH spin around the transition is tiny, although this +tiny variation can be important to determine the evolu- +tion of the cloud. This result makes robust the constraint +on the existence of an axion field obtained through the +BH parameter distribution measured by GWs from bi- +nary systems. Second, we studied the influence of the +transition on the inspiral GW waveform. We found that +even for extremely small cloud mass, the backreaction to +the orbital motion works effectively, and the frequency +stagnates around the resonance frequency. In particular, +the combination f ¨f/ ˙f 2 is affected by the transition to a +detectable level. Therefore, for example, the GWs from +an intermediate mass BH associated with a small mass +satellite can be a good target for the axion search. We +need more extensive analysis to conclude the observabil- +ity of axion clouds with the modification of the wave- +form. Furthermore, the generalization of the inspiral or- +bit and the discrimination from other environmental ef- +fects would be important. We leave them as future work. +ACKNOWLEDGMENTS +T. Takahashi was supported by JST, the establishment +of university fellowships towards the creation of science +technology innovation, Grant Number JPMJFS2123. +TT is supported by JSPS KAKENHI Grant Number +JP17H06358 (and also JP17H06357), A01: Testing grav- +ity theories using gravitational waves, as a part of the in- +novative research area, “Gravitational wave physics and +astronomy: Genesis”, and also by JP20K03928. HO is +supported by Grant-in-Aid for JSPS Fellows JP22J14159. +Appendix A: Timescales +In this appendix, we summarize the timescales involved +in our problem. +Binary evolution: +The timescale of the binary evolution due to the +GW radiation at the resonance frequency Ω0 is +given by +τbin = Ω0 +γ = 5 +96M (1 + q)1/3 +q +(MΩ0)−8/3 , +(A1) +where γ is defined by Eq. (22). +Transition: +The resonance bandwidth can be estimated as +∆Ω ∼ 2η. Hence, if one can neglect the instability +of the mode of the transition destination and lin- +earize the orbital evolution, the timescale for pass- +ing through the resonance band is given by +τtrans = 2η +γ . +(A2) +Decay of the secondary cloud: +The secondary cloud decreases as ∼ e−2|ω(2) +I +|t, and +the timescale is given by +τinst = |ω(2) +I |−1 . +(A3) +GW emission of the primary cloud: +From the energy conservation +˙Mc = − ˙EGW (see +Eq. (14)), one can obtain +Mc(t) = +Mc,0 +1 + (t − t0)/τGW +. +(A4) +Here, the timescale is given by +τGW = 1 +C +M 2 +Mc,0 +α−14 . +(A5) +Parameter dependencies of the timescales mentioned +above are summarized in Fig. 12 and Table I. +Appendix B: Toy model for adiabatic elimination +In this appendix, we discuss the approximation used in +Sec. III C with a simplified toy model. Consider the two +level transition described by the Schr¨odinger equation +i d +dt +� +c1 +c2 +� += +� +0 +η +η ∆(t) − iωI +� � +c1 +c2 +� +. +(B1) +Let η and ωI be constants and ∆(t) = 2γt (γ is constant). +This model is a simplification of ignoring all backreac- +tions, GW emissions and linearizing the binary evolution +in the problem we investigate. +If ωI = 0, this model +is known as Landau-Zener problem [27, 51, 52]. Now, +we want to study the level transition to the decaying +mode (ωI > 0). For this problem, we have an exact ana- +lytic solution with the initial conditions c1(−∞) = 1 and +c2(−∞) = 0 as [53, 54] + +13 +10-6 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +0.001 +1 +1000 +106 +109 +1012 +q +Timescale [yrs] +Binary evolution (hyperfine) +Binary evolution (Bohr) +Transition (hyperfine) +Transition (Bohr) +Decay |ω21-1 +-1 +GW +FIG. 12. +Timescales involved in the resonant transition of axion clouds in binary systems for α = 0.1 and M = M⊙. Blue +and yellow solid lines show the timescale of the binary evolution and the transition at the hyperfine resonance, respectively. +Blue and yellow dashed lines show the same quantities, but for the typical Bohr transition (|211⟩ → |31 − 1⟩). Green and red +lines show the timescales of decay of the secondary cloud (|21 − 1⟩) and of the GW emission of the primary cloud (|211⟩) for +Mc,0 = 0.1M, respectively. +TABLE I. Timescales involved in the hyperfine resonance of axion clouds. +process +time +Binary evolution +τbin = 2.2 × 1013 s(1 + q)1/3 +q +� M +M⊙ +� � χ +0.4 +�−8/3 � α +0.1 +�−16 +Transition +τtrans = 1.3 × 1010 s +1 +(1 + q)2/3 +� M +M⊙ +� � χ +0.4 +�−5/3 � α +0.1 +�−13 +Decay of the secondary mode +τinst ≃ 5.9 × 105 s +� M +M⊙ +� � χ +0.4 +�−1 � α +0.1 +�−9 +GW emission +τGW = 2.0 × 1011 s +� M +M⊙ +� �Mc,0/M +0.1 +�−1 � α +0.1 +�−14 +|c1(t)|2 = e−ωIt− π +2 +η2 +2γ +���Diη2/2γ +� +ei 3π +4 ( +� +2γt − iωI/ +� +2γ) +���� +2 +, +(B2) +|c2(t)|2 = e−ωIt− π +2 +η2 +2γ η2 +2γ +���Diη2/2γ−1 +� +ei 3π +4 ( +� +2γt − iωI/ +� +2γ) +���� +2 +, +(B3) +where Dν(z) is the parabolic cylinder function. +Carrying out the adiabatic elimination as Sec. 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A 55, 2982 +(1997). + diff --git a/rdFPT4oBgHgl3EQf9zV6/content/tmp_files/load_file.txt b/rdFPT4oBgHgl3EQf9zV6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f688524944861f1d2a3559fc6483c97fffe1d5da --- /dev/null +++ b/rdFPT4oBgHgl3EQf9zV6/content/tmp_files/load_file.txt @@ -0,0 +1,1103 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf,len=1102 +page_content='Evolution of binary systems accompanying axion clouds in extreme mass ratio inspirals Takuya Takahashi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' ∗ Hidetoshi Omiya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' † and Takahiro Tanaka1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' ‡ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Kyoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Kyoto 606-8502,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Japan 2Center for Gravitational Physics and Qunatum Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Yukawa Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Kyoto University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Kyoto 606-8502,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Japan (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 2023) Superradiant instability of rotating black holes (BHs) leads to the formation of a cloud of ultralight bosons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' such as axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' When the BH with the cloud belongs to a binary system and is in an inspiraling orbit, the resonant transition between the axion’s bound states can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We study the history of the evolution of the binary system accompanying the cloud composed of the fastest growing mode, and its impact on the observational signatures, especially for small mass ratio cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, the hyperfine resonance, which has a very small resonance frequency, is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Therefore, due to the long timescale, we should take into account the decaying process of axions in the transition destination mode, the backreaction to the orbital motion and the central BH, and gravitational emission from the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We present a formulation to examine the evolution of the system around the resonance and useful expressions for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As a result, we found the mass of the cloud that can remain after the resonance is, at most, about 10−5 of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The maximum remaining cloud mass is achieved when the mass ratio of the binary is q ∼ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In addition, we show that the resonant transition hardly changes the BH mass and spin distribution, while the associated modification of the gravitational wave frequency evolution when the binary pass through the resonance can be a signature of the presence of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' INTRODUCTION Ultralight bosons, such as axions or axion-like parti- cles, can cause various phenomena in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Such particles are universally predicted by string theory [1, 2] and can be a candidate for dark matter [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' They can be weakly coupled to the Standard Model particles, but even in such a case the gravitational interaction with black holes (BHs) and related gravitational waves (GWs) can provide a new avenue to explore them observation- ally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The existence of massive bosonic fields induces the su- perradiant instability around rotating BHs [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Bosons with mass in the range 10−20 ∼ 10−10 eV have the Comp- ton wavelength comparable to the size of astrophysical BHs, and extract energy and angular momentum effi- ciently to form a condensate [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We refer to the condensate as an axion cloud and the composing parti- cles simply as axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The cloud formation makes astro- physical observable imprints, such as a forbidden region in the distribution of mass and spin of BHs [11–13] and continuous GW emission [14–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this paper, we focus on the cases where BHs with clouds belong to binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' GWs from the binary inspiral can be a signature to examine the environment around BHs including the cloud [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Axion clouds occupy a quasi-bound state of axions, which is usually the fastest growing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' During the inspiral phase, the ∗ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='takahashi@tap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='scphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='jp † omiya@tap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='scphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='jp ‡ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='tanaka@tap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='scphys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='jp tidal interaction from the companion acts as an oscil- lating tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It induces the resonant transition to another mode when the orbital frequency coincides with the phase velocity difference between the original mode of the cloud and the other [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The change of the orbital motion of the binary and the associated GW fre- quency due to the backreaction can also be a signature of the presence of the cloud [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' To clarify the impact on the observational signatures, it is important to un- derstand the history of the evolution during the inspiral phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If the separation of the binary is sufficiently small, the cloud configuration is tidally disrupted [31, 32], and the transition to unbound states occurs [31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, for binary systems formed with a sufficiently large sep- aration, the resonant transition should first occur with the smallest possible resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The frequency spectrum of axion eigenmodes possesses the structure of hyperfine splittings due to the rotation of the central BH [34], and the resonance frequency associated with the hyperfine splitting is the smallest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [31], we showed that, for nearly equal mass binaries, this hy- perfine resonance can be neglected since the resonance condition is not maintained long enough because of the decrease of the angular momentum of the cloud itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We also showed that, before the transition caused by the leading quadrupole moment of the tidal potential occurs, the cloud is disrupted by the effects of higher multipole moments, and finally the cloud is depleted as a result of transitions to unbound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In contrast to nearly equal mass binaries, for small mass ratio binaries, the hyperfine resonance should be considered because of a large backreaction to the orbital motion, which maintains the orbital frequency within the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='13213v1 [gr-qc] 30 Jan 2023 2 resonance band for a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It has great importance to examine the dynamics of small mass ratio binaries, because they are one of the main targets for future GW observations, such as LISA [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, because of the very long timescale of the binary evolution due to the radiation reaction, some effects that can be neglected for the transition for nearly equal mass binaries become relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' First, the decay of non-superradiant transition destina- tion modes and the backreaction to the central BH mass and spin become relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Since the resonance band is broadened corresponding to the imaginary part of the frequencies of decaying destination modes, the transition timescale staying within the resonance band becomes even longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Therefore, we should also take into account the GW emission from the cloud during the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We develop a formulation that includes all of these effects within the adiabatic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It is difficult to solve the originally obtained set of equations throughout the whole period across the resonance band, since the solu- tion oscillates rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' To overcome this difficulty, we also present a method to give an approximate solution with sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this paper, we consider axion clouds in a non- relativistic regime, and neglect the self-interaction of ax- ions, for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For a relativistic regime, the energy spectrum deviates significantly from the one obtained by non-relativistic approximation, and the transition to be considered can change [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In addition, the self- interaction can play an important role during the for- mation of the cloud [38–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we leave considering these effects as future work, to focus on the tidal effect in binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' II, we review the elements involved in the evolution of axion clouds in binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' III, we present a formulation for examining the hyperfine resonance in small mass ratio bi- naries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' IV, we discuss the results obtained using our formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Finally, we give a summary and conclu- sion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Throughout this paper, we use the unit with c = ¯h = G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' ELEMENTS INVOLVED IN THE EVOLUTION OF AXION CLOUDS In this section, we summarize the elements involved in describing the evolution of axion clouds, especially dur- ing the binary inspirals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Consider a scalar field (axion) of mass µ around a rotating BH belonging to a binary sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We denote the central BH mass by M and angular momentum by J = aM = χM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Formally, we can write the equation of motion for axion on a spacetime with the metric ˜gµν = gµν + hµν as (˜gµν ˜∇µ ˜∇ν − µ2)φ = 0 , (1) where gµν is the Kerr metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We consider the tidal field from the binary companion and the decay due to the gravitational wave emission from the cloud as contribu- tions to the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As we will see later, since there is a hierarchy of frequencies between them, we can treat them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We first review the features of ax- ion clouds in the unperturbed background, and later the effects of the tidal interaction and the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Energy spectrum and superradiance In the non-relativistic regime, it is appropriate to in- troduce a new complex scalar field variable ψ by φ = 1 √2µ � e−iµtψ + eiµtψ∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (2) We assume that ψ changes slowly in time compared to the timescale determined by µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, we can ignore the ∂2 t ψ term and rewrite the background equation of motion (1) as i ∂ ∂tψ = H0ψ , H0 = − 1 2µ∇2 − α r + O(α2) , (3) where we have introduced the gravitational fine struc- ture constant α ≡ Mµ, and this approximation is well justified for α ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Solving this equation with the in- going boundary condition at the BH horizon and the ex- ponentially decaying boundary condition at infinity, we have the quasi-bound eigenstate ϕnlm(r) that satisfies H0ϕnlm = (ωnlm − µ)ϕnlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' They are labeled by the principal, azimuthal and magnetic quantum numbers like a hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The eigenfrequency is approximately given by ωnlm = (ωR)nlm + i(ωI)nlm , (4) with [26, 34] (ωR)nlm = µ � 1 − α2 2n2 − α4 8n4 + (2l − 3n + 1)α4 n4(l + 1/2) + 2mχα5 n3l(l + 1/2)(l + 1) � , (5) (ωI)nlm = 2(r+/M)Cnlm(a, α)(mΩH − ωnlm)α4l+5 , (6) where r+ = M + √ M 2 − a2 is the horizon radius, ΩH = a/2Mr+ is the angular velocity of the BH horizon and the explicit form of Cnlm(a, α) can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [26]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As one can see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (6), the eigenfrequency of a mode satisfying ωR < mΩH has a positive imaginary part, and the cloud grows exponentially by the superra- diance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The mode |nlm⟩ = |211⟩ is the fastest growing 1 It was first derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [9], and corrected by a factor of 1/2 [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 3 mode for α ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The BH spin decreases as the cloud grows until the superradiance condition is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The critical spin at which the superradiance terminates is ap- proximately given by χcrit = 4mα m2 + 4α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (7) The real part of the eigenfrequency can be regarded as eigenenergy, and its degeneracy among the modes with only m being different is solved due to the rotation of the BH at the order of O(α5), which is called the “hyperfine” splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Tidal interaction When a BH accompanied by an axion cloud belongs to a binary system, the tidal field from the companion introduces a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The general state of the cloud can be expressed by ψ = � i ci(t)ϕi , (8) as a superposition of orthonormal eigenfunctions ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Un- der the same approximation taken in the preceding sub- section, the equation of motion with the perturbation is given by idci dt = � j � (ωj − µ)δij + � d3x ϕ∗ i V∗ϕj � cj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (9) For simplicity, we assume that the binary orbit is quasi- circular and on the plane perpendicular to the central BH spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' By multipole expansion, we can write the tidal field from the companion of mass M∗ at r(t) = (R∗(t), Θ∗(= π/2), Φ∗(t)) as V∗ = 1 2µhtt tidal = −qα � l∗m∗ 4π 2l∗ + 1 rl∗ < rl∗+1 > Y ∗ l∗m∗(Θ∗, Φ∗)Yl∗m∗(θ, φ) , (10) where q ≡ M∗/M is the mass ratio, r>(r<) is the larger (smaller) of r and R∗, and Ylm are the spherical har- monics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The angular velocity of the binary is defined by ˙Φ∗(t) = ±Ω(t), and the upper (lower) sign represents the case of co-rotating (counter-rotating) orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Since this interaction oscillates quasi-periodically, it works ef- ficiently only when the orbital angular velocity is close to the difference between the phase velocity of the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Therefore, it is sufficient to consider a two-mode subspace [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Tidal field mixes two modes, and the time evolution of particle number in each mode is, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (9), given by i ˙c = Hc (11) with H = � −∆E/2 + iω(1) I ηei∆mΦ∗ ηe−i∆mΦ∗ ∆E/2 + iω(2) I � , (12) where ∆E = ω(2) R − ω(1) R , ∆m = m2 − m1, and η(t) = ��� d3x ϕ∗ 2V∗ϕ1 ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' To remove the rapidly oscil- lating term, we perform the unitary transformation as c → U−1c and H → U†H U − i U† ˙U with the matrix U(t) = diag(ei∆mΦ∗/2, e−i∆mΦ∗/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As a result, we can describe the level transition due to the tidal field by H = � ± ∆m 2 (Ω − Ωres) + iω(1) I η η ∓ ∆m 2 (Ω − Ωres) + iω(2) I � , (13) where we defined the “resonance” frequency by Ωres = ±∆E/∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Now, we are interested in the time evolution of the occupation number of each state, |ci(t)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Gravitational wave emission After an axion cloud forms, it dissipates through the emission of GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we assume that the cloud is composed of a single mode as ψ = c1ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, we can neglect the GW emission due to the spontaneous level transition, and GWs are sourced by the pair-annihilation of axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The frequency of GWs is given by ωGW = 2ωR ∼ 2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The energy flux of GWs from the l = m = 1 cloud is given by [16] dEGW dt = C �Mc M �2 α14 , (14) where C is a numerical factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In our analysis, we adopt C = (484 + 9π2)/23040 calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, Mc is the mass of the cloud defined by Mc = − � d3x T tt, where T tt is the t-t component of the energy momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' According to this, the wave function ψ is normal- ized as |c1|2(= � d3x|ψ|2) = Mc/µ at the leading order in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' When we consider only the effect of GW emission, en- ergy conservation implies that ˙Mc = − ˙EGW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We set the initial mass of the cloud to Mc,0 at t = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we define the normalized particle number by n1(t) = µ|c1(t)|2/Mc,0, and write Mc(t) = Mc,0n1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Energy conservation reads dn1 dt = − C M �Mc,0 M � n2 1α14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (15) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' FORMULATION In this section, we first explain the setup of the problem that we consider and then give a formulation to investi- gate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='30 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='100 1 α q FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Parameter region where the hyperfine resonance is relevant to dissipate the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In the shaded region, the reso- nance sustains longer because the effect of the backreaction to the orbital motion is stronger than the effect of the reduction of the hyperfine splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The initial angular momentum of the cloud is set to Jc,0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [31] for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Setup We focus on the fastest growing mode |nlm⟩ = |211⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We consider the situation in which the cloud is initially composed of the single mode |211⟩, and the hyperfine level transition between |211⟩ and |21 − 1⟩ subsequently occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Note that this transition occurs only for co- rotating orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [31], we found that when the bi- nary mass ratio q is not too small, this transition does not significantly contribute to the dissipation of the cloud because of the reduction of the hyperfine splitting asso- ciated with the transfer of the angular momentum of the cloud to the orbital motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, when the mass ratio is somewhat small, the resonant tidal interaction at this hyperfine splitting frequency would largely affect the dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We show the parameter re- gion where we should consider the hyperfine resonance as a process that contributes to the cloud dissipation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We investigate the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For the transition between |211⟩ and |21 − 1⟩, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (5), the resonance frequency is given by2 Ωres = µ 12χα5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (16) This is smaller by a factor of α3 than that of the “Bohr” transition between modes with different values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' When we study the Bohr transition, ωI in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (13) 2 We do not include the contribution from the angular momentum of the cloud itself, focusing on the case where it is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' and GW flux are so small in the timescale for passing through the resonance band that we can usually neglect them3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, for hyperfine transition, binary evolu- tion around the resonance frequency is very slow and the timescale for passing through the resonance band can be large, especially for q ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In addition, since the angu- lar momentum of the cloud is transferred to the orbital motion, the timescale becomes even larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As a result, we should take into account not only the backreaction to the orbital motion, but also the backreaction to the mass and spin of the central BH and the effect of the GW emission from the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We summarize the timescales involved in the current problem in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In the following, we label the quantities associated with the mode |211⟩ by 1, and those with |21 − 1⟩ by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For these modes, the imaginary parts of the eigenfrequencies are given by ω(i) I = 1 24 r+ M �� 1 − χ2� + 4r2 +(miΩH − ωR)2� ×(miΩH − ωR)α9 , (17) where i is 1 or 2, and m1 = 1 and m2 = −1 represent the magnetic quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The mixing term in the Hamiltonian (13) is given by η = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 q 1 + q MΩ2 α3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (18) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Evolution of the system The dynamical timescale of the cloud can be estimated by ω−1 R ≃ µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It is always short compared to the growth/decay rate of the cloud, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', (ω(i) I )−1 ≫ µ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, we describe the evolution of the cloud and the cen- tral BH within the adiabatic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The local energy and angular momentum conservation at the BH horizon reads dM dt + 2ω(1) I M (1) c + 2ω(2) I M (2) c = 0 , (19) dJ dt + 2ω(1) I µ M (1) c − 2ω(2) I µ M (2) c = 0 , (20) with M (i) c = Mc,0ni(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we used the relation be- tween the energy flux and the angular momentum flux for each mode ˙J(i) c = (mi/ω(i) R ) ˙E(i) c and the approxima- tion ωR = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We denote the initial mass and angular momentum of the BH just before entering the resonance band by M0 and J0, and accordingly α0 = M0µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 3 When we consider a higher l mode, the transition to the mode with smaller l is allowed by the selection rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In that case, the decay rate of the second mode can be large, and it would be important [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 5 Next, we consider the evolution of the binary sys- tem at the leading post-Newtonian order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In clean bi- nary systems, angular momentum conservation implies ˙Jorb = −TGW, where Jorb = q(1 + q)−1/3M 5/3 0 Ω−1/3 is the orbital angular momentum and TGW is the torque caused by the radiation reaction due to the GW emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It can be rewritten as [49, 50] dΩ dt = γ � Ω Ω0 �11/3 , (21) γ Ω2 0 = 96 5 q (1 + q)1/3 (M0Ω0)5/3 , (22) where the reference frequency is chosen as Ω0 = (µ/12)(J0/M 2 0 )α5 0 (which is the “initial” resonance fre- quency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we add the cloud and the BH contri- butions to the total angular momentum conservation as ˙Jorb + ˙J + ˙J(1) c + ˙J(2) c + ˙JGW = −TGW, where ˙JGW = (1/µ) ˙EGW is the angular momentum flux of the GW from the cloud in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Note that we consider GW emis- sion only from the first mode |211⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (As we will see later, the particle number occupying the second mode, which is non-superradiant, is always tiny and does not contribute to the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=') Then, we obtain4 dΩ dt =γ � Ω Ω0 �11/3 + R � Ω Ω0 �4/3 Ω0 M 2 0 × � d dt � J + J(1) c + J(2) c � + 1 µ dEGW dt � , (23) with R = 3(1 + q)1/3q−1(M0Ω0)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We take Ω(t0) = Ω0(1 + (8/3)(γ/Ω0)|t0|)−3/8 as the initial value so that Ω = Ω0 at t = 0 when there are no clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Finally, we describe the level transition between two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It is described by the Schr¨odinger equation with the Hamiltonian (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Note that the particle number oc- cupying the first mode decreases due to the GW emission by pair annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Since the frequency of the emitted GW (ωGW ∼ 2µ) is much larger than that of the tidal field (Ωres ∼ µα6), we can treat them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, we add the effect of the GW emission into the Schr¨odinger equation as idc1 dt = � −(Ω − Ωres) + iω(1) I − iΓGW � c1 + ηc2 , (24) idc2 dt = ηc1 + � (Ω − Ωres) + iω(2) I � c2 , (25) where |ci(t)|2= Mc,0ni(t)/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, ΓGW represents the decay rate through the GW emission, whose explicit ex- pression does not become necessary below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 4 Strictly speaking, we should take the mass of the one paired with the companion as M + Mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, since the cloud mass is small compared to the central BH mass, we approximated it as M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' From the above, the variables in this problem are {M, J, Ω, c1, c2}, and we should solve the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (19), (20), (23), (24), and (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, because of the highly oscillatory behavior of the solutions for Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (24) and (25), it is difficult to solve these equations for a long time with sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' To overcome this difficulty, we derive a set of approximate equations that can be solved easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Adiabatic elimination Here, we take advantage of the fact that the decay rate of the second mode |ω(2) I | is large compared to the transi- tion rate due to the mixing term η around the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Indeed, their ratio is estimated as5 |ω(2) I | η ∼ 8 × 102 �10−3 q � �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (26) In this case, we can carry out an adiabatic elimination of the second mode and discuss with only the particle number of the first mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' First, we redefine the variables as ˜ci(t) = e −i � t dt′� (Ω−Ωres)−iω(1) I +iΓGw � ci(t) , (27) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, we can rewrite Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (24) and (25) as id˜ci dt = � j=1,2 ˜Hij˜cj , ˜H = � 0 η(t) η(t) ∆(t) + iΓ(t) � , (28) with ∆(t) = 2 (Ω(t) − Ωres(t)) , (29) Γ(t) = ω(2) I (t) − ω(1) I (t) + ΓGW(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (30) Redefined particle numbers |˜ci|2 are related to |ci|2 as |˜ci(t)|2= e−2 � t dt′(ω(1) I −ΓGW)|ci(t)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (31) Now, we write ˜c2(t) = y(t)e−i � t −∞ dt′(∆+iΓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (32) Substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (28), we have dy dt = −iη˜c1ei � t −∞ dt′(∆+iΓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (33) By integrating this, we formally obtain y(t) = −i � t −∞ dt′ η˜c1ei � t′ −∞ dt′′(∆+iΓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (34) 5 Here, we approximate |ω(2) I |≃ 1 48 µχα8 and χ = χcrit ≃ 4α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6 Total w/o backreaction GW only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 -500000 0 500000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 106 10-15 10-11 10-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 2 γ t n1(t) Total w/o backreaction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 500000 0 500000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='02 2 γ t Ω(t)/Ω0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Evolution of the normalized particle number of the first mode n1(t) (left) and the orbital frequency Ω(t) (right) around the resonance frequency for q = 10−4, α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 and Mc,0 = 10−3M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Blue solid lines show the results of solving all equations, and orange dashed lines show the results without taking into account the backreaction to the orbital motion and the mass and spin of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Green dashed line, in the left panel, shows the evolution of n1(t) considering only the effect of the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106-500000 0 500000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 106 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 2 γ t (M(t)-M0)/M0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106-500000 0 500000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 106 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 2 γ t (J(t)-J0)/M02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106-500000 0 500000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 10-9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-10 0 2 γ t χ(t)- χcrit(t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Evolution of the central BH mass M(t) (left), the angular momentum J(t) (middle) and the deviation from the critical spin χ(t) − χcrit (right) for q = 10−4, α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 and Mc,0 = 10−3M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Due to the absorption of particles belonging to the primary cloud, the mass and the angular momentum of the BH increase slightly, but it maintains the BH spin parameter slightly below the threshold value of the superradiance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If |∆ + iΓ|≫ η, we can assume that the change rate of ˜c1(t) is much slower than |∆+iΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, we can carry out repeated integration by parts of the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (34), to obtain an expansion in the inverse power of |∆ + iΓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' At the leading order of this expansion, we have y(t) = − η ∆ + iΓ˜c1ei � t −∞ dt′(∆+iΓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (35) Then, substituting this expression for y(t) into ˜c2 in the equation for d˜c1/dt, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (28), and integrating it, we ob- tain ˜c1(t) = exp � i � t −∞ dt′ η2 ∆ + iΓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (36) From the above expressions, we find that the change rate of the amplitude ˜c1 is much smaller than |Γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (26), the assumed conditions are all satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Finally, we can write the redefined particle number for each mode as |˜c1(t)|2 = exp � 2 � t −∞ dt′ Γη2 ∆2 + Γ2 � , (37) |˜c2(t)|2 = η2 ∆2 + Γ2 |˜c1(t)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (38) Under this approximation, the equations that we need to solve are Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (19), (20) , (23), and dn1 dt = 2ω(1) I n1 + 2Γη2 ∆2 + Γ2 n1 − 1 Mc,0 dEGW dt , (39) with n2 = η2 ∆2 + Γ2 n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (40) The last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (39) comes from the iΓGW in the exponential of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (31), and can be identified with the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In practical calculations, ΓGW should be so small compared to |ω(2) I | that we can neglect it in Γ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (30)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We also neglect the time deriva- tive of η and the higher order term of |∆ + iΓ|−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Now, the set of variables to be solved are {M, J, Ω, n1}, and we can easily solve the equations numerically for a wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' RESULTS In this section, we show the evolution of the system obtained by solving the equations we formulated in the 7 30 20 10 0 10 20 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-4 t/ts Mc(t)/M0 α0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Dependence of the evolution of the cloud on the gravitational fine structure constant α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Each line shows the evolution of the cloud mass for q = 10−4, Mc,0 = 10−3M0 and various α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The cloud mass at a late epoch monotonically increases, as α0 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 30 20 10 0 10 20 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-4 t/ts Mc(t)/M0 log10q 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Dependence of the cloud on the mass ratio q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Each line shows the evolution of the cloud mass for α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1, Mc,0 = 10−3M0 and various q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As q becomes smaller, the timescale of the binary evolution becomes longer, and thus the decay due to the GW emission becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' preceding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In addition, we discuss their implica- tions for observable signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Time evolution We first discuss the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' To form a some- what large cloud, the BH must have a large spin when it is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, the growth timescale of the cloud is much faster than the timescale of the binary evolution, and hence the BH spin will be quickly reduced to the threshold value for the superradiance of the dominant cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, we set the initial BH spin to the threshold value, J0 = acritM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Also, how to choose the initial time is not trivial because of the decay of the cloud through the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' From the analysis of the simplified toy model in Appendix B, we can estimate the “start time” at which the tidal field begins to be relevant as t ∼ − � 1 + η2 γ � |ω(2) I | 2γ ≡ −ts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (41) We adopt t0 = −30ts evaluated with α = α0, Ω = Ω0 and a = acrit as the the initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Now, the initial condition of this system is param- eterized by {q, α0, Mc,0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' First, let us discuss the re- sults for the fiducial set of parameters: {q = 10−4, α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1, Mc,0 = 10−3M0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The time evolution of the nor- malized particle number of the primary cloud n1(t) and that of the binary’s orbital frequency Ω(t) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Before reaching the resonance frequency, the par- ticle number decreases mainly through the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, since the resonance band is widened due to the presence of rapid decay of the secondary mode, charac- terized by ω(2) I , the orbital frequency is slightly modified by the effect of transition, even in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, when the orbital frequency gets close to the res- onance frequency, the tidal interaction works more effi- ciently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The particles in the first mode are transferred to the second mode, and the number n1 decreases dramat- ically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' With the transition, the angular momentum of the cloud is transferred to the binary orbital motion, and the orbital frequency stagnates around the resonance fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we should note that, because of this stag- nation, the duration to pass through the resonance band becomes much longer and the net transition rate is much larger than the case when the backreaction is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' After the resonance, the particle number is exponen- tially reduced owing to the backreaction to the central BH shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Let us explain the reason why it can give such a large influence on the cloud decay after the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Initially, the superradiance condition of the primary cloud is saturated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', ω(1) I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, once even a small number of particles are transferred to the second mode, which has an angular momentum in the opposite direction to the central BH spin, and is ab- sorbed by the BH, the BH spin decreases slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, the first mode becomes a non-superradiant mode, and the particles belonging to the primary cloud also begin to be absorbed by the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, the BH mass and angu- lar momentum gradually increase maintaining the spin parameter slightly below the threshold value until the resonant transition becomes more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' At around the peak of the resonance, the particle num- ber of the second mode increases, and the flux to the BH of the second mode with negative angular momentum dominates that of the first mode with a positive spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' After passing the resonance frequency, the flux of the first mode dominates again, but at that time there are not enough particles left to spin-up the BH beyond the superradiance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As a result, the BH spin settles to a value slightly below the threshold for the first mode to be superradiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Although the deviation from the crit- ical spin is tiny, |ω(1) I | is sufficiently large to eliminate the cloud within the timescale of the binary inspiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In summary, the cloud, first, dissipates through the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, the particle number of the first mode decreases dramatically with the resonant transi- tion, and the transferred particles to the second mode are absorbed by the BH immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' After that, the pri- 8 30 20 10 0 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-4 t/ts Mc(t)/M0 α0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2, q=10-3 log10Mc,0/M0 10 8 6 4 2 30 20 10 0 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-7 0 t/ts χ(t)- χcrit(t) log10Mc,0/M0 10 8 6 4 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Evolution of the cloud mass (top) and the BH spin (below) for α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2, q = 10−3 and various initial cloud mass Mc,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Black dotted line in the below panel shows the minimum value of the spin estimated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, the particle number after the transition is too small to spin-up the BH after the transition for a large initial cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The small initial cloud mass such that the BH spin- down is negligible gives the largest final cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' mary cloud decreases exponentially due to the BH spin- down below the superradiance threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We also show the parameter dependence of this sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 5, we show the evolution of the particle number for the same parameter but varying α0 and q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If we neglect the backreaction and GW emission, and approximate the binary orbital fre- quency evolution by a linear function of t, the survival probability of the primary cloud is analytically evaluated as exp(−πη2/γ) [27]6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This means that the efficiency of the tidal effect is determined by the product of the amplitude of the tidal perturbation η and the timescale passing through the resonance band η/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This measure of the tidal effect η2/γ is proportional to qα−11/3 0 for q ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, the cloud mass after the resonance becomes tiny, when α0 is small and q is somewhat large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Initial and final cloud mass In terms of observation, it is interesting to clarify how much of the cloud can remain after the satellite passes 6 Surprisingly, this result is not changed by the presence of ω(2) I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 30 20 10 0 10 20 30 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 t/ts Mc(t)/M0 α0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2, q=10-4 log10Mc,0/M0 10 8 6 4 2 30 20 10 0 10 20 30 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-9 0 t/ts χ(t)- χcrit(t) log10Mc,0/M0 10 8 6 4 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The same plot as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6, but for q = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, since the transition rate is not large, there is enough particle number left to spin-up the BH after the transition for a large initial cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, the cloud mass does not decrease at the late epoch, and the largest initial cloud mass gives the largest final cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' through the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The evolution of this system and the fate of the cloud also depend on the ini- tial cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If there are no processes that prevent the cloud’s growth and the BH has nearly extremal spin when it forms, the cloud mass can be estimated as ∼ αM [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In reality, however, there can be other dissipation pro- cesses besides GW emission, such as dissipation due to axion’s self-interaction [40, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Therefore, it is worth discussing the dependence on the initial cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We find that the initial value of the cloud mass that maximizes the final cloud mass is mainly determined by the value of the BH spin after the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It can be classified into two cases, which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We show the example of the cloud mass and BH spin evolu- tion for the initial cloud mass from 10−10M0 to 10−1M0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6 (case 1) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 7 (case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we take the final time as τbin/4, where τbin = Ω0/γ is the timsescale of the binary evolution (see also Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In case 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6), for large initial cloud mass, the cloud mass decreases exponentially due to the BH spin-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, since the transition rate is large, there are not enough particles left to spin-up the BH after the res- onance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' On the other hand, for somewhat small initial cloud mass, the particle number is too small to spin-down the BH efficiently from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, the absorption to the BH can be neglected, and the final mass is determined only by the transition due to the tidal in- 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='×10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='×10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='010 α0 q log10Mc,fin/M0 14 13 12 11 10 9 8 7 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Possible maximum final mass of the cloud after the hyperfine resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The area above the red boundary be- longs to the case1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6), and the area below it belongs to the case2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' teraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, the case with such a small initial cloud mass gives the maximum final cloud mass, for example, Mc,0 = 10−9M0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In case 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 7), for the largest initial cloud mass (Mc,0 = 10−1M0), the cloud mass does not decrease at the late epoch in this timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This is because the tran- sition rate is small and there are enough particles left to spin-up the BH by almost the threshold value of the su- perradiance after the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, in this case, the largest initial cloud mass simply gives the maximum final cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We summarize the possible maximum final mass Mc,fin of the cloud after the resonance in the parameter space (α0, q) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We take 10−1M0 as the largest initial cloud mass, and contours below 10−15M0 are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The area above the red boundary belongs to the case1, and the area below it belongs to the case2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In case1 re- gion, the final cloud mass is mainly determined by the transition rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', the strength of the tidal interaction characterized by η2/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, for small α0 and somewhat large q, the cloud hardly remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In case2 region, the fi- nal cloud mass is mainly determined by the GW emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For small α0 and q, the timescale of the binary evolution becomes large, and thus the cloud has small mass by the time orbital frequency reaches around the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As a result, we find that the largest final mass of the cloud is ∼ 10−5M0, which is achieved at α0 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2 and q ∼ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' BH spin-down In this subsection, we discuss the impact on the statis- tical distribution of BH spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If axions exist, most of BHs which experienced sufficiently large spin-up in the past 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='×10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='×10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='100 1 α0 q log10( χcrit- χmin) 14 12 10 8 6 4 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Approximate minimum value of the spin parameter of the central BH obtained by dχ/dt = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It gives an estimation of the upper limit of the deviation from the critical spin, which can be reached by the BH spin-down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Blue solid line shows the boundary below which the hyperfine transition is relevant as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' are expected to remain at the critical spin corresponding to the threshold for the superradiance (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Such an accumulation of the spin distribution can be an ob- servational signature of the existence of axions [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, as we saw in the preceding subsections, axions transferred to the mode with m = −1 by the tidal inter- action make the BH spin smaller than the critical spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, the question is, how small can the BH spin be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' To answer it, we analyze the evolution of the BH spin parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (19) and (20), we have dχ dt = −2 χ M dM dt + 1 M 2 dJ dt = 4χMc,0 M (ω(1) I n1 + ω(2) I n2) + 2 α Mc,0 M (−ω(1) I n1 + ω(2) I n2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (42) For χ < χcrit (, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' ω(1,2) I < 0), the first term on the right-hand side is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Near the resonance, the flux of the second mode can be dominant, at which point the second term is also negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' On the other hand, when n2 decreases and the flux of the first mode becomes dominant, the second term becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, we can estimate the minimum value of the BH spin parameter achieved by the reabsorption of trans- ferred axions as χmin satisfying dχ/dt = 0 around the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Here, we use the approximation obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (40) for n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In particular, near the resonance, we can write n2 ≃ η2 Γ2 n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (43) 10 Substituting it in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (42) and approximating M ≃ M0 and α ≃ α0, we can find the root of dχ/dt = 0 numeri- cally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 9, we show the deviation of χmin obtained in this way from the critical spin χcrit for the param- eter space (α0, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, it is important to stress that the deviation obtained here is only an approximate upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In fact, if the cloud mass is too small, χcrit(dχ/dt)−1 can be larger than the timescale of binary evolution as the cloud mass decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In that case, the BH spin-down stops before reaching the χmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 7, we show the evolution of the BH spin, with the dotted line corresponding to χmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' When the cloud mass is somewhat large, the BH spin can only go down to about χmin at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' On the other hand, if the cloud mass is too small, BH spin-down terminates before reach- ing χmin, and the absorption to the BH is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In particular, although it seems that the deviation of χmin from the critical spin for large α0 and small q can be O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1) from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 9, in that region, the timescale of the binary evolution becomes small and there are no enough time to spin-down the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Therefore, although the spin- down due to the absorption can be sufficiently large to deplete the cloud, it would not affect the constraints on axions from the BH spin measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Modification of the orbital frequency Next, we discuss the modification of the GW frequency evolution at around the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The GW frequency at which resonance occurs is given by [26] fres = Ω0 π = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2 mHz 1 1 + 4α2 0 � α0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �7 �10M⊙ M � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (44) For typical binary systems with a supermassive BH hav- ing an extreme mass ratio companion, the resonance fre- quency is too low to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, GWs at around the resonance frequency from an intermediate mass BH accompanied by a stellar mass or an even smaller mass exotic compact object could be observed by space-based GW detectors, such as LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Around the resonance frequency, the orbital frequency stagnates due to the angular momentum transfer asso- ciated with the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This backreaction effect also causes the delay of the rapid decrease of the cloud and enhances the transition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 10, we show the evolution of the cloud mass and the orbital frequency for α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1, q = 10−5 and various initial cloud mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' When the cloud mass is large enough, this backreaction greatly affects the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We can estimate the thresh- old value of the cloud mass before the transition for the backreaction works effectively from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For sim- plicity, neglecting the GW emission from the cloud and considering only the primary cloud, the orbital evolution around the resonance is approximated as dΩ dt ≃ γ + R Ω0 M 2 0 dJ(1) c dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (45) Here, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (39), the time derivative of J(1) c is given by dJ(1) c dt ≃ Mc µ 2η2 ω(2) I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (46) For the orbital frequency to stagnate, in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (45), the first term γ (GW radiation reaction) and the second term must be comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, we can estimate the threshold value of the cloud mass required for the backreaction to work by equating these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We denote it as Mc,float, and it is given as Mc,float = γ|ω(2) I |α 2RΩ0η2 M ≃ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 × 10−8M0(1 + q)4/3 � α0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �16/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (47) Therefore, even with a small mass of the cloud, we can expect that this modification can be a clear signature of the presence of an axion condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Unfortunately, the timescale of the binary evolution τbin is typically much longer than the observation time (≲ 10 yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' At first glance, it seems difficult to resolve the degeneracy with the uncertainties in the chirp mass and the mass ratio by observing the time derivatives of the GW frequency ˙f and ¨f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, we point out that f ¨f/ ˙f 2 can be a good indicator of deviation from clean binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If the binary system is clean and the mass ratio is sufficiently small, q ≪ 1, this non-dimensional quantity becomes a model-independent constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', f ¨f/ ˙f 2 = 11/3 in the early stage of the inspiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' It would be natural to assume that the cloud mass is bounded from above by the mass where the GW emis- sion timescale τGW equals the timescale of the binary evolution τbin (see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, around the reso- nance, the cloud mass, reduced only by the GW emission, is bounded by Mc,GW = M 2 τbin α−14 C ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='9 × 10−4M0 q (1 + q)1/3 � α0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �14/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (48) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 11, we show the value of the indicator f ¨f/ ˙f in the presence of a cloud with Mc,0 = Mc,GW, for α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 and α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' They show that the deviation from clean binaries can be larger than O(1), even if the axion cloud has only a tiny fraction of the mass of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' While q dependence on the indicator for the same cloud mass is weak7, Mc,GW is approximately linearly proportional to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, when the mass ratio q is too small, the effect of the angular momentum transfer due to the tidal interaction also becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 7 In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (23), the main contribution to the square brackets in the second term of the right-hand side is ˙J(1) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (39), ˙J(1) c is roughly proportional to q2 around the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Thus, ˙Ω ≃ O(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' One can find that ¨Ω ≃ O(q2) by differentiating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 11 30 20 10 0 10 20 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' × 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='01 t/ts Mc(t)/M0 α0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1, q=10-5 log10Mc,0/M0 10 8 6 4 2 30 20 10 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='010 t/ts Ω(t)/Ω0 α0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1, q=10-5 log10Mc,0/M0 10 8 6 4 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Evolution of the cloud mass (left) and the orbital frequency (right) for α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1, q = 10−5 and various initial cloud mass Mc,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Black dotted line in the left panel shows the threshold value of the cloud mass required for the backreaction to work effectively, obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Black dashed line in the right panel shows the evolution of the orbital frequency in the clean binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 4 2 0 2 4 50 0 50 100 t/ts f f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='. / f\uf110 2 α0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 log10q 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 4 2 0 2 4 5 0 5 10 15 20 t/ts f f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='. / f\uf110 2 α0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2 log10q 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Indicator in GW frequency of the presence of the cloud f ¨f/ ˙f 2 around the resonance t = 0 for various q, α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 (left) and α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' The initial cloud mass is set where the timescale of the GW emission and the binary evolution are equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', Mc,0 = Mc,GW in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If the binary system is clean, f ¨f/ ˙f 2 = 11/3 model-independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Around the resonance, this quantity can be largely changed with the level transition of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' SUMMARY AND DISCUSSION In this paper, we have investigated the evolution of in- spiralling binary systems accompanying an axion cloud before and after the orbital frequency crosses the hyper- fine resonance frequency, focusing on small mass ratio (q ≪ 1) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Our main interest is how the hyperfine level transition proceeds and affects the observational sig- natures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' From the comparison of timescales, we found it necessary to take into account the following components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' the decaying process of the axion in the destination mode of the hyperfine transition (imaginary part of the eigen- frequency), the GW emission from the cloud, and the backreaction to the orbital motion and that to the mass and spin of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We presented a formulation to examine the evolution of the cloud, the central BH, and the orbital motion including all these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In particu- lar, carrying out the adiabatic elimination of the degree of freedom of the amplitude of the second mode allows us to examine a wide parameter region numerically, and gives useful expressions for analyzing the behavior of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Our results show that the cloud mass is typically signif- icantly reduced by the GW emission before the resonant transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If q is sufficiently large or α is suffi- ciently small, axions in the m = 1 fastest growing mode are almost transferred to the m = −1 mode, which has angular momentum in the opposite direction to the BH spin and is easily absorbed by the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Then, the pri- mary cloud becomes non-superradiant and can fall into the BH, which results in the increase of the BH angular momentum, counter-intuitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, the increase of the BH mass dominates to maintain the first mode to be non-superradiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As a result, the cloud almost completely disappears by the absorption to the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' On the other hand, if q is extremely small or α is sufficiently large, the transition rate due to tidal interaction is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In such cases, since there are enough particles left to spin- up the BH again after the transition, the absorption to the BH at the late epoch can be neglected, and the cloud does not disappear completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' However, it dissipates 12 mainly owing to the GW emission before the transition, and the maximum mass of the cloud that can remain af- ter the resonance is ∼ 10−5M0 at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' How much of axion clouds can remain after the resonance might have an implication to the survey of the cloud as an environ- ment around the BH, such as [22, 23, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We also discussed the implication to the observational signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' First, we confirmed that the time variation of the BH spin around the transition is tiny, although this tiny variation can be important to determine the evolu- tion of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This result makes robust the constraint on the existence of an axion field obtained through the BH parameter distribution measured by GWs from bi- nary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Second, we studied the influence of the transition on the inspiral GW waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We found that even for extremely small cloud mass, the backreaction to the orbital motion works effectively, and the frequency stagnates around the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In particular, the combination f ¨f/ ˙f 2 is affected by the transition to a detectable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Therefore, for example, the GWs from an intermediate mass BH associated with a small mass satellite can be a good target for the axion search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We need more extensive analysis to conclude the observabil- ity of axion clouds with the modification of the wave- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Furthermore, the generalization of the inspiral or- bit and the discrimination from other environmental ef- fects would be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We leave them as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' ACKNOWLEDGMENTS T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Takahashi was supported by JST, the establishment of university fellowships towards the creation of science technology innovation, Grant Number JPMJFS2123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' TT is supported by JSPS KAKENHI Grant Number JP17H06358 (and also JP17H06357), A01: Testing grav- ity theories using gravitational waves, as a part of the in- novative research area, “Gravitational wave physics and astronomy: Genesis”, and also by JP20K03928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' HO is supported by Grant-in-Aid for JSPS Fellows JP22J14159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Appendix A: Timescales In this appendix, we summarize the timescales involved in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Binary evolution: The timescale of the binary evolution due to the GW radiation at the resonance frequency Ω0 is given by τbin = Ω0 γ = 5 96M (1 + q)1/3 q (MΩ0)−8/3 , (A1) where γ is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Transition: The resonance bandwidth can be estimated as ∆Ω ∼ 2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Hence, if one can neglect the instability of the mode of the transition destination and lin- earize the orbital evolution, the timescale for pass- ing through the resonance band is given by τtrans = 2η γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (A2) Decay of the secondary cloud: The secondary cloud decreases as ∼ e−2|ω(2) I |t, and the timescale is given by τinst = |ω(2) I |−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (A3) GW emission of the primary cloud: From the energy conservation ˙Mc = − ˙EGW (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (14)), one can obtain Mc(t) = Mc,0 1 + (t − t0)/τGW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (A4) Here, the timescale is given by τGW = 1 C M 2 Mc,0 α−14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (A5) Parameter dependencies of the timescales mentioned above are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 12 and Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Appendix B: Toy model for adiabatic elimination In this appendix, we discuss the approximation used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' III C with a simplified toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Consider the two level transition described by the Schr¨odinger equation i d dt � c1 c2 � = � 0 η η ∆(t) − iωI � � c1 c2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (B1) Let η and ωI be constants and ∆(t) = 2γt (γ is constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' This model is a simplification of ignoring all backreac- tions, GW emissions and linearizing the binary evolution in the problem we investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' If ωI = 0, this model is known as Landau-Zener problem [27, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Now, we want to study the level transition to the decaying mode (ωI > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For this problem, we have an exact ana- lytic solution with the initial conditions c1(−∞) = 1 and c2(−∞) = 0 as [53, 54] 13 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='100 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 1 1000 106 109 1012 q Timescale [yrs] Binary evolution (hyperfine) Binary evolution (Bohr) Transition (hyperfine) Transition (Bohr) Decay |ω21-1 1 GW FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Timescales involved in the resonant transition of axion clouds in binary systems for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 and M = M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Blue and yellow solid lines show the timescale of the binary evolution and the transition at the hyperfine resonance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Blue and yellow dashed lines show the same quantities, but for the typical Bohr transition (|211⟩ → |31 − 1⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Green and red lines show the timescales of decay of the secondary cloud (|21 − 1⟩) and of the GW emission of the primary cloud (|211⟩) for Mc,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Timescales involved in the hyperfine resonance of axion clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' process time Binary evolution τbin = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2 × 1013 s(1 + q)1/3 q � M M⊙ � � χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='4 �−8/3 � α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �−16 Transition τtrans = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='3 × 1010 s 1 (1 + q)2/3 � M M⊙ � � χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='4 �−5/3 � α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �−13 Decay of the secondary mode τinst ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='9 × 105 s � M M⊙ � � χ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='4 �−1 � α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �−9 GW emission τGW = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 × 1011 s � M M⊙ � �Mc,0/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �−1 � α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='1 �−14 |c1(t)|2 = e−ωIt− π 2 η2 2γ ���Diη2/2γ � ei 3π 4 ( � 2γt − iωI/ � 2γ) ���� 2 , (B2) |c2(t)|2 = e−ωIt− π 2 η2 2γ η2 2γ ���Diη2/2γ−1 � ei 3π 4 ( � 2γt − iωI/ � 2γ) ���� 2 , (B3) where Dν(z) is the parabolic cylinder function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Carrying out the adiabatic elimination as Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' III C, we obtain the approximate solution for the particle number as |c1(t)|2 ≃ exp � 2 � t −∞ dt′ ωIη 4γ2t ′2 + ω2 I � = exp � −η2 γ � arctan 2γt ωI + π 2 �� , (B4) and |c2(t)|2≃ η2 4γ2t2 + ω2 I |c1(t)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (B5) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 13, we compare the approximate solution obtained by the adiabatic elimination with the exact one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' As one can confirm from the figure, the two solutions agree quite well when ωI/η is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' We can estimate the time when the perturbation starts to work from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (B4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' For |t|≫ ωI/2γ (t < 0), one can expand |c1(t)|2 with respect to 1/|t| as |c1(t)|2∼ exp � − η2ωI 2γ2|t| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (B6) If η2/γ ≫ 1, the exponent can be O(1), even for |t|≫ ωI/2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, the proper choice of the time for the onset of the perturbation would be t ∼ −η2ωI/2γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' On the other hand, if η2/γ ≲ 1, the exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' (B4) vanishes for |t|≫ ωI/2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' In this case, it is enough to choose the starting time at t ∼ −ωI/2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Combining them, we have the estimation of the time when the per- turbation starts to work as t ∼ −(1 + η2/γ)(ωI/2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 14 Exact Approx 100 50 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='0 |c1(t) 2 100 50 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='005 2 γ t |c2(t) 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Time evolution of the particle number of each mode for η/√2γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='5 and ωI/√2γ = 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=', ωI/η = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Blue solid line shows the exact solution and red dashed line shows the approximate solution obtained by the adiabatic elimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Arvanitaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Dimopoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Dubovsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Kaloper, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' March-Russell, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content=' D81, 123530 (2010), arXiv:0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdFPT4oBgHgl3EQf9zV6/content/2301.13213v1.pdf'} +page_content='4720 [hep-th].' metadata={'source': 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b/sdE1T4oBgHgl3EQf3QVe/content/tmp_files/2301.03487v1.pdf.txt @@ -0,0 +1,360 @@ +arXiv:2301.03487v1 [cs.CC] 9 Dec 2022 +A Critique of Sopin’s “PH = PSPACE”∗ +Michael C. Chavrimootoo, Ian Clingerman, and Quan Luu +Department of Computer Science +University of Rochester +Rochester, NY 14627, USA +December 9, 2022 +Abstract +We critique Valerii Sopin’s paper “PH = PSPACE” [Sop14]. The paper claims to resolve one +of the major open problems of theoretical computer science by leveraging the Skolemization of +existential quantifiers of quantified boolean formulas to show that QBF (a well-known PSPACE- +complete problem) is in Πp +4, and thus PH = PSPACE. In this critique, we highlight problems +in that paper and conclude that it fails to establish that PH = PSPACE. +1 +Introduction +In this paper, we critique Valerii Sopin’s paper “PH = PSPACE” [Sop14]. The paper introduces +two theorems. The first theorem is about Skolemization of quantified boolean formulas, while the +second theorem attempts to leverage the first theorem to collapse the polynomial hierarchy and +prove that PH = PSPACE. Before going into more detail, we first discuss the importance of such +a result. +In computational complexity theory, the polynomial hierarchy is defined as PH = P ∪ NP ∪ +NPNP∪NPNPNP ∪· · · , and PSPACE refers to the class of problems that can be solved using at most +polynomial space. Both are fundamentally important and intensely studied classes. It is known +that PH ⊆ PSPACE. However, whether PH equals PSPACE to this day remains one of the central +open problems in theoretical computer science. Informally, this is because most known techniques +cannot settle this question.1 +There are many consequences to proving PH = PSPACE [Far20]. +Hypothetically, if one were able to show that PSPACE is no more complex than some level k of +PH, it would indicate that all levels above k collapse to the kth level (since they are all contained +in PSPACE). Even for a large value of k, a collapse would help advance research into other open +questions of complexity theory. For example, PH = PSPACE would yield the existence of PH- +complete problems, a currently unresolved issue (that is in fact equivalent to the issue of whether +∗Supported in part by NSF grant CCF-2006496. +1This is because a proof resolving PH vs. PSPACE cannot relativize, as follows from the results of Baker, Gill, +and Solovay [BGS75] and Yao [Yao85]. +1 + +PH collapses, i.e., for some i, PH = Σp +i ). +Another serious implication of this result would be +resolving the relationship between L and classes like PH or NP.2 +The approach used in Sopin’s paper to purportedly prove its theorems is as follows: first apply +Skolemization to an arbitrary quantified boolean formula and then construct an algorithm to solve +QBF from one that solves Π4SAT. However, we will show how the proofs for these theorems are +flawed. +In Section 2, we introduce the definitions and notations used within Sopin’s proofs. In Sections 3 +and 4, we summarize the key points of Theorems 1 and 2 and analyze each proof. We will use +Section 5 to point out a key observation on the use of Skolemization in the context of the paper. +Finally, in Section 6, we conclude our critique. +2 +Preliminaries +We present here many standard concepts in complexity theory. Equivalent definitions can be found +in most modern textbooks on complexity [HO02, AB09, Sip13]. +As to notation, given a string w, let |w| denote the length of w. Additionally, let N = {0, 1, 2, . . .} +and let N+ = {1, 2, 3, . . .}. +Throughout this paper, we will speak of boolean formulas and boolean variables. We will assume +that our (boolean) variables are always assigned 1 (true) or 0 (false). A boolean formula φ with +boolean variables x1, . . . , xn is denoted by φ(x1, . . . , xn). Quantified boolean formulas are of the +form (Q1x1) · · · (Qnxn)[φ(x1, . . . , xn)], such that for each i ∈ {1, . . . , n}, it holds that Qi is either ∃ +or ∀. +Let us now define the polynomial hierarchy. Fix i ≥ 1. A language L is in Σp +i exactly if there +is a polynomial q and a polynomial-time computable predicate R such that for all x it holds that +x ∈ L ⇐⇒ (∃w1 : |w1| ≤ q(|x|))(∀w2 : |w2| ≤ q(|x|) · · · (Qiwi : |wi| ≤ q(|x|))[R(x, w1, . . . , wi)], +where Qi is ∃ if i is odd and ∀ if i is even. Additionally, a language L′ is in Πp +i exactly if L′ ∈ Σp +i . +The polynomial hierarchy is defined as the class PH = � +i∈N+ Σp +i .3 This definition yields the same +classes as the other common definition, drawn on in our introduction, i.e., Σp +3 = NPNPNP. +For each of the abovementioned classes (except PH), we can define the following canonical com- +plete problems. The definition that follows is adapted from one given by Arora and Barak [AB09]. +For each i ≥ 1, the problem ΣiSAT, which is complete for Σp +i , is defined as the set of quantified +boolean formulas of the form (∃u1) · · · (Qiui)[ψ(u1, . . . , ui)] that are true, where ψ is a boolean +formula, each of u1, u2, . . . , ui denotes a list of a boolean variables, and Qi is ∃ if i is odd and ∀ if +i is even. The problem ΠiSAT, which is complete for Πp +i , is defined analogously, except that Qi is +∀ when i is odd, and ∃ if i is even. +PSPACE is the class of languages accepted by a Turing machine that runs in polynomial space. +Formally, PSPACE = � +k∈N+ DSPACE[nk]. Additionally, “PSPACE embodies the power of polyno- +mially bounded quantifiers” [HO02]. And so, QBF, the set of all true quantified boolean formulas, +2L is the class of decision problems solvable by a deterministic Turing machine in logarithmic space. It is known +that L ⊆ P. +Thus if L = NP, then L = P = NP = PH. +It is also true by the space hierarchy theorem that +L ̸= PSPACE. Consequently, if PH = PSPACE, then L ̸= PH, and thus L ̸= NP. +3Readers familiar with the concepts will notice that we omitted explicit mention of Σp +0, but it is of course contained +in Σp +1 and thus not actually omitted. This “omission” occurs simply because we do not need to refer to that class in +the rest of this paper, and so it saves us the trouble of defining it. +2 + +is a canonical PSPACE-complete problem. It’s easy to see that QBF = � +i∈N ΣiSAT ∪ ΠiSAT, and +thus PH ⊆ PSPACE. +Skolemization is the process of removing existentially quantified variables from a quantified +boolean formula, and replacing them with Skolem constants or Skolem functions, which are boolean +functions that are only over the universally quantified variables that appear before the existentially +quantified variable being removed in the quantifier order. +Interested readers may consult the +textbook by Chang and Lee [CL73] for more details. +3 +On Theorem 1 +We will now focus on Theorem 1 of Sopin’s paper. The theorem has been reformulated for the sake +of clarity. +Theorem 1 ([Sop14]). For an arbitrary n ∈ N+, let Φ = (Ω1x1)(Ω2x2) · · · (Ωnxn)[φ(x1, . . . , xn)] be +an arbitrary boolean formula where (Ω1, . . . , Ωn) ∈ {∃, ∀}n. Let I = {i | i ≤ n ∧ Ωi = ∃}. Then, Φ +is a true quantified boolean formula if and only for every i ∈ I, there is a boolean function yi over +the variables in {xj | j < i ∧ Ωj = ∀} such that the quantified boolean formula that results from +substituting each xi with yi is a tautology. +To the best of our understanding, the theorem is essentially proposing that Skolemization +preserves satisfiability, since the process described by the theorem resembles that of Skolemization. +However, it is well-known that Skolemization does in fact preserve satisfiability (for reference, see +textbooks by Chang and Lee [CL73] and Loveland [Lov78]). In order to leverage this technique to +decide QBF, it follows that one would at the very least need a polynomial upper bound on the space +complexity of Skolemization. Unfortunately, no such result is currently known. We elaborate this +point further in Section 5. Additionally, we notice several issues with the given proof of Theorem 1 +and discuss them in the rest of this section. +In their proof of Theorem 1, the author introduces a recursive algorithm to test for membership +in QBF. +It works by removing quantifiers sequentially and producing, based on the removed +quantifier, a new and logically equivalent formula. After that, the author adds a note on how “a +[boolean] function determines the truth table” [Sop14] of its variables. The proof then follows with +two examples to help illustrate its main argument. +First, we comment on the structure of the proof. It is unclear how each part of the proof relates +to the other and to the main argument. Therefore, throughout our analysis, we will attempt to +understand the role of each part of the proof. +Sopin presents a recursive algorithm to decide QBF and claims that the proof of Theorem 1 +follows from it, however this algorithm is not well-defined. The recursive step is the only defined +aspect of the algorithm: It involves pulling off the first quantified variable and checking both values +for that variable. Depending on the quantifier, a new equation is formed using these two equations +with both possible values of first variable. In the definition of this algorithm, there is neither a base +case nor a recursive call. A base case is needed in order to show when the algorithm terminates and +a recursive call is needed in order to show where the recursive step is applied. So it is ambiguous as +to whether the algorithm presented is indeed recursive, or if only the first quantifier is to be pulled +off. It appears to us that is a common algorithm to decide QBF, but it is unclear how the proof of +Theorem 1 follows from it. +3 + +We would now like to turn our attention to a statement made at the end of the algorithm’s +description: “Notice that a [boolean] function determines the truth table (one-to-one correspon- +dence)” [Sop14]. The term “truth table” in this context is ambiguous. If the intention of the paper +is indeed to use truth-tables to represent Skolem functions, as in Example 1 of Theorem 1 (discussed +later), then this is a matter of concern, as the size of a truth-table is exponential in the number +of its variables. On the other hand, it’s possible that the quoted statement is simply a fact of the +relationship between boolean functions and truth-tables. Still, the paper fails to show the potential +space complexity of this transformation, which is crucial when dealing with PSPACE-complete +problems. +At the end of the proof of Theorem 1, two examples are presented in order to add clarity +to the theorem. +Example 1 provides an explicit description of a Skolem function, where exis- +tentially quantified variables can be rewritten as functions of the variables that come before it +in the formula. At the end of the example, it is mentioned that the Skolem function “is indeed +the truth table, where values of x1, . . . , xk−1 determine the value of xk” [Sop14]. As mentioned +before, the use of truth-table in this context would lead to an exponential size blow-up. Exam- +ple 2 is just a restatement of the theorem using an explicit quantified boolean formula. Example 2 +states that “∀x1∃z1∀x2∃z2∀x3∃z3 is a true [quantified boolean formula] if and only if there ex- +ist such boolean functions y1 : {0, 1} → {0, 1}, y2 : {0, 1}2 → {0, 1}, y3 : {0, 1}3 → {0, 1} that +φ(x1, y1(x1), x2, y2(x1, x2), x3, y3(x1, x2, x3)) is tautology” [Sop14]. This example just shows how +a formula looks after Skolemization has occurred. The formula must be a tautology in order to +be a true quantified boolean formula because only universally quantified variables are left. The +replacement of existentially quantified variables with boolean functions over the correct variables +appears to have been performed correctly but there is neither a proof that these functions are +Skolem functions nor a direct connection to the proof of Theorem 1. +Thus while the statement is true, the proof given does not support the theorem. It’s unclear, +based on our observations about Skolemization not being known to be computable using polynomial +space, how this theorem will become useful in the rest of Sopin’s paper (and indeed, we discuss in +the next section that we see no such connection). +4 +On Theorem 2 +In this section, we will focus our attention to the following theorem of Sopin’s paper. +Theorem 2 ([Sop14]). Πp +4 = PSPACE. +As one would expect, the purported proof takes an arbitrary quantified boolean formula Φ and, +from it, constructs a quantified boolean formula Φ′ (which has a specific syntactic form that we +specify later in this section) in polynomial time, such that Φ ∈ QBF +⇐⇒ +Φ′ ∈ Π4SAT. Let +us preface that the proof presented in the paper is often unclear about what it means and makes +several logical leaps, and so we approach the arguments in the proof using our best understanding +of what the author could have meant. +Our first comment is about clarity and touches on the form of the quantified boolean formulas. +The author assumes that quantified boolean formulas are of the form (which we will often refer to +as the “standard” form in this critique) +(∀x1)(∃y1) · · · (∀xn)(∃yn)[φ(x1, y1, . . . , xn, yn)], +4 + +for some n ∈ N+, which at a glance seems incorrect. For example, the formula (∃x, y)(∀z)[(x∨y∨z)] +is certainly in QBF, but is not included in the paper’s treatment. Indeed, the general form given +seems to miss formulas with an odd number of variables, formulas that start with an existential +quantifier, and formulas that have multiple variables bound to the same quantifier. However, what +the paper does not make clear, is that all such formulas can be converted to the “standard” form in +polynomial-time. The key to putting arbitrary quantified boolean formulas into “standard” form +is the use of dummy variables.4 Our example, +(∃x, y)(∀z)[(x ∨ y ∨ z)], +introduced earlier in this paragraph exhibits all the issues that seem to exist with the “standard” +form. And so, we shall present, in an informal manner (since the issue is rather simple and does +not warrant much more than an example), how to convert the above formula to “standard” form. +Let us first make the first quantifier be a universal quantifier by introducing the dummy variable +α. This yields the formula +(∀α)(∃x, y)(∀z)[(x ∨ y ∨ z)]. +Next, we separate the variables x and y so that each quantifier is only bound to one variable by +introducing the dummy variable β. This yields the formula +(∀α)(∃x)(∀β)(∃y)(∀z)[(x ∨ y ∨ z)]. +To finish, since we need an even number of variables, we introduce the dummy variable γ and +obtain +(∀α)(∃x)(∀β)(∃y)(∀z)(∃γ)[(x ∨ y ∨ z)]. +It is not hard to see that if the original formula has n variables, then the new formula will have at +most 2n + 2 variables. +We will now focus on the correctness of the proof of Theorem 2. For the rest of this section, +we will fix, for some n ∈ N+ and some boolean formula (with no quantifiers) φ that is over 2n +variables, the following quantified boolean formula +Φ = (∀x1)(∃y1) · · · (∀xn)(∃yn)[φ(x1, y1, . . . , xn, yn)]. +The paper seeks to construct, from Φ, the following (purportedly logically equivalent) formula +(which is not in “standard” form) +Φ′ =(∀x1, . . . , xn)(∃y1, . . . , yn)[φ(x1, y1, . . . , xn, yn)∧ +(∀ˆxn)(∃zn)[φ(x1, y1, . . . , xn−1, yn−1, ˆxn, zn)]∧ +(∀ˆxn−1, ˆxn)(∃zn−1, zn)[φ(x1, y1, . . . , xn−2, yn−2, ˆxn−1, zn−1, ˆxn, zn)] ∧ · · · ∧ +(∀ˆx2, . . . , ˆxn)(∃z2, . . . , zn)[φ(x1, y1, ˆx2, z2, . . . , ˆxn, zn)]], +such that Φ ∈ QBF ⇐⇒ Φ′ ∈ Π4SAT. +The attempted proof is by induction on the number of variables in the formula. We will not +repeat the entire argument presented there, and we urge interested readers to consult the original +4For our purposes, a dummy variable is one that is quantified over, but does not appear in the boolean formula. +That is certainly legal. For example, we can say that the formula (∀x)(∃y)(∀z)[(x ∨ y)] is over the set of variables +{x, y, z}. In this case, z is a dummy variable. +5 + +paper directly. The gist of the purported inductive proof is as follows: We can swap the positions +of quantifiers inside the original formula, and then we can detect in polynomial-time whether the +formula’s truth value has been affected by the changes. We note, off the bat, two major issues with +this approach: (1) it does not leverage the implications of the statement of Theorem 1, and (2) the +induction does not actually prove the logical equivalence. +Let us address (1) first. +The only mention made to Theorem 1 in the purported proof of +Theorem 2 is in the case where the number of variables, m, is greater or equal to 3. The paper +states that by “taking off the first quantifier and checking both possible values for the first variable +in [the] way we did in Theorem 1, we come to the m − 1 case” [Sop14]. It is worth reiterating +that the attempted proof of Theorem 1 simply presents a version of the (well-known) recursive +algorithm to decide QBF in polynomial space. +Thus while this approach of “eating” variables +one at a time may help decide if the formula is true, it potentially uses an exponential amount of +(nondeterministic) time, and there is no clear passage in the paper that clarifies the use of that +algorithm. +Let us now address (2). +In the purported inductive proof, the paper claims that for any +quantified boolean formula ψ over two variables, x and y, it holds that (∀x)(∃y)[ψ(x, y)] ≡ +(∃y)(∀x)[ψ(x, y)] if and only if ψ is neither the XOR function nor the negation of the XOR function, +which is true. And thus, the argument in the paper states that as long as there is no way for ψ +to be the XOR function (or its negation) when the values of all but two variables, with one being +universally quantified over and the other being existentially quantified over, are fixed, then the +induction holds. The additional clauses in Φ′ are meant to play a role in supporting this argument. +However, the paper not only fails to explain how these additional clauses work and why they work, +but it also seems to be missing cases. Indeed, the paper states that the above check can be done in +polynomial time since “there are [only] n2 such formulas” [Sop14]. We believe this was derived by +selecting, from Φ, one of the n variables that are universally quantified over and one of the n vari- +ables that are existential quantifier over. However, missing from the argument is the fact that each +of the remaining 2n−2 variables can have one of two values (0 or 1), thus creating n222n−2 possible +formulas to check. (It’s worth noting that the above check can easily be done in nondeterministic +polynomial time. However, the paper only states “polynomial time,” which, as is standard, implies +“deterministic polynomial time.”) We thus conclude that the induction presented in that proof is +incorrect. +We mention briefly in passing that there are other minor errors, but we bear them no attention +as they do not carry as much importance as the ones we pointed out. Additionally, in the final +parts of that proof, the paper mentions the formula’s algebraic normal form as being crucial to the +argument and provides an example, but it is not clear why the algebraic normal form is crucial. +Thus we are not certain as to whether the author was hoping to achieve something different than +what is being conveyed through the technical report. +5 +A Note on Skolemization +We add this section with the hopes that the use of Skolemization can be better explained. Our +expectation is that the size of the Skolem functions must have a role to play in the proof. Indeed, +if the size of such functions is not polynomially-bounded, then it is unclear how any skolemized +formula can even be computed in polynomial space (and be a useful approach in this context). On +the other hand, if there is a polynomial upper bound on the size of Skolem functions, then one can +6 + +show something shocking. +Proposition 3. If there is a polynomial p : N → N+ such that, for each quantified boolean formula +Φ, the Skolemization of Φ produces no Skolem function of size greater than p(|Φ|), then Σp +2 = +PSPACE. +Proof. The ⊆ relationship is well-known, so it suffices to show that under the assumptions of the +above statement, PSPACE ⊆ Σp +2. We will do so by showing that QBF ∈ Σp +2. Let p be a polynomial +as defined by the proposition’s statement. Without loss of generality, let us assume that our input +is a quantified boolean formula (as we can easily detect that in polynomial time if it is not and +immediately reject). Fix an arbitrary boolean formula Φ with n variables as our input. Let E +denote the set of variables in Φ that are existentially quantified. Since Skolemization preserves +satisfiability, it follows that Φ ∈ QBF ⇐⇒ +there are ∥E∥ Skolem functions such that for each +assignment to the variables not in E, the boolean formula that results from replacing each variable +in E with the corresponding Skolem function is true under the current assignment. Because each +Skolem function has size bounded by p(|Φ|), substituting the Skolem functions into Φ and checking +the truth value of the resulting formula, given a specific assignment, can be computed in polynomial +time. Per our definition of Σp +2 in Section 2, this implies that QBF ∈ Σp +2. +This strengthens a result by Akshay et al. [ACG+18] who under similar assumptions conclude +that Σp +2 = Πp +2 = PH (by leveraging the Karp–Lipton Theorem). +6 +Conclusion +In this critique, we pointed out the errors in “PH = PSPACE” [Sop14] and concluded that Sopin’s +paper fails to show that PSPACE and the polynomial hierarchy coincide. The primary issue is that +Sopin’s paper does not account for a potentially exponential amount of work needed to perform +one of its checks that it claims can be done in polynomial time. We additionally find that the use +of Skolemization in the attempted proof is not clear. Indeed, it would be shocking if a machine +could compute the Skolemization using only a polynomial amount of space as that would yield +ground-breaking results (as proved in our Proposition 3). +Acknowledgements +We would like to thank Erin Gibson, David E. Narv´aez, and Lane A. +Hemaspaandra for their helpful comments on prior drafts. The authors are responsible for any +remaining errors. +References +[AB09] +S. Arora and B. Barak. Complexity Theory: A Modern Approach. Cambridge University +Press, 2009. +[ACG+18] S. Akshay, S. Chakraborty, S. Goel, S. Kulal, and S. Shah. What’s hard about boolean +functional synthesis? +In Computer Aided Verification, pages 251–269. Springer Inter- +national Publishing, July 2018. +[BGS75] +T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal +on Computing, 4(4):431–442, 1975. +7 + +[CL73] +C. Chang and R. Lee. Symbolic Logic and Mechanical Theorem Proving. Academic +Press, 1973. +[Far20] +A. Farago. What would be the consequences of PH = PSPACE? Theoretical Computer +Science Stack Exchange, https://cstheory.stackexchange.com/questions/21191, +2020. +[HO02] +L. Hemaspaandra and M. Ogihara. +The Complexity Theory Companion. +Springer- +Verlag, 2002. +[Lov78] +D. Loveland. Automated Theorem Proving: A Logical Basis. North-Holland, 1978. +[Sip13] +M. Sipser. Introduction to the Theory of Computation. Cengage Learning, 3rd edition, +2013. +[Sop14] +V. Sopin. PH = PSPACE. Technical Report arXiv:1411.0628v20 [cs.CC], Computing +Research Repository, arXiv.org/corr/, November 2014. Revised November 2022. +[Yao85] +A. Yao. Separating the polynomial-time hierarchy by oracles. In Proceedings of the 26th +IEEE Symposium on Foundations of Computer Science, pages 1–10. IEEE Computer +Society Press, October 1985. +8 + diff --git a/sdE1T4oBgHgl3EQf3QVe/content/tmp_files/load_file.txt b/sdE1T4oBgHgl3EQf3QVe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..675cff610ea3d60dad4eff9cf0a20a5ce297d1e1 --- /dev/null +++ b/sdE1T4oBgHgl3EQf3QVe/content/tmp_files/load_file.txt @@ -0,0 +1,307 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf,len=306 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='03487v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='CC] 9 Dec 2022 A Critique of Sopin’s “PH = PSPACE”∗ Michael C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Chavrimootoo, Ian Clingerman, and Quan Luu Department of Computer Science University of Rochester Rochester, NY 14627, USA December 9, 2022 Abstract We critique Valerii Sopin’s paper “PH = PSPACE” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The paper claims to resolve one of the major open problems of theoretical computer science by leveraging the Skolemization of existential quantifiers of quantified boolean formulas to show that QBF (a well-known PSPACE- complete problem) is in Πp 4, and thus PH = PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In this critique, we highlight problems in that paper and conclude that it fails to establish that PH = PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 1 Introduction In this paper, we critique Valerii Sopin’s paper “PH = PSPACE” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The paper introduces two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The first theorem is about Skolemization of quantified boolean formulas, while the second theorem attempts to leverage the first theorem to collapse the polynomial hierarchy and prove that PH = PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Before going into more detail, we first discuss the importance of such a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In computational complexity theory, the polynomial hierarchy is defined as PH = P ∪ NP ∪ NPNP∪NPNPNP ∪· · · , and PSPACE refers to the class of problems that can be solved using at most polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Both are fundamentally important and intensely studied classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It is known that PH ⊆ PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, whether PH equals PSPACE to this day remains one of the central open problems in theoretical computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Informally, this is because most known techniques cannot settle this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='1 There are many consequences to proving PH = PSPACE [Far20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Hypothetically, if one were able to show that PSPACE is no more complex than some level k of PH, it would indicate that all levels above k collapse to the kth level (since they are all contained in PSPACE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Even for a large value of k, a collapse would help advance research into other open questions of complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For example, PH = PSPACE would yield the existence of PH- complete problems, a currently unresolved issue (that is in fact equivalent to the issue of whether ∗Supported in part by NSF grant CCF-2006496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 1This is because a proof resolving PH vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' PSPACE cannot relativize, as follows from the results of Baker, Gill, and Solovay [BGS75] and Yao [Yao85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 1 PH collapses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=', for some i, PH = Σp i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Another serious implication of this result would be resolving the relationship between L and classes like PH or NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='2 The approach used in Sopin’s paper to purportedly prove its theorems is as follows: first apply Skolemization to an arbitrary quantified boolean formula and then construct an algorithm to solve QBF from one that solves Π4SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, we will show how the proofs for these theorems are flawed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In Section 2, we introduce the definitions and notations used within Sopin’s proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In Sections 3 and 4, we summarize the key points of Theorems 1 and 2 and analyze each proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We will use Section 5 to point out a key observation on the use of Skolemization in the context of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Finally, in Section 6, we conclude our critique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 2 Preliminaries We present here many standard concepts in complexity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Equivalent definitions can be found in most modern textbooks on complexity [HO02, AB09, Sip13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' As to notation, given a string w, let |w| denote the length of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Additionally, let N = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='} and let N+ = {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Throughout this paper, we will speak of boolean formulas and boolean variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We will assume that our (boolean) variables are always assigned 1 (true) or 0 (false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' A boolean formula φ with boolean variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn is denoted by φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Quantified boolean formulas are of the form (Q1x1) · · · (Qnxn)[φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn)], such that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , n}, it holds that Qi is either ∃ or ∀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let us now define the polynomial hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Fix i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' A language L is in Σp i exactly if there is a polynomial q and a polynomial-time computable predicate R such that for all x it holds that x ∈ L ⇐⇒ (∃w1 : |w1| ≤ q(|x|))(∀w2 : |w2| ≤ q(|x|) · · · (Qiwi : |wi| ≤ q(|x|))[R(x, w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , wi)], where Qi is ∃ if i is odd and ∀ if i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Additionally, a language L′ is in Πp i exactly if L′ ∈ Σp i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The polynomial hierarchy is defined as the class PH = � i∈N+ Σp i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='3 This definition yields the same classes as the other common definition, drawn on in our introduction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=', Σp 3 = NPNPNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For each of the abovementioned classes (except PH), we can define the following canonical com- plete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The definition that follows is adapted from one given by Arora and Barak [AB09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For each i ≥ 1, the problem ΣiSAT, which is complete for Σp i , is defined as the set of quantified boolean formulas of the form (∃u1) · · · (Qiui)[ψ(u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , ui)] that are true, where ψ is a boolean formula, each of u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , ui denotes a list of a boolean variables, and Qi is ∃ if i is odd and ∀ if i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The problem ΠiSAT, which is complete for Πp i , is defined analogously, except that Qi is ∀ when i is odd, and ∃ if i is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' PSPACE is the class of languages accepted by a Turing machine that runs in polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Formally, PSPACE = � k∈N+ DSPACE[nk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Additionally, “PSPACE embodies the power of polyno- mially bounded quantifiers” [HO02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' And so, QBF, the set of all true quantified boolean formulas, 2L is the class of decision problems solvable by a deterministic Turing machine in logarithmic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It is known that L ⊆ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Thus if L = NP, then L = P = NP = PH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It is also true by the space hierarchy theorem that L ̸= PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Consequently, if PH = PSPACE, then L ̸= PH, and thus L ̸= NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 3Readers familiar with the concepts will notice that we omitted explicit mention of Σp 0, but it is of course contained in Σp 1 and thus not actually omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' This “omission” occurs simply because we do not need to refer to that class in the rest of this paper, and so it saves us the trouble of defining it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 2 is a canonical PSPACE-complete problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It’s easy to see that QBF = � i∈N ΣiSAT ∪ ΠiSAT, and thus PH ⊆ PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Skolemization is the process of removing existentially quantified variables from a quantified boolean formula, and replacing them with Skolem constants or Skolem functions, which are boolean functions that are only over the universally quantified variables that appear before the existentially quantified variable being removed in the quantifier order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Interested readers may consult the textbook by Chang and Lee [CL73] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 3 On Theorem 1 We will now focus on Theorem 1 of Sopin’s paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The theorem has been reformulated for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Theorem 1 ([Sop14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For an arbitrary n ∈ N+, let Φ = (Ω1x1)(Ω2x2) · · · (Ωnxn)[φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn)] be an arbitrary boolean formula where (Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , Ωn) ∈ {∃, ∀}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let I = {i | i ≤ n ∧ Ωi = ∃}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Then, Φ is a true quantified boolean formula if and only for every i ∈ I, there is a boolean function yi over the variables in {xj | j < i ∧ Ωj = ∀} such that the quantified boolean formula that results from substituting each xi with yi is a tautology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' To the best of our understanding, the theorem is essentially proposing that Skolemization preserves satisfiability, since the process described by the theorem resembles that of Skolemization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, it is well-known that Skolemization does in fact preserve satisfiability (for reference, see textbooks by Chang and Lee [CL73] and Loveland [Lov78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In order to leverage this technique to decide QBF, it follows that one would at the very least need a polynomial upper bound on the space complexity of Skolemization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Unfortunately, no such result is currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We elaborate this point further in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Additionally, we notice several issues with the given proof of Theorem 1 and discuss them in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In their proof of Theorem 1, the author introduces a recursive algorithm to test for membership in QBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It works by removing quantifiers sequentially and producing, based on the removed quantifier, a new and logically equivalent formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' After that, the author adds a note on how “a [boolean] function determines the truth table” [Sop14] of its variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The proof then follows with two examples to help illustrate its main argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' First, we comment on the structure of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It is unclear how each part of the proof relates to the other and to the main argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Therefore, throughout our analysis, we will attempt to understand the role of each part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Sopin presents a recursive algorithm to decide QBF and claims that the proof of Theorem 1 follows from it, however this algorithm is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The recursive step is the only defined aspect of the algorithm: It involves pulling off the first quantified variable and checking both values for that variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Depending on the quantifier, a new equation is formed using these two equations with both possible values of first variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In the definition of this algorithm, there is neither a base case nor a recursive call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' A base case is needed in order to show when the algorithm terminates and a recursive call is needed in order to show where the recursive step is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' So it is ambiguous as to whether the algorithm presented is indeed recursive, or if only the first quantifier is to be pulled off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It appears to us that is a common algorithm to decide QBF, but it is unclear how the proof of Theorem 1 follows from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 3 We would now like to turn our attention to a statement made at the end of the algorithm’s description: “Notice that a [boolean] function determines the truth table (one-to-one correspon- dence)” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The term “truth table” in this context is ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' If the intention of the paper is indeed to use truth-tables to represent Skolem functions, as in Example 1 of Theorem 1 (discussed later), then this is a matter of concern, as the size of a truth-table is exponential in the number of its variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' On the other hand, it’s possible that the quoted statement is simply a fact of the relationship between boolean functions and truth-tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Still, the paper fails to show the potential space complexity of this transformation, which is crucial when dealing with PSPACE-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' At the end of the proof of Theorem 1, two examples are presented in order to add clarity to the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Example 1 provides an explicit description of a Skolem function, where exis- tentially quantified variables can be rewritten as functions of the variables that come before it in the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' At the end of the example, it is mentioned that the Skolem function “is indeed the truth table, where values of x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xk−1 determine the value of xk” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' As mentioned before, the use of truth-table in this context would lead to an exponential size blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Exam- ple 2 is just a restatement of the theorem using an explicit quantified boolean formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Example 2 states that “∀x1∃z1∀x2∃z2∀x3∃z3 is a true [quantified boolean formula] if and only if there ex- ist such boolean functions y1 : {0, 1} → {0, 1}, y2 : {0, 1}2 → {0, 1}, y3 : {0, 1}3 → {0, 1} that φ(x1, y1(x1), x2, y2(x1, x2), x3, y3(x1, x2, x3)) is tautology” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' This example just shows how a formula looks after Skolemization has occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The formula must be a tautology in order to be a true quantified boolean formula because only universally quantified variables are left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The replacement of existentially quantified variables with boolean functions over the correct variables appears to have been performed correctly but there is neither a proof that these functions are Skolem functions nor a direct connection to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Thus while the statement is true, the proof given does not support the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It’s unclear, based on our observations about Skolemization not being known to be computable using polynomial space, how this theorem will become useful in the rest of Sopin’s paper (and indeed, we discuss in the next section that we see no such connection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 4 On Theorem 2 In this section, we will focus our attention to the following theorem of Sopin’s paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Theorem 2 ([Sop14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Πp 4 = PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' As one would expect, the purported proof takes an arbitrary quantified boolean formula Φ and, from it, constructs a quantified boolean formula Φ′ (which has a specific syntactic form that we specify later in this section) in polynomial time, such that Φ ∈ QBF ⇐⇒ Φ′ ∈ Π4SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let us preface that the proof presented in the paper is often unclear about what it means and makes several logical leaps, and so we approach the arguments in the proof using our best understanding of what the author could have meant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Our first comment is about clarity and touches on the form of the quantified boolean formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The author assumes that quantified boolean formulas are of the form (which we will often refer to as the “standard” form in this critique) (∀x1)(∃y1) · · · (∀xn)(∃yn)[φ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn, yn)], 4 for some n ∈ N+, which at a glance seems incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For example, the formula (∃x, y)(∀z)[(x∨y∨z)] is certainly in QBF, but is not included in the paper’s treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Indeed, the general form given seems to miss formulas with an odd number of variables, formulas that start with an existential quantifier, and formulas that have multiple variables bound to the same quantifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, what the paper does not make clear, is that all such formulas can be converted to the “standard” form in polynomial-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The key to putting arbitrary quantified boolean formulas into “standard” form is the use of dummy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='4 Our example, (∃x, y)(∀z)[(x ∨ y ∨ z)], introduced earlier in this paragraph exhibits all the issues that seem to exist with the “standard” form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' And so, we shall present, in an informal manner (since the issue is rather simple and does not warrant much more than an example), how to convert the above formula to “standard” form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let us first make the first quantifier be a universal quantifier by introducing the dummy variable α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' This yields the formula (∀α)(∃x, y)(∀z)[(x ∨ y ∨ z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Next, we separate the variables x and y so that each quantifier is only bound to one variable by introducing the dummy variable β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' This yields the formula (∀α)(∃x)(∀β)(∃y)(∀z)[(x ∨ y ∨ z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' To finish, since we need an even number of variables, we introduce the dummy variable γ and obtain (∀α)(∃x)(∀β)(∃y)(∀z)(∃γ)[(x ∨ y ∨ z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It is not hard to see that if the original formula has n variables, then the new formula will have at most 2n + 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We will now focus on the correctness of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For the rest of this section, we will fix, for some n ∈ N+ and some boolean formula (with no quantifiers) φ that is over 2n variables, the following quantified boolean formula Φ = (∀x1)(∃y1) · · · (∀xn)(∃yn)[φ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn, yn)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The paper seeks to construct, from Φ, the following (purportedly logically equivalent) formula (which is not in “standard” form) Φ′ =(∀x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn)(∃y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , yn)[φ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn, yn)∧ (∀ˆxn)(∃zn)[φ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn−1, yn−1, ˆxn, zn)]∧ (∀ˆxn−1, ˆxn)(∃zn−1, zn)[φ(x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , xn−2, yn−2, ˆxn−1, zn−1, ˆxn, zn)] ∧ · · · ∧ (∀ˆx2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , ˆxn)(∃z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , zn)[φ(x1, y1, ˆx2, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' , ˆxn, zn)]], such that Φ ∈ QBF ⇐⇒ Φ′ ∈ Π4SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The attempted proof is by induction on the number of variables in the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We will not repeat the entire argument presented there, and we urge interested readers to consult the original 4For our purposes, a dummy variable is one that is quantified over, but does not appear in the boolean formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' That is certainly legal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' For example, we can say that the formula (∀x)(∃y)(∀z)[(x ∨ y)] is over the set of variables {x, y, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In this case, z is a dummy variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 5 paper directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The gist of the purported inductive proof is as follows: We can swap the positions of quantifiers inside the original formula, and then we can detect in polynomial-time whether the formula’s truth value has been affected by the changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We note, off the bat, two major issues with this approach: (1) it does not leverage the implications of the statement of Theorem 1, and (2) the induction does not actually prove the logical equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let us address (1) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The only mention made to Theorem 1 in the purported proof of Theorem 2 is in the case where the number of variables, m, is greater or equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The paper states that by “taking off the first quantifier and checking both possible values for the first variable in [the] way we did in Theorem 1, we come to the m − 1 case” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' It is worth reiterating that the attempted proof of Theorem 1 simply presents a version of the (well-known) recursive algorithm to decide QBF in polynomial space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Thus while this approach of “eating” variables one at a time may help decide if the formula is true, it potentially uses an exponential amount of (nondeterministic) time, and there is no clear passage in the paper that clarifies the use of that algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let us now address (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In the purported inductive proof, the paper claims that for any quantified boolean formula ψ over two variables, x and y, it holds that (∀x)(∃y)[ψ(x, y)] ≡ (∃y)(∀x)[ψ(x, y)] if and only if ψ is neither the XOR function nor the negation of the XOR function, which is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' And thus, the argument in the paper states that as long as there is no way for ψ to be the XOR function (or its negation) when the values of all but two variables, with one being universally quantified over and the other being existentially quantified over, are fixed, then the induction holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The additional clauses in Φ′ are meant to play a role in supporting this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, the paper not only fails to explain how these additional clauses work and why they work, but it also seems to be missing cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Indeed, the paper states that the above check can be done in polynomial time since “there are [only] n2 such formulas” [Sop14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We believe this was derived by selecting, from Φ, one of the n variables that are universally quantified over and one of the n vari- ables that are existential quantifier over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, missing from the argument is the fact that each of the remaining 2n−2 variables can have one of two values (0 or 1), thus creating n222n−2 possible formulas to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' (It’s worth noting that the above check can easily be done in nondeterministic polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' However, the paper only states “polynomial time,” which, as is standard, implies “deterministic polynomial time.”) We thus conclude that the induction presented in that proof is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We mention briefly in passing that there are other minor errors, but we bear them no attention as they do not carry as much importance as the ones we pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Additionally, in the final parts of that proof, the paper mentions the formula’s algebraic normal form as being crucial to the argument and provides an example, but it is not clear why the algebraic normal form is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Thus we are not certain as to whether the author was hoping to achieve something different than what is being conveyed through the technical report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 5 A Note on Skolemization We add this section with the hopes that the use of Skolemization can be better explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Our expectation is that the size of the Skolem functions must have a role to play in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Indeed, if the size of such functions is not polynomially-bounded, then it is unclear how any skolemized formula can even be computed in polynomial space (and be a useful approach in this context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' On the other hand, if there is a polynomial upper bound on the size of Skolem functions, then one can 6 show something shocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' If there is a polynomial p : N → N+ such that, for each quantified boolean formula Φ, the Skolemization of Φ produces no Skolem function of size greater than p(|Φ|), then Σp 2 = PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The ⊆ relationship is well-known, so it suffices to show that under the assumptions of the above statement, PSPACE ⊆ Σp 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We will do so by showing that QBF ∈ Σp 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let p be a polynomial as defined by the proposition’s statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Without loss of generality, let us assume that our input is a quantified boolean formula (as we can easily detect that in polynomial time if it is not and immediately reject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Fix an arbitrary boolean formula Φ with n variables as our input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Let E denote the set of variables in Φ that are existentially quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Since Skolemization preserves satisfiability, it follows that Φ ∈ QBF ⇐⇒ there are ∥E∥ Skolem functions such that for each assignment to the variables not in E, the boolean formula that results from replacing each variable in E with the corresponding Skolem function is true under the current assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Because each Skolem function has size bounded by p(|Φ|), substituting the Skolem functions into Φ and checking the truth value of the resulting formula, given a specific assignment, can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Per our definition of Σp 2 in Section 2, this implies that QBF ∈ Σp 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' This strengthens a result by Akshay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' [ACG+18] who under similar assumptions conclude that Σp 2 = Πp 2 = PH (by leveraging the Karp–Lipton Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 6 Conclusion In this critique, we pointed out the errors in “PH = PSPACE” [Sop14] and concluded that Sopin’s paper fails to show that PSPACE and the polynomial hierarchy coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' The primary issue is that Sopin’s paper does not account for a potentially exponential amount of work needed to perform one of its checks that it claims can be done in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' We additionally find that the use of Skolemization in the attempted proof is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Indeed, it would be shocking if a machine could compute the Skolemization using only a polynomial amount of space as that would yield ground-breaking results (as proved in our Proposition 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Acknowledgements We would like to thank Erin Gibson, David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Sopin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' PH = PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Technical Report arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='0628v20 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='CC], Computing Research Repository, arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content='org/corr/, November 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Revised November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' [Yao85] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' Separating the polynomial-time hierarchy by oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, pages 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' IEEE Computer Society Press, October 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE1T4oBgHgl3EQf3QVe/content/2301.03487v1.pdf'} diff --git a/tNAzT4oBgHgl3EQfBfpr/content/tmp_files/2301.00943v1.pdf.txt b/tNAzT4oBgHgl3EQfBfpr/content/tmp_files/2301.00943v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..767a995659af8cf0545b6d64cd44ae7d2a47ce5f --- /dev/null +++ b/tNAzT4oBgHgl3EQfBfpr/content/tmp_files/2301.00943v1.pdf.txt @@ -0,0 +1,2661 @@ +Characterizing Architecture Related Posts and Their Usefulness in Stack +Overflow +Musengamana Jean de Dieua, Peng Lianga,∗, Mojtaba Shahinb, Arif Ali Khanc +aSchool of Computer Science, Wuhan University, 430072 Wuhan, China +bSchool of Computing Technologies, RMIT University, 3000 Melbourne, Australia +cM3S Empirical Software Engineering Research Unit, University of Oulu, 90014 Oulu, Finland +Abstract +Context: Stack Overflow (SO) has won the intention from software engineers (e.g., architects) to learn, +practice, and utilize development knowledge, such as Architectural Knowledge (AK). But little is known +about AK communicated in SO, which is a type of high-level but important knowledge in development. +Objective: This study aims to investigate the AK in SO posts in terms of their categories and charac- +teristics as well as their usefulness from the point of view of SO users. +Method: We conducted an exploratory study by qualitatively analyzing a statistically representative +sample of 968 Architecture Related Posts (ARPs) from SO. +Results: The main findings are: (1) architecture related questions can be classified into 9 core cate- +gories, in which “architecture configuration” is the most common category, followed by the “architecture +decision” category, and (2) architecture related questions that provide clear descriptions together with +architectural diagrams increase their likelihood of getting more than one answer, while poorly structured +architecture questions tend to only get one answer. +Conclusions: Our findings suggest that future research can focus on enabling automated approaches +and tools that could facilitate the search and (re)use of AK in SO. SO users can refer to our proposed +guidelines to compose architecture related questions with the likelihood of getting more responses in SO. +Keywords: +Architectural Knowledge, Architectural Level Element, Architecture Solution, Stack +Overflow, Usefulness +1. Introduction +Technical Questions and Answers (Q&A) sites, such as Stack Overflow (SO), have revolutionized how +users seek knowledge on the Internet [1]. SO has shown to be the most prominent community Q&A site +for knowledge sharing and learning in software development, and SO leverages the knowledge and skills of +its users, such as developers, to share their thoughts and experience by asking various types of technical +questions related to development and providing answers to these questions. Also, SO users can learn +novel techniques and tools from SO [2]. SO is predominately being used to solve coding problems [3], and +these problems are often not relevant or less interesting to architects because they focus on lower-level +implementation details [3]. However, ever since this site started growing and being popular, architects +have begun to share their competencies, experience, and design problems by asking architecture related +questions or providing architecture solutions, such as architecture tactics. In our recent industrial survey +on how developers search for architectural information [4], practitioners acknowledged Q&A sites (e.g., +SO) as the most useful source of architectural information (e.g., benefits and drawbacks of architecture +solutions). Hence, similar to searching and (re)using existing coding related answers provided in SO to +solve programming related problems, software engineers (e.g., architects and developers) also search and +(re)use existing architecture solutions in SO for addressing their design concerns. Thus, SO not only +∗Corresponding author at: School of Computer Science, Wuhan University, China. Tel.: +86 27 68776137; fax: +86 27 +68776027. +Email addresses: mjados@outlook.com (Musengamana Jean de Dieu), liangp@whu.edu.cn (Peng Liang), +mojtaba.shahin@rmit.edu.au (Mojtaba Shahin), arif.khan@oulu.fi (Arif Ali Khan) +Preprint submitted to Journal of Systems and Software +January 4, 2023 +arXiv:2301.00943v1 [cs.SE] 3 Jan 2023 + +accumulates code examples, but also curates a large number of architecture solutions provided to a wide +range of architecture related questions or design problems [5] [6]. +Although SO users discuss high-level knowledge in SO, for instance, architecture tactics and qual- +ity attributes knowledge [5], architecture knowledge for technology decisions [6], to date the majority +of the existing studies mainly focus on analyzing programming related knowledge in SO posts from +different perspectives. For example, Diamantopoulos et al. [7] employed source code information to +improve question-answering in SO, and Zhang et al. [8] investigated the quality of code examples in SO +programming related posts. Little work has focused on analyzing architectural knowledge provided in +Architecture Related Posts (ARPs) in SO. For instance, Bi et al. [5] mined posts from SO and structured +the design relationships between architectural tactics and quality attributes used in practice. Liu et al. +[9] extracted SO posts and mined the design pattern use scenarios and related design pattern pairs. +Soliman et al. [10] developed a search approach (i.e., a domain specific search approach) for searching +architecture knowledge in SO. In another work, Soliman et al. [11] conducted an empirical study with +50 software engineers, who used Google to make design decisions using Attribute Driven Design [12], +and they determined how effective web search engines are to find relevant architectural information from +various sources (including SO) and to capture AK. Malavolta et al. [13] extracted data from five open +source software repositories (including SO), and mined architectural tactics for energy-efficiency applied +by practitioners in real robotics projects. Tian et al. [14] studied SO users’ conception of architectural +smells using SO posts. The abovementioned studies extracted ARPs from SO and investigated archi- +tecture knowledge (high-level concepts) from different aspects. However, prior work is only based on +architecture related questions and their associated answers. In contrast, our work covers the entire ARP, +including its question, all comments under the question, all answers associated with the question, and +all comments under the answers. In addition, no prior study has specifically investigated architectural +knowledge provided in ARPs (answers) with regard to their usefulness. Moreover, there has been no +comprehensive research on exploring architectural knowledge communicated by SO users in terms of +their types, design contexts, characteristics, and usefulness, which is the focus of this study. Analyzing +and understanding how SO users deal with architecture design concerns in online developer communities, +such as SO, brings three benefits: (1) it provides key insights about the types of design problems SO +users face during their architecture design and the types of architecture solutions discussed as well as +their usefulness, (2) it can help to know the design contexts in which architecture problems are raised, +and (3) it can help to know the characteristics of architecture problems and solutions discussed. These +benefits provide an opportunity to develop new techniques and tools that can help SO users search and +(re)use architectural knowledge shared in online developer communities. Therefore, this study aims to +complement prior works by analyzing the characteristics and categories of ARPs in SO as well as their +usefulness from the point of view of SO users. In this study, we treated usefulness (one quality criterion +of posts, e.g., answers, in Q&A sites [15]) using the definition in [15] (i.e., are the answers useful to +address the questions?). +To achieve the goal of this study (see Section 3.1), we conducted an exploratory study to investigate +various aspects (e.g., categories and characteristics) of ARPs in SO. More specifically, we extracted +32,182 posts from SO. We went on to manually filter out irrelevant posts and got 10,423 candidate +ARPs. Since 10,423 candidate ARPs were a quite large dataset, and it was not easy to manually analyze +this size of dataset with human effort and get accurate and comprehensive results, we used the power +statistics and calculated a representative sample size [16] of these 10,423 ARPs. With a 95% confidence +level and 3% margin of error, the final representative sample size calculated was 968 ARPs. Then, we +randomly selected 968 ARPs from the 10,423 ARPs and analyzed them for answering a set of research +questions (see Table 1). Specifically, we manually analyzed the 968 ARPs using open coding and constant +comparison from Grounded Theory (GT) [17] to answer those research questions. The main results +and findings of this study are that: (1) SO users ask a broad spectrum of architecture related questions +ranging from architecture tool to architecture configuration, architecture implementation to architecture +deployment. +(2) The useful architecture solutions are classified into seven categories as a taxonomy +(see Figure 4), such as solution for architecture configuration, solution for architecture implementation, +architecture tactic, and architecture pattern. One observation is that the identified categories of these +posts (questions and answers) cover almost all the architecting activities that span from the initial +stages (i.e., architectural analysis and synthesis [18]) of architectural creation as well as the later stages +(i.e., architectural implementation and maintenance & evolution [19]) in a system lifecycle. Thus our +identified categories of ARPs can support the mentioned architecting activities during the architecture +lifecycle, and SO can be considered as one of the sources of architectural knowledge [6]. +We found +2 + +that architecture related questions that provide clear descriptions together with architectural diagrams +increase their likelihood of getting more than one answer, while poorly structured architecture questions +tend to only get one answer. (3) SO users frequently use two terms related to usefulness (i.e., useful +and helpful) to explicitly communicate about the usefulness of certain architecture solutions provided to +their associated architecture related questions. +This study makes the following three contributions: (1) a classification and characterization of ar- +chitecture related questions that SO users (e.g., developers) asked in SO; (2) a list of identified design +contexts in which architecture related questions were raised; and (3) a classification (i.e., a proposed +taxonomy) and characterization of useful architecture solutions in SO. Our study findings can be ben- +eficial to various stakeholders. For example, researchers can refer to our proposed taxonomy of useful +architecture solutions in SO as a guidance to develop new automated approaches and tools that could +mine and locate architecture solutions (e.g., solution for architecture configuration, see Figure 4) for +addressing similar design concerns (e.g., questions that ask about architecture configuration, see Table +5). This can facilitate SO users to check the questions and solutions that are relevant to their design +concerns. SO can use our results to better adjust its answers and comments organization mechanisms +and enhance the search and (re)use of useful architecture solutions in SO. +The rest of this paper is structured as follows: Section 2 presents the background of the study. +Section 3 describes the research methodology, and Section 4 elaborates the study results. Section 5 +analyzes the results and discusses their implications. Section 6 presents the threats to the validity of the +study results, and Section 7 summarizes the related work. Finally, Section 8 concludes this work with +potential areas of future research. +2. Background +In this section, we introduce the background concepts used in this study, including Stack Overflow, +architecture knowledge, architecture problem, design context, and architecture solution. +2.1. Stack Overflow +Stack Overflow is one of the websites that make Stack Exchange1 network, which provides a Q&A +platform for its users to share knowledge across various domains (e.g., programming, design, statistics, +mathematics). SO users exchange knowledge related to software development by asking questions or +providing answers to existing questions. Among other development knowledge, architecture knowledge, +such as drawbacks and benefits of architecture solutions (e.g., patterns and tactics) in certain application +domains, has been shared at SO to support architecting activities [20][21]. Mining architecture knowledge +in Q&A websites, specifically in SO, has been the subject of the architecture research community in recent +years, such as architectural knowledge for technology decisions [6] and architecture tactics and quality +attributes [5], in order to support the architecting process. +2.2. Architecture knowledge +Software Architecture (SA) is a set of structures comprising software elements, the relationships +among them, and the properties of the elements and relationships [22]. Building an architecture of a +software system often requires knowledge, especially architecture knowledge [23], and skills. Architecture +knowledge, such as architecture decisions and their rationale [24], benefits and drawbacks of architecture +solutions [6], is one of the most important types of knowledge in software development [22]. Architectural +knowledge is often described in various formats, such as textual and graphical representation [25] and this +knowledge is recorded in various sources, such as books [22], technical blogs and tutorials [11], developer +mailing lists (e.g., ArgoUML [26]), Q&A sites (e.g., SO [5]). In this study, we investigated architecture +knowledge discussed in SO from various aspects, such as categories and characteristics of ARPs in SO, +SO users’ discussions on the usefulness of architecture solutions provided in SO. +2.3. Architecture problem +Architecture problems (such as “any testable architecture or design pattern for an MFC applica- +tion?”2) arise during development when addressing specific architecture design concerns (e.g., quality +1https://stackexchange.com/sites +2https://tinyurl.com/2z69uzs5 +3 + +attributes) and their trade-offs [22]. There are various problems related to architecture design that are +asked in SO. In this study, we investigated the categories of architecture problems/questions, specifically, +the categories of architecture related questions asked in SO (see Section 4.1). +2.4. Design context +Design context of a software system comprises the knowledge that an architect needs to have about +the environment (e.g., a hardware platform) in which a system is expected to operate [27]. Design contexts +can be seen as “conditions that influence design decisions but are not specified explicitly as requirements” +[28]. Harper and Zheng suggested that design contexts are forces that influence stakeholders’ concerns +[29]. +There are some works that categorize design contexts. +Bedjeti et al. +identified four context +categories of an architecture viewpoint (i.e., platform context, user context, application context, and +organizational context) [27]. Petersen and Wohlin provided a checklist for documenting design contexts +from six perspectives: product, processes, practices and techniques, people, organization, and market [30]. +Groher and Weinreich studied environmental factors that influence architecture decision making [31], and +they identified eight categories: company size, business factors, organizational factors, technical factors, +cultural factors, individual factors, project factors, and decision scope. In our study, we investigated +the design contexts that were discussed in architectural related posts in SO, and we referred to the +classification of design contexts proposed in two existing studies [27][30] (see Section 4.2). +2.5. Architecture solution +Architecture solutions are the fundamental building blocks in modern software design and they +are used to address architecture design concerns [22]. There are various architecture solutions, such as +patterns, tactics, and frameworks for addressing different design concerns. Architecture patterns (e.g., +Model–View–Controller, Client-Server, Publish-Subscribe patterns) are reusable solutions to commonly +occurring problems in architecture design within given contexts [22]. Contrarily to changing implemen- +tation (e.g., low-level code), once an architecture solution (e.g., an architecture pattern) is adopted and +implemented, it is quite difficult and costly to change it [22]. Architecture patterns determine the overall +structure and behavior of a software system [32] and are typically selected early during development for +achieving multiple system requirements (e.g., quality attributes) [22]. In this study, we studied SO users’ +discussions on the usefulness of architecture solutions, for example, patterns, tactics, frameworks (see +Section 4.5), as well as the categories and characteristics (in Section 4.6) of architecture solutions that +were considered useful in SO. +3. Research design +We carried out an exploratory study on various aspects (e.g., categories and characteristics) of ARPs +in SO. In the following subsections, we describe the details of the research design of this study, including +the goal and Research Questions (RQs) in Section 3.1 and the execution of this study in Section 3.2. +3.1. Goal and research questions +The overall goal of this study based on Goal-Question-Metric approach [33] is “to analyze the ARPs +(questions and answers) in SO for the purpose of investigating their categories, characteristics, and +usefulness from the point of view of SO users in the context of software development in practice”. +Following the goal of this research, we derived six research questions (see Table 1) that aim to examine +four aspects that highlight the question and answer threads of SO posts, including (1) categorization of +architecture related questions, (2) the design contexts in which architecture related questions were raised, +(3) characterization of architecture related questions that have more than one answer and characteristics +of architecture related questions that only have one answer, and (4) categorization and characterization +of architecture solutions that are considered useful. +3.2. Study Execution +In this subsection, we describe the process of data collection and analysis of ARPs. Figure 1 shows +an overview of the two processes (i.e., data collection and analysis). +4 + +Table 1: Research questions and their rationale +Research Question +Rationale +RQ1. +What architecture related +questions are asked in SO? +Architecture related questions (design problems) are mainly asked to ad- +dress certain design concerns (e.g., quality attributes of a system) during +architecting activities, for example, architectural analysis. SO curates dif- +ferent types of architecture related questions that are raised with various +design issues. The answer to this RQ can help researchers to be aware of +the areas of interest of SO users in architecture design and help practition- +ers to get an insight into the architecture related questions asked in SO so +that they can provide practical contributions. +RQ2. +What are the design con- +texts in which architecture related +questions were raised? +Design contexts comprise the knowledge about the environments in which +systems are expected to operate [27]. Design contexts are indispensable +ingredients that can drive the architecture design of a system [27]. A system +of similar functionalities can operate differently in different contexts [30]. +Although the importance of considering design contexts during architecture +design has been recognized, there is limited understanding on what design +contexts are considered in architecture design. The answer to this RQ can +help researchers and practitioners be aware of typical design contexts in +which architecture related questions are raised in SO. +RQ3. What are the characteristics +of architecture related questions in +SO that have more than one an- +swer? +One major challenge during architecture design is choosing the right archi- +tecture solutions to address the requirements of the systems [34]. Although +different architecture solutions act as alternative solutions to similar ar- +chitecture problems, they differ in terms of their qualities [34]. Therefore, +providing more than one answer (e.g., alternative solutions) to architecture +problems/questions is important as they provide a wide range of possibili- +ties for making architecture design decisions. With this RQ, we identify and +examine the characteristics of architecture related questions that get more +than one answer. By characteristics of an architecture related question, +we mean certain features, such as architectural diagrams, in the content of +the question or question formulation [35], that distinguish the architecture +related question to another or make the architecture related question get +attraction from SO users and get more than one answer. The answer to this +RQ can help researchers and practitioners know what motivates SO users +to provide more solutions to these questions, and consequently improve or +prevent unanswered architecture related questions in SO. +RQ4. What are the characteristics +of architecture related questions in +SO that only have one answer? +Some architecture related questions fail to continuously get attention from +SO users by answering them. Similar to RQ3, we want to examine the +factors behind this situation. We study the characteristics of architecture +related questions that only get one answer. The answer of this RQ can help +researchers and practitioners know what demotivates SO users to continue +answering these questions and design general guidelines for SO users to +compose architecture related questions with the likelihood of getting more +responses in SO. +RQ5. +What are the types of ar- +chitecture solutions provided in SO +that are considered useful by SO +users? +There are many architecture solutions (e.g., tactics and patterns) to ad- +dress architecture related questions provided in SO. However, the quality of +solutions/answers provided in SO has been a major concern for researchers +and practitioners. As elaborated in the related work (see Section 7), this +is evident in the growing number of studies, in which the focus is on an- +alyzing the quality of the content in SO posts from different perspectives, +for example, code and text. The results of this RQ can help researchers +and practitioners be aware of types (a taxonomy) of architecture solutions +considered useful in SO. +RQ6. What are the characteristics +of architecture solutions in SO that +are considered useful by SO users? +Zhang et al. [8] argued that accepted, highly voted, and frequently viewed +SO posts are not always reliable or useful in SO. Identifying the features +of architecture solutions that are considered useful helps to better under- +stand what SO users consider when accepting architecture solutions as +useful ones, thus providing insights for improving the current answering +mechanism of architecture related questions and helping SO users retrieve +their desired architecture solutions. +5 + +Phase I: Gathering Architecture Related Posts (ARPs) +Stack Overflow +Query with keywords +Returned posts +Returned posts +Query with keywords +32,182 + retrieved posts +Filter ARPs from others posts +(e.g., programming and +hardware related posts) +Is it an +ARP? +Stack +Exchange +API +Pilot +filtering +Disagreements +& +Discussions +Formal +filtering +Apply inclusion & +exclusion criteria +10,423 valid +ARPs +Consensus on +inclusion & +exclusion +criteria +Phase II: Determining the representative sample size +968 ARPs (RQ1, RQ2) +Identifying questions with +more than one answer +Identifying ARPs with +useful information +Data extraction +and analysis +650 questions (RQ3) +ARP +categories, characteristics, +and design contexts +Identifying questions with +only one answer +318 questions (RQ4) +324 ARPs (RQ5, RQ6) +Figure 1: An overview of data collection and analysis +3.2.1. Data collection +Our data collection is divided into two phases, namely Phase I: Gathering architecture related posts +and Phase II: Determining the representative sample size, as detailed below: +Phase I: Gathering architecture related posts +a) Search terms: Before we decided the most suitable terms for capturing posts relevant to architec- +ture design, we first performed a pilot search with several terms, namely “architect*” (i.e., “architect”, +“architecture”, “architectural”, and “architecting”) and “design*” (i.e., “design” and “designing”), within +SO. The process was carried out by using a SQL query through the query interface provided by StackEx- +change Data Explorer3, which is a web interface that allows the execution of SQL queries on data from +Q&A sites, including SO. After the pilot search with the mentioned terms, we saw that SO users mostly +use the terms “design*” (i.e., “design” and “designing”) in the programming context in SO, for instance, +“singleton design pattern”4. Moreover, we were aware that Soliman et al. [6] identified distinctive terms +between ARPs and pure programming posts from their studied sample of SO posts. However, in our +study, we did not use those distinctive terms to search ARPs in SO due to the following two reasons: +(1) The purposes of our work and Soliman et al.’s work in [6] are different. The purpose of the work +in [6] is technology related architecture knowledge extraction from SO. Specifically, the authors in [6] +identified and analyzed ARPs that mainly discuss architectural knowledge for technology decisions, such +as the pros and cons of a technology solution in a certain application. In addition, the authors in [6] +claimed that they did not find many pure architectural concepts (such as architectural pattern or tactic) +in their dataset of ARPs. In contrast, our study takes the problems from a wider scope. Specifically, +our study aims to identify and analyze ARPs from SO by looking at various architectural information, +including architecture patterns, tactics. Therefore, using the specific distinctive terms, such as versus, +alternative, pros, cons, xmpp, that were found in [6] may lead to missing other relevant ARPs, which +may affect the completeness of the retrieved ARPs. +(2) Using the distinctive terms found in Soliman et al.’s work [6] may lead to bias in the search +results. The relevancy and completeness of extracted ARPs may affect the correctness of the answers to +our six RQs (see Table 1). Thus, including the specific distinctive terms that were found in [6] in the +search queries may lead to the situation that the search results are biased to those terms. +3https://data.stackexchange.com/stackoverflow/query/new +4https://tinyurl.com/8yks7nhm +6 + +Therefore, we selected the general terms “architect*” (i.e., “architect”, “architecture”, “architec- +tural”, and “architecting”) to be used in our search. It is worth mentioning that we did not use the +search terms to search exclusively through tags only because tags can sometimes be less informative +and ineffective [36]. There are several disadvantages of using tags as the only approach to determine +whether a post is related to a topic. This is due to the reason that a user who created a post could +be unsure about the title of the most appropriate tag for their discussion, which can lead to the use +of incorrect or irrelevant tags [37]. For example, in this architecture related post5 that asked for an +architecture pattern that can be used in the design of a single webform application, a developer used +tags (“jc#”, “asp.net”, and “web”), and these tags cannot immediately tell in which contexts (e.g., +architecture or programming context) they are really used. Another problem with user-defined tags is +that users may try to add as many tags as possible (SO allows up to 5 tags) to raise the number of views +and probably increase the probability of getting responses quickly [38]. Thus, while tags can be helpful +to capture posts related to architecture design, using tags exclusively may miss important posts on this +topic. Hence, we decided to add the title and body of the questions into the search. For example, by +following the criteria of the query interface provided by Stack Exchange6, for the term “architect”, we +searched in the title, tags, and body of posts by using this query: SELECT p.Id, p.Tags, p.Title, +p.Body as “Questions Body”, p.Score as “Questions Score”, p.Answercount as “Answer +Count” FROM Posts p WHERE (p.Body like ‘%architect%’ or p.Title like ‘%architect%’ or +p.Tags like ‘%architect%’) AND p.Score >0 and p.AnswerCount <>0 ORDER BY p.Score DESC. +In our replication package [39], we provided the complete SQL query used to search ARPs in SO, such +as how the title, tags, and body of a post were combined during the search. The searching process +resulted in 32,182 posts (see Figure 1). Note that we used the mentioned search terms (i.e., “architect”, +“architecture”, “architectural”, and “architecting”), not for the purpose of accumulating all ARPs in +SO, but for gathering sufficient data for a relatively comprehensive analysis to achieve the goal of this +study. +b) Filtering ARPs from other posts (i.e., programming and hardware related posts): We found that +SO users use the term “architecture” not only in the context of software architecture, but also in other +contexts, such as hardware architecture context (e.g., ARM6 CPU architecture7) and programming con- +text (e.g., array architecture8), when describing their concerns in the SO posts. Therefore, we need to +filter the retrieved 32,182 posts and exclude those posts related to programming and hardware architec- +ture. To do so, we performed context analysis and applied our defined inclusion and exclusion criteria +(see Table 2) to accurately filter and separate software ARPs from other types of posts mentioned above. +Before the formal post filtering (manual inspection), to reach an agreement about the inclusion +and exclusion criteria (see Table 2), a pilot filtering was performed whereby the first author took a +random sample of 1,000 posts from the 32,182 posts. +He manually checked them with our defined +criteria (see Table 2). The other three authors checked and examined the results so that all the authors +(four authors) of this study could get a consensus on the understanding of the defined inclusion and +exclusion criteria. Thereafter, we got controversy and misunderstanding on 51 posts from the filtered +results. Such controversy and misunderstanding were discussed between the four authors of this study +till a consensus was reached. The first author carried on with the formal post filtering based on the +inclusion and exclusion criteria. The process continued till all the 32,182 posts were manually checked. +This step resulted in 10,423 candidate ARPs (see Figure 1). The results from this round were checked +and verified by the other three authors of this study, and it took us twenty one full days to identify and +separate ARPs from programming and hardware architecture related posts in these 32,182 posts. +Phase II: Determining the representative sample size +The 10,423 candidate ARPs (filtered from the previous phase (i.e., Phase I)) are a quite large +dataset, and it is not realistic to analyze this size of dataset with human effort and get accurate and +comprehensive results. Thus, in order to get statistically significant results, we used the power statistics +and calculated a representative sample size [16] of these 10,423 ARPs. At a confidence level of 95%, +we set a margin of error (i.e., how much we can expect our analysis results to reflect the view of the +overall dataset) to 3% for the whole 10,423 ARPs. The final representative sample size calculated is 968 +5https://tinyurl.com/mb9y37z4 +6https://data.stackexchange.com/stackoverflow/query/new +7https://tinyurl.com/f8sjvzz2 +8https://tinyurl.com/3ps7b3ek +7 + +Table 2: Inclusion and exclusion criteria for filtering ARPs from programming and hardware related posts +Inclusion criteria +I1. An ARP should contain a discussion on software architecture, for example, architecture design and +architecture tactics. +I2. An ARP should contain at least one answer attached to its architecture related question as we aim to +study the factors that make these questions have more than one answer or only have one answer and the +usefulness of their answers. +I3. An ARP should contain at least one data item that can be extracted according to the data items +defined in Table 4. +Exclusion criteria +E1. An ARP that has a score (i.e., medium number of down/upvote) that is less than 1 is excluded since +we want to make sure that all studied posts have attracted enough attention from the community [40]. +ARPs. Then, we randomly selected 968 ARPs from the 10,423 ARPs and analyzed them for answering +the six RQs (see Table 1). To be more specific, except for RQ1 and RQ2 on which we used 968 (a +representative sample size of ARPs) to answer them, we used subsets of the 968 ARPs that satisfy our +defined criteria (in the following steps) with respect to the purposes of the remaining RQs (i.e., RQ3, +RQ4, RQ5, and RQ6) (see Table 1). We followed the following steps to further divide our calculated +representative sample size of ARPs (968) into subsets of the ARPs that are relevant to answering the +remaining RQs: +Step 1: Identification of questions that have more than one answer (RQ3) and questions that only +have one answer (RQ4). As stated in the rationale of these two RQs (see Table 1), we want to examine +the factors that make such architecture questions get more than one answer or only get one answer. +Thus we considered comments posted on architecture related questions by referring to these two studies +[41][42], in which the authors argued that the quality of an answer is a combination of both the answer +and its associated comments as comments may provide additional information about the answer, for +example, improvement of answers [42] and obsoleted answers [41]. Therefore, in our study, we included +comments posted on architecture related questions and studied what makes these posts get more than +one answer (RQ3) or only get one answer (RQ4). More specifically, the first author manually checked +968 ARPs and their comments, and got 650 architecture related questions (wherein each question has +more than one answer). These questions were used to answer RQ3 (see Figure 1). On the other hand, +the first author followed the same procedure (manual inspection of the 968 ARPs for RQ3) and got 318 +architecture related questions (wherein each question has only one answer). These questions were used +to answer RQ4 (see Figure 1). +Step 2: Identification of ARPs with useful knowledge. For answering RQ5 and RQ6, we need to +identify ARPs with useful knowledge from the representative random sample of ARPs (968 ARPs). We +referred to these two studies (i.e., [41, 42]) to identify ARPs with usefulness knowledge. These studies +by Zhang et al. +[41, 42] argued that the quality of an answer is a combination of both the answer +and its associated comments as comments may provide additional information to support the answer, +such as improvement of answers [42] and obsoleted answers [41]. Therefore, in this study, we included +the information in comments to gain a deep understanding of how SO users discuss the usefulness of +architecture solutions provided to their architecture related questions in SO. In this study, we did not +consider vote score for answering RQ5 and RQ6 since even highly voted SO posts are not always reliable +or useful as argued by Zhang et al. [8]. We observed that SO users occasionally use terms related to +usefulness, such as “helpful”, in the comments to indicate that certain architecture solutions provided to +their architecture related questions are useful (see Figure 2). Thus, based on this observation along with +the aid of our defined selection criteria in Table 3, we manually checked and filtered the 968 ARPs to +identify solution threads with useful knowledge (each solution thread includes all solutions to a question +(i.e., accepted & not-accepted solutions) and all the comments that are associated with the solutions. +Specifically, to filter out ARPs that do not discuss the usefulness of architecture solutions and reach an +agreement about the criteria defined in Table 3, two authors (i.e., the first and second authors) did a pilot +ARPs filtering. They independently and manually examined a random sample of 20 ARPs from the 968 +8 + +ARPs. Similar procedure (i.e., selecting a random sample of data from a large dataset and subsequent +manual filtering) has also been employed in recent studies, such as [43]. To measure the inter-rater +agreement between the first two authors, we calculated the Cohen’s Kappa coefficient [44] and got an +agreement of 0.898. Note that before a solution was finally included as a relevant one (i.e., a useful +solution), the first and second authors first read the solution that was commented to be useful/helpful in +order to verify if it is really useful to address the question. Disagreements on the ARPs were discussed +between the two authors till a consensus was reached. Then the first author carried on to check and +filter the remaining ARPs. The number of resulting ARPs (with useful knowledge) that were used to +answer the two RQs (RQ5 and RQ6) is 324 ARPs (see Figure 1). +Figure 2: An example answer that was commented to be helpful +Table 3: Inclusion and exclusion criteria for identifying ARPs with useful knowledge +Inclusion criterion +I1. A comment in an answer thread must contain one of the keywords related to usefulness, such as “useful”, +“helpful”, “beneficial”, “handy”, and “effective”, and this comment is used to signify the usefulness of the +answer. +Exclusion criteria +E1. The keyword related to usefulness, for example, “useful”, “helpful”, “beneficial”, is used to talk about +something else (e.g., a question or answer itself is related to a “usefulness” topic rather than being a sign +that the answer is likely useful). +E2. An ARP with controversy discussions on the answer (i.e., if there are two comments in the same +answer thread, and one states the usefulness of the answer while another states its uselessness) is not +included. +3.2.2. Data extraction and analysis +(1) Data extraction: We performed the data extraction process by identifying the relevant informa- +tion to be extracted from 968 ARPs to answer our defined RQs (see Table 1). In Table 4, we present +the data items for which the relevant information was extracted from the candidate ARPs. It also shows +the RQs that are supposed to be answered using the extracted data. The data extraction was subse- +quently followed by data analysis, and these two processes were conducted and recorded with the aid of +MAXQDA (a qualitative data analysis tool)9. +9https://www.maxqda.com/ +9 + +Okay so as an electrical engineer who works with software involving C# and serial ports almost +every day, here is some advice that I can offer you from an architectural perspective. +1 +1. Use this architecture in the situation where you need to control the motors across several +application domains. Have some process/service running in the background that controls +your port 1oo% of the time (via your APl). You can launch an application then talk to the +background service (responsible for controlling the motor) via TCP sockets. This way you can +launch as many applications as you want and everybody will get access to the APl without +having to worry about serial port access issues. +2. Use this architecture in the situation where you need to control the motors in a single +application domain. This one's similar to what you're already proposing in your question, +which by the way I think is a pretty good way of doing things. Instantiate the class to control +the motors from your APl and then use constructor/property injection, or some kind of Dl to +pass a reference to the controller to everybody who needs it. +Share Edit Follow +answered Feb 24 '17 at 17:22 +Snoop +95311127 +Thanks, Snoopy! I've been working on this project for a week now, and I've learned a ton so far. I decided to +go with a static service for controlling all of the different devices from my APl, and just running any +commands I need to send to them in background threads. It seems to be working really well, and the result is +even cooler (l can control motors over the web, so cool). Thanks for the tips, definitely helpful! +Brian CorbinFeb 27 '17 at 0:13 Table 4: Data items to be extracted from the ARPs with their description, analysis approaches, and relevant RQs +# +Data item +Description +Data analysis approach +RQs +D1 +Content +of +the +question +The main content of the +question in the ARP +Open coding & constant com- +parison +RQ1 +D2 +Design context +The design context elabo- +rated in the content of the +question in the ARP +Predefined classifications in [27] +and [30] +RQ2 +D3 +Content +of +the +question +and +its +comments +The main content of the +question and a summary of +the question’s comments in +the ARP +Open coding & constant com- +parison +RQ3, RQ4 +D4 +Content of the an- +swer +The main content of the an- +swer in the ARP +Open coding & constant com- +parison +RQ5 +D5 +Content of the an- +swer and its com- +ments +The main content of the an- +swer and a summary of the +answer’s comments in the +ARP +Open coding & constant com- +parison +RQ6 +(2) Data analysis: +Similarly to several existing studies (e.g., [4]), we used open coding & constant comparison to answer +RQ1 and RQ3-RQ6 in our study. Open coding & constant comparison are two widely used techniques +from Grounded Theory [17] during qualitative data analysis. Grounded Theory (GT) is a bottom-up +approach and focuses on theory generation, rather than extending or verifying existing theories [17]. +Open coding generates codes for incidents that can be further classified into concepts and categories [17]. +Constant comparison is a continuous process for verifying the generated concepts and categories. Both +concepts and categories evolve and saturate until they fit the data [17]. Thus, in this study, we employed +open coding & constant comparison techniques from GT to generate the concepts and categories for +answering RQ1 and RQ3-RQ6. Specifically, we used open coding to encode the extracted data items +for RQ1 and RQ3-RQ6 (see Table 4) to generate codes. Afterwards, we applied constant comparison to +compare the codes identified in one piece of data with the codes that emerge from other data to identify +the codes which have similar semantic meanings. We proceeded to group similar codes into high-level +concepts and categories. On the other hand, we employed predefined classifications of design contexts +in [27] and [30] to answer RQ2. We followed the same procedure (encoding and grouping similar codes +into high-level categories) to answer RQ2. +Before the formal data analysis, the first author conducted a pilot data analysis for each RQ. +Specifically, this analysis process involved the following steps: (1) The first author selected a random set of +100 ARPs from the representative sample size calculated (i.e., 968 ARPs). (2) The first author coded the +extracted data (see Table 4) for each RQ. When such posts were unclear and the first author got confused +while coding the extracted data, physical meetings with the second author were scheduled to solve such +confusion. (3) The first author applied constant comparison and grouped all the codes into higher-level +concepts and turned them into categories and subcategories. The grouping process was iterative, in +which the first author continuously went back and forth between the concepts, categories, subcategories, +and contents of the questions, answers, and comments to revise and refine the concepts, categories, and +subcategories. (4) Thereafter, other three authors (the second, third, and fourth authors) checked and +validated the results from the pilot data analysis (i.e., concepts, categories, and subcategories). The +disagreements were resolved in a meeting using a negotiated agreement approach [45] to improve the +reliability of the pilot data analysis results. The first author carried on with the formal data analysis +and followed similar steps used during the pilot data analysis. In the following paragraphs, we provide +details of the formal data analysis process: +a) For analyzing RQ1, RQ3, and RQ4 +As abovementioned, we used open coding and constant comparison [17] to manually analyze the +extracted data (i.e., content of the question for RQ1 and content of the question and its comments for +10 + +RQ3 and RQ4) as shown in Table 4. With these RQs, we investigated architecture related questions from +two aspects, namely categorization (RQ1) and characterization (RQ3 and RQ4) of these questions (see +Table 1). Specifically, regarding the categorization of the questions, the first author studied the content +of each architecture related question (from the ARPs that are relevant to answer RQ1 (see Figure 1)) +by exploring and identifying their main purposes (e.g., design concerns), such as asking for help on +how to refactor the architecture of a system (e.g., refactoring of circular dependencies). Thereafter, +the first author summarized each question’s purpose in a short sentence. Firstly, the first author went +on to encode the summarized sentence. This process was iterative, in which he continuously applied +this technique till all ARPs in the dataset were encoded. Secondly, the first author applied constant +comparison to compare the codes identified in one summarized sentence with the codes that emerged +from other summarized sentences to check the codes which have similar semantic meanings. The first +author proceeded to group similar codes into high-level concepts, categories and subcategories. The +grouping process was iterative, in which the first author continuously went back and forth between the +concepts, categories, subcategories, and contents of the questions to revise and refine both the concepts, +categories and their subcategories. To mitigate the personal bias during the formal data analysis, the +other authors (second, third, and fourth authors) of this study participated in the validation of the +generated codes, concepts, categories, and subcategories. The disagreements were resolved in a meeting +using the negotiated agreement approach [45] to improve the reliability of the analysis results for RQ1 as +during the pilot data analysis. We finally got 9 high-level categories and 21 subcategories as the results +of RQ1, and these results are fully elaborated in Section 4.1. +As mentioned in Section 3.2.1, for answering RQ3 and RQ4, we manually checked the 968 ARPs to +identify ARPs with more than one answer (RQ3) and ARPs with only one answer (RQ4). This led to two +subsets of the 968 ARPs (i.e., 650 ARPs and 318 ARPs) relevant to answering RQ3 and RQ4 (see Figure +1). Specifically, in the formal data analysis, the first author analyzed the questions and their attached +comments (from the ARPs of the two mentioned subsets, 650 ARPs and 318 ARPs, of the 968 ARPs) +by encoding the extracted data for the two RQs (i.e., content of the question and comments as shown in +Table 4). He wanted to study if there might be such factors, for example, question formulation [35] or +certain features in the question (e.g., architecture diagram) that contribute to such architecture related +questions having more than one answer or only having one answer. For example, when investigating RQ3 +(questions with more than one answer), one community member posted a comment under an architecture +related question saying that “+1 great question, very well-articulated”10, for this comment, he picked +a phrase (i.e., a summary of that comment) “well-articulated”. Subsequently, he went on to study the +content of the question (e.g., how architectural information is stated in the question) under which this +comment was commented, and then he came up with one code that fits this question. He followed the +same processes (e.g., grouping similar codes into high-level concepts, categories) that the used when +analyzing RQ1 to analyze these two RQs (RQ3 and RQ4). To mitigate the personal bias, the results +from this analysis were checked and validated by other three authors of this study. As in the analysis +of RQ1, we held a meeting and followed the negotiated agreement approach [45] to discuss and resolve +any disagreement, therefore improving the reliability of the analysis results for RQ3 and RQ4. In the +final analysis, we generated four characteristics of architecture related questions that have more than +one answer and five characteristics of architecture related questions that are that only have one answer +as the results of RQ3 and RQ4, respectively. The details about these four characteristics for RQ3 and +five characteristics for RQ4 are provided in Section 4.3 and Section 4.4. +b) For analyzing RQ2 +We employed pre-defined classifications in [27] and [30] to answer RQ2. Specifically, the first author +manually analyzed the extracted data for RQ2 (i.e., design contexts, see Table 4) from the questions +in the ARPs relevant to answering RQ2 (see Figure 1). The first author then examined the extracted +data to investigate the design contexts in which architecture related questions were raised. By referring +to the categories of design contexts presented in the abovementioned studies, three main categories and +eight subcategories were generated from the analyzed ARP questions. The personal bias was mitigated +through the validation of the generated categories and subcategories with the other three authors of +this study. As in the analysis of the previous RQs, the disagreements were discussed and resolved in a +meeting using the negotiated agreement approach [45] to improve the reliability of the analysis results +for RQ2. The results of RQ2 are presented in Section 4.2. +10https://tinyurl.com/dfw27h38 +11 + +c) For analyzing RQ5 and RQ6 +These two RQs aim to investigate the usefulness of architecture solutions in SO. As discussed in +Section 3.2.1, in order to answer these two RQs, we defined a set of criteria (see Table 3) and filtered +ARPs with useful knowledge. For clarity, we examined these two RQs from two aspects: +We investigated how SO users discuss the usefulness of architecture solutions attributed to their +associated architecture related questions. We needed to gain insights into ways (e.g., terms) SO users +may use to communicate the usefulness of architecture solutions in SO. In addition, understanding SO +users’ discussions on the usefulness of these solutions is important to direct Q&A platform owners in +creating the mechanisms that can help their users to efficiently and effectively search and (re)use such +useful architecture solutions. To achieve this, we manually checked comments attached to the solutions +in the 968 ARPs with the aid of our defined criteria (see Table 3). We found that SO users occasionally +use terms related to usefulness, such as “helpful”, in the comments along with other terms (e.g., “very”, +“super”) to explicitly convey how useful they found certain architecture solutions (e.g., see Figure 2). As +stated in Section 3.2.1, the number of resulting ARPs (with useful knowledge) that were used to answer +the two RQs (RQ5 and RQ6) is 324 ARPs (see Figure 1). It is worth noting that we did not count on +the occurrence of the terms, e.g., “useful” (and similar) stated in comments to measure the usefulness of +such architecture solution given to an architecture related question in our study. As explained above, we +referred to information in the comments attacked to the solutions to examine SO users’ discussions on the +usefulness of architecture solutions. We mean the reaction of SO users after seeing and using architecture +solutions given to their associated architecture related questions, for example, see a comment in Figure +2. Moreover, before such ARPs (solutions) were finally included for analysis, we first read the solutions +commented to be helpful and their associated questions to check if they are really useful (i.e., are the +solutions/answers useful to address the questions? [15]). In Section 4.5, we provide more details about +the identified terms related to usefulness (e.g., “helpful”) along with other terms (e.g., “extremely”, +“very”) (from 324 ARPs) that SO users use in comments to explicitly signify how useful they found +architecture solutions provided to their associated questions. +During the data analysis for these two RQs (i.e., the taxonomy of architecture solutions considered +useful (RQ5) and their characteristics (RQ6)), the first author followed the same procedures (e.g., coding +and grouping similar codes into high-level categories) that were used when analyzing RQ1. One thing to +elucidate when analyzing RQ5 to construct a taxonomy is that the first round of grouping yielded seven +main categories. In the second round, the all the authors of this study proceeded to further generate +subcategories and types from these seven main categories, ensuring that these main categories, their +subcategories, and types follow an “is a” relationship. The grouping process was iterative, in which +the authors continuously went back and forth between categories, subcategories, types, and solutions +to refine the taxonomy. The final results of this analysis yielded a taxonomy of 7 main categories, 20 +subcategories of which 1 were encoded as “Others” (i.e., refer to codes that did not fit into the already +generated subcategories), and 85 types. Note that the negotiated agreement approach [45] was used to +discuss and resolve any disagreements. The final taxonomy as the result of RQ5 is elaborated in Section +4.5. Four characteristics were distilled as the results of RQ6 and are detailed in Sections 4.6. +Note that while categorizing and characterizing ARPs in SO by reading through those posts, we +observed that a single ARP may contain multiple types of architecture knowledge. For example, in +this ARP11 from our dataset, an SO user asked about alternative architecture patterns for Model View +Controller (MVC) pattern (i.e., alternative architecture solutions). In the question body, the user asked +the reasons that could drive someone to decide to use those alternative architecture patterns over MVC +(i.e., architecture decisions and their rationale), the types of systems that the alternative architecture +patterns are typical used for (i.e., design context), and the pros and cons that come along with using those +alternative architecture patterns (i.e., benefits and drawbacks of architecture solutions). We encoded such +a post with multiple types of architecture knowledge accordingly. Moreover, while analyzing ARPs in our +dataset, we noted down and then discussed the results (e.g., categories of architecture related questions +and taxonomy of useful architecture solutions) during the qualitative data analysis. This has led to +several interesting findings and actionable implications for various stakeholders, which are presented in +Section 4 and Section 5, respectively. The dataset collected and used in this study and the details of +data analysis (e.g., coding in MAXQDA) are available online for replication and validation purposes [39]. +11https://tinyurl.com/2d8r6w8m +12 + +4. Results +In this section, we present the results to our RQs that we got from data analysis (see Section +3.2.2). The result of each RQ is presented in a dedicated subsection, ending with the key findings of the +corresponding results. +4.1. Categories of architecture related questions (RQ1) +Categories of architecture related questions that SO users ask in SO were determined using the open +coding and constant comparison techniques described in Section 3.2.2. We examined questions in the 968 +ARPs to answer this RQ (see Figure 1). Our data analysis yielded 9 main categories and 21 subcategories +of architecture related questions. Table 5 shows the mentioned categories, their subcategories, their +percentages of occurrence (out of 968 ARP questions), and count information. As shown in Table 5, +architecture configuration (27%, 261 out of 968 ARP questions), architecture decision (19%, 181 out +of 968 ARP questions), and architecture concept (15%, 142 out of 968 ARP questions) are the top +three categories of most frequently asked architecture related questions. In the following, we report +those categories and subcategories. Where required, we provide an SO question example to support the +understanding of the categories and their subcategories. +Table 5: Categories of architecture related questions, their subcategories, and their counts & percentages +Category +Subcategory +Count +Architecture configuration (27%, 261) +Architecture configuration with technologies support +144 +Architecture pattern configuration +117 +Architecture decision (19%, 181) +Technology decision +104 +Behavioral decision +77 +Architecture concept (15%, 142) +Architecture overview +62 +Basic architectural concept +32 +Architecture component functionality +26 +Specific architecture pattern +22 +Architecture implementation (12%, 119) +Architecture component implementation +79 +Architecture pattern implementation +40 +Architecture tool (10%, 99) +Architecture modeling tool +34 +Model-based code generation tool +30 +Usage of architecture tool +21 +Code-based model generation tool +14 +Architecture evolution (6%, 55) +Architecture extension to meet new requirements +42 +Component extension to meet new requirements +13 +Architecture refactoring (5%, 45) +Refactoring of circular dependencies +21 +Refactoring of large components +13 +Refactoring of big ball of mud +11 +Architecture deployment (4%, 34) +Application deployment to meet quality attributes +24 +Application deployment to meet functional requirements +10 +Architecture documentation (3%, 32) +32 +(1) Architecture configuration questions in this category ask about how to configure compo- +nents and connectors in software systems. The types of components and connectors could either belong +to certain technologies (e.g., Windows Communication Foundation (WCF) and Windows Presentation +Foundation (WPF)) or other architectural concepts (e.g., architecture patterns). This category is the +most common (27%, 261 out of 968 ARP questions) category of architecture related questions in SO (see +13 + +Table 5). We further classified this category into two subcategories, in which architecture configuration +with technologies support surpasses half of the questions (144 out of 261 ARP questions of the architecture +configuration category) that SO users ask in this category (see Table 5). +• Architecture configuration with technologies support is concerned about how to configure an ar- +chitecture of an application with specific technologies (e.g., WPF, WCF). For instance, in this +question12, a developer asked about how to configure or build a scalable Web-based application by +using WCF: “Does anyone have any experience with how well web services build with Microsoft’s +WCF will scale to a large number of users? The level I’m thinking of is in the region of 1000+ client +users connecting to a collection of WCF services providing the business logic for our application +(...)”. +• Architecture pattern configuration seeks practical guidance on how to configure a specific architec- +ture pattern (e.g., Model View Controller and hexagonal architecture patterns) when designing an +application to achieve certain requirements (e.g., functional requirements). For example, in this +question13, a developer asked about how to configure an application that conforms to a Hexagonal +architecture pattern by stating that: “I’m looking for some guidance or best practices for how +to configure and structure an application which conforms to Hexagonal architecture that supports +multiple (driver) adapters simultaneously (...)”. +(2) Architecture decision: SO users ask this type of questions mostly when they want to decide +between two or more alternative architecture solutions when deigning their software systems. Among +two subcategories identified in this category, the technology decision subcategory contains the majority +of questions (104 out of 181 ARP questions of the architecture decision category) that SO users ask in +this category (see Table 5). +• Technology decision is mainly concerned about choosing between two or more technology solutions +(e.g., frameworks, databases) to meet certain requirements at the architecture level. Moreover, +various aspects can be considered during this choice, such as technology features, benefits, and +drawbacks [34]. For instance, in this question14, a developer asked about the reasons that could +drive him or her to decide to use Cassandra over HBase for his/her application by stating that: +“HBase is known for being a key-value store and random reads with .get and .put functions based on +the key. Is Cassandra a better choice for suiting a requirement of key-value store? Can it support +random reads based on key? If so, in which conditions should I choose Cassandra over HBase in a +Spark Streaming application?”. +• Behavioral decision is concerned with deciding how certain elements in a system would interact +together to provide some functionality or to satisfy certain quality attributes [46]. For example, +a developer wanted to decide on either to let clients connect directly to the database or let the +connection go through the web service by asking this question15: “Recently I have been developing +a system to run a high secured database (using vb.net and SQL Server 2005). I want to increase +the security of the database so no connection will be made directly to the database but instead +a HttpWebRequest is sent to a web service which then connects to the database and returns the +requested data table in XML format. My concern is just about the performance, I cannot decide +either to let clients connect directly to the database or let the connection go through the web service”. +(3) Architectural concept includes theoretical related questions about software architecture. We +divided this category into four subcategories, among which architecture overview contains the majority +of questions (62 out of 142 ARP questions of the architecture concept category) that SO users ask in this +category. +• Architecture overview questions are concerned with the information about the general working +mechanism or overview of certain existing architecture. For instance, in this question16, a developer +asked about architectural overview of Drupal version 7: “Could someone provide an architectural +overview of the Drupal 7 control flow? Perhaps in the sense of a flowchart about how a page gets +generated (...)”. +12https://tinyurl.com/yuxjp2su +13https://tinyurl.com/4kn6t27e +14https://tinyurl.com/k9xzkman +15https://tinyurl.com/4pjh3ufk +16https://tinyurl.com/2a9hb3ek +14 + +• Basic architectural concept refers to questions that seek explanations about basic concepts in soft- +ware architecture. For instance, in this question17, a developer was seeking explanations about +several architecture concepts, such as a architecture pattern: “Is MVC a pattern or architecture or +framework? What is a pattern? What is an architecture? (...)”. +• Architecture component functionality is concerned with the use, purpose, or functionality of certain +components in the architecture. For example, in this question18, a developer was asking about the +use or purpose of the lifecycle aware component in Android based application: “We already have a +Lifecycle in our Activity/Fragment then why will we use Lifecycle aware component & kindly guide +me the main purpose of it. And if we use lifecycle aware then why we use lifecycle that we knew +already”. +• Specific architecture pattern questions ask about particular architecture patterns that are commonly +used in the design of certain applications to address functional or non-functional requirements. For +instance, in this question19, a developer asked about commonly used architecture patterns for three- +dimensional (3D) video game applications: “What are some of the more common design patterns +used when developing 3D games? Are there any high-level architectural design patterns that are +commonly used? (...)”. +(4) Architecture implementation questions ask about how to implement a certain software +system according to its architecture design. The architecture design is refined in detailed design, and +then implemented in code. Architecture implementation category has two subcategories, among which +architecture component implementation occupies the majority of questions (79 out 119 ARP questions +of the architecture implementation category) that SO users ask in this category. +• Architecture component implementation is concerned with how components should be implemented +in the system. For instance, in this question20: “How to implement a single component sharing in +different modules in Angular 7 while using lazy loading?”. +• Architecture pattern implementation questions are about the ways certain architecture patterns are +implemented with regard to the fundamental design principles. For example, in this question21: +“How to implement MVC in Swift? I’ve been building Swift apps where basically all the functionality +is in the ViewController. I know this isn’t the optimal way to do it because design patterns help +you expand the app but I don’t really understand them (...). How do I go about turning this into a +Model-View-Controller design?”. +(5) Architecture tool: There are various architecture tools (e.g., Enterprise Architect, Archi, +Cloudcraft) that can be used to assist in the architecture design of a software system. With our dataset, +we found architecture related questions in which SO users ask about these tools and classified them +in the architecture tool category. +We further classified this category into four subcategories, among +which architecture modeling tool contains the majority of questions (34 out of 99 ARP questions of the +architecture tool category) that SO users ask in this category. +• Architecture modeling tool questions ask about tools that can enable the creation or drawing of +architectural diagrams to model or represent an architecture of a software system during the design. +For example, in this question22: “I am a newbie in TOGAF and I need to start a first trial. I +am trying to model my architecture. Which tool do you advise me to use in order to model my +architecture using TOGAF?”. +• Model-based code generation tool refers to questions that ask about architecture tools that can +enable the generation of code from architectural models. For instance, in this question23: “Please +suggest me any open source tool to generate C# code from UML designer (...). My requirement is +to have a code generation tool for C#”. +• Usage of architecture tool questions look for instructions on how to use certain architecture tools +17https://tinyurl.com/5burxuca +18https://tinyurl.com/3x35jzu6 +19https://tinyurl.com/2u2df8z5 +20https://tinyurl.com/kp73y3wk +21https://tinyurl.com/mpmvwb5c +22https://tinyurl.com/ndf7mrnc +23https://tinyurl.com/jwkvtzwc +15 + +(e.g., Archi, Microsoft Visio). For instance, in this question24: “How to add UML/layer diagram +to an existing solution in VS 2015 community? There is no architecture menu there?”. +• Code-based model generation tool is concerned with tools that assist in architectural models gen- +eration or recovery from the codebase. For example, in this question25: “I need to make a UML +class diagram for a project (...) I do not really want to write all the classes/functions manually, +so I was trying to generate the diagram from the source code but can’t seem to find a way or tool +to do it. (...)”. +(6) Architecture evolution: SO users ask this type of architecture related questions when seeking +help on how they can re-architect and expand their existing architecture for the purposes of achieving +certain new requirements (functional or non-functional requirements). Among two subcategories iden- +tified in this category, the architecture extension to meet new requirements subcategory contains the +majority of questions (42 out of 55 ARP questions of the architecture evolution category) that SO users +ask in this category (see Table 5). +• Architecture extension to meet new requirements is concerned with practical guidance for expanding +an existing architecture of a system to address certain new functional or non-functional require- +ments. The changes do not only happen in one component, but they may happen in almost the +whole architecture of the system. For example, in this question26: “I am expanding/converting a +legacy Web Forms application into a totally new MVC application. The expansion is both in terms +of technology as well as business use case (...). The new project has two primary goals: Extensibil- +ity (for currently and future pipeline requirements) and Performance (...). Is there a way in DDD +to achieve both, Extensibility that DDD provides and performance that DBDD provides?”. +• Component extension to meet new requirements includes questions that ask about the extension of +certain architectural components to meet some new functional or non-functional requirements in +existing and running software systems. This is different from the above subcategory, as here the +change or extension happens in local to a specific component or layer, rather than affecting the +whole architecture of the system. For instance, in this question27: “We are currently evaluating +CQRS and Event Sourcing architecture (...). What happens if, after an application has been up +and running for a while, there is a new requirement to add an additional field to a ViewModel on +the ReadModel database? Say, the Customer Zip Code is required on the CustomerList ViewModel, +where it was not previously”. +(7) Architecture refactoring: SO users ask this type of architecture related questions when +they want to restructure architecture of systems aiming at improving non-functional attributes of those +systems without modifying their external behaviors. This category includes three subcategories, where +most of the questions (21 out of 45 ARP questions) are related to the subcategory refactoring of circular +dependencies. +• Refactoring of circular dependencies is concerned with techniques and tools that can help remove +undesirable circular or cyclic dependency issues among modules so that layering violations can +be addressed and dependency structure can be improved in the systems. For instance, in this +question28: “I am working on the MVC project where I am following the layered architecture (...). +Now, my Business Logic Layer(BLL) is depending on the Data Access Layer (DAL) which is +depending on BLL because domain objects are inside BLL. So, both are having reference to each +other (...). How can I overcome the circular dependency?”. +• Refactoring of large components is concerned with approaches that can help refactor large compo- +nents in software systems. For instance, in this question29: “I have a pretty large table component +and I want to separate its body section into new component. Each time I am trying to do this, the +styling of table gets broken (...). I would like to have exactly this same page after this refactoring. +Does anyone know how to pass styling to this new child component, or how to make thing styling +work again ?”. +24https://tinyurl.com/52zffbw3 +25https://tinyurl.com/n7rz24mu +26https://tinyurl.com/ymbyzcvz +27https://tinyurl.com/3rh9xmhs +28https://tinyurl.com/475dvbp5 +29https://tinyurl.com/475dvbp5 +16 + +• Refactoring of big ball of mud: Big ball of mud occurs when a software system lacks a perceivable, +flexible, and appropriate architecture [47]. This subcategory includes questions that ask about +approaches and tools for big ball of mud refactoring. For instance, in this question30: “What step +would you take to refactor a ball of mud CF app into something modern and maintainable”. +(8) Architecture deployment collects architecture related questions that ask about how certain +software systems should be deployed in the hosting environments to meet requirements (e.g., functional +and non-functional requirements). According to our studied dataset, we divided this category into two +subcategories, among which application deployment to meet quality attributes contains the majority of +questions (24 out of 34 ARP questions of the architecture deployment category) that SO users ask in this +category. +• Application deployment to meet quality attributes includes architecture related questions that ask +about methods and tools that assist in the deployment of applications in the hosting environments +to meet quality attributes (e.g., availability and performance). For instance, in this question31, a +developer asked how to deploy a microservice based system with zero downtime: “At the moment +I’m working on an application which will be based on the Microservice architecture. +As main +technologies, we planned to use Spring Boot and Docker for each Micro Service development. One +of the goals/requirements is to provide a Zero Downtime Deployment feature for the users (...). +Any suggestions on the Zero Downtime Deployment process? If you have any great ideas for a +different architecture or maybe you’ve used tools which can help us here (...)”. +• Application deployment to meet functional requirements covers architecture related questions that +ask about methods and tools that assist in the deployment of software systems to meet functional +requirements. +For example, in this question32, a developer asked about the method s/he can +follow in order to deploy his/her microservices based application in the production environment so +that each service of the application can call each other: “I am trying to deploy my microservices +architecture to production env. Now I have 15 services, 1 Facade Layer, Facade Layer calls services, +gets data, aggregates them, and generates the final result. Also, services call each other(rarely but +yes, they call each other) (...). So I have decided that I will have 5 Boxes (5 high-end servers). +A, B, C, D, E A will be LVS (for Load Balancing) B & C will host the Facade layer. So when +the request came for Facade, it will come from A and load balanced to B & C (...). So B & C box +will contain each one haproxy instance also since when Facade Layer calls services, it will be load +balanced (...). But my question is how should I allow my services to call each other? (...)”. +(9) Architecture documentation: This is the only category of architecture related questions +with no subcategories in our studies dataset. The architecture documentation category includes ques- +tions that ask about methods and tools that assist in the documentation of architecture of software +systems. +For instance, in this question33, a developer asked about the best practices and tools for +documenting architecture of different types of systems: “What are the best practices and software tools +for documenting software design and architecture for PC based applications based on Java or .NET? +Embedded Applications based on VxWorks or Embedded Linux or Windows CE? (...)”. +Key Findings of RQ1 +Finding 1: SO users ask a broad range (9 categories) of architecture related questions, among +which architecture configuration (27%, 261 out of 968 ARP questions), architecture decision (19%, +181 out of 968 ARP questions), and architecture concept (15%, 142 out of 968 ARP questions) +are the top three categories of most frequently asked architecture related questions. +4.2. Categories of design contexts (RQ2) +This RQ aims to investigate the categories of design contexts in which architecture related questions +were raised. As described in Section 3.2.2, to answer this RQ, we used a predefined classifications of +design contexts from [27] and [30]) when analyzing the extracted data for RQ2 (i.e., design contexts) +30https://tinyurl.com/j6rdefeb +31https://tinyurl.com/4tyd4yt6 +32https://tinyurl.com/37dmd6av +33https://tinyurl.com/msnbc7xb +17 + +from the 968 ARP questions (see Figure 1). We found that most (71%, 687 out of 968) of our analyzed +ARP questions describe their design contexts (i.e., the knowledge about the environments in which the +systems are expected to operate [27]), and then the responders provided potential solutions with rationale +based on the given design issues and design contexts. In addition, we identified three main categories +and eight subcategories of design contexts. We report the mentioned categories, their subcategories, +their percentages of occurrence (out of 687 ARP questions), and count information in Table 6. It is also +evident from Table 6 that application context is the most common (54%, 377 out of 687 ARP questions) +category of design contexts, and organizational context is the least significant category (8%, 56 out of +687 ARP questions). +Table 6: Categories of design contexts, their subcategories, and their counts & percentages +Design Context +Subcategory +Count +Application context (55%, 377) +Application domain context +313 +External service context +64 +Platform context (37%, 254) +Software context +139 +Hardware context +115 +Organizational context (8%, 56) +Development schedule context +36 +Stakeholders context +13 +Resources context +7 +(1) Application context refers to the software system or product that is to be designed. It is +accessed through a device (platform entity) to deliver services to end-users [27]. This category includes +two subcategories, in which the application domain context subcategory is the most (313 out of 377 ARP +questions) common one. +• Application domain context describes the domain/type of the application that is being developed +(such as E-commerce system, banking system, distributed system) [30]. Some SO users like to +reveal in their architecture related questions what kind of application domains they are about +to design in order to get potential and relevant architecture solutions that fit their application +domains. For example, in this question34, a developer mentioned that s/he was designing an E- +commerce system: “I am designing an E-commerce using microservices architecture. Suppose that +I have two contexts: a product catalog, inventory and pricing. It’s seems clear to me that they have +a clear responsibility. But to serve the show case (the product list) I need to make a request for the +product catalog, get a list of ID’s and then use it to query the Inventory micro services to check +inventory status (in stock or stock out). Besides that I need to make a request to Pricing to get the +price of each product (...). I have been reading about microservices architecture and when you are +dealing with many ‘joins’ it’s possible that the these contexts should be a single one (...). We can +use a domain event to notify ‘search’ microsecond that something has changed. So we can resolve +show case with a single request. This look like a CQRS. Is there a correct approach? Which one is +better ? Trade-offs?”. +• External service context refers to specifications of external software services that the application +uses [27]. For example, in this question35, a developer mentioned that s/he was designing a system +that will require to use Azure or Amazon cloud services: “Basically my question is on the application +architecture. Designing for hosting is easy but cloud computing adds new challenges (...). I am +not certain what I should do in designing an application for safety engineers, so a high uptime is +important. So, if my application is written in ASP.NET, using SQL Server, it would seem that my +best bet is to design for Azure, but would Amazon’s solution be a good choice? How would I decide +if I should just have everything on the same system or have the data on Amazon’s cloud and the +ASP.NET on Azure? (...). I decide on the language, does that lock me into a cloud solution?”. +(2) Platform context comprises the hardware technology a user employs to access an application, +34https://tinyurl.com/2p93nesu +35https://tinyurl.com/2p8phmr6 +18 + +the software it runs, and the network capabilities of such technology [27]. In our dataset, we identified +two platform contexts (i.e., software and hardware context). +• Software context comprises information about the software elements of the device, such as the Op- +erating System (OS) or other installed applications. This subcategory collects the ARP questions +that describe the software elements of the device (e.g., OS) on which the planned software system +will need to run in production [48]. We found that some SO users provide this kind of information +when asking architecture related questions. For example, in this question36: “I need to build one +mobile application starts with windows phone 7 and then need to convert the application to other +platforms like Android, iOS. The application contains many screens with data capture and all the +data stores it in local storage and finally, it is passed to a central server. I would like to know how +the architecture needs to be designed (...)”. +• Hardware context comprises the platform entity which defines the device through which the user +accesses and uses the application, and can be of different types, such as desktop, laptop computers, +and wearable mobile devices [27]. The hardware context category gathers ARP questions that +mention hardware technologies (e.g., desktop computers) through which the users access and utilize +the planned applications. For example, in this question37, a developer mentioned that s/he was +developing a desktop application: “We want to start develop an intermediate desktop software. We +decided to use the WPF. We don’t want to use the MVVM pattern. Because we are not familiar with +MVVM. Is it true to develop WPF application without MVVM pattern (using 3 layer architecture +but without MVVM) although does it have better performance than win forms yet?”. +(3) Organizational context refers to the development schedule (e.g., time-to-market), the people +(i.e., stakeholders), or the resources that could influence the development of software systems. This +category includes two subcategories, in which the development schedule context subcategory is the most +(36 out of 56 ARP questions) common one. +• Development schedule describes the time put on software development. For example, in this ques- +tion38 a developer mentioned that the development time for a software project was restricted to +only three months: “For personal and university research reasons I am thinking of building a simple +CRM using a service-oriented architecture (...). The architecture that I’m designing defines: - We- +bGUI (a client of the other services) - AnalyticsService (a service that receives data, analyzes, and +collects it) - CustomerCareService (a service that uses RESTful APIs to apply CRUD operations +(...). What sort of authentication is more suitable for a client (user token vs OAuth or similar). +I’ve about 3 months to do it (...)”. +• Stakeholders context describes the people who are involved in the development of a software system, +for example, project managers, owners, architects, developers, users, among others. For instance, +in this question39 an asker mentioned a number of developers that was involved in the development +of 3D Map application by stating that: “I’m trying to develop 3D Map, and I found 3 solutions: +Use game engine (like unity) or Use 3D graphic API (OpenGL, etc) or Web app. Is there another +way to do it? And which one of those three solutions (design decision) is better? (with reason) (..) +Developers: 3 programmers”. +• Resources context denotes the lack (or availability) of resources (e.g., financial or technological +competencies) at disposal to develop an application [49]. For instance, one developer needed to +update an application with a tight budget and stated in this question40 that: “I have to build a +database/image-rich application that’s only going to increase in size (scalability). I am on a budget, +but do have a rather good 3Ghz Xeon server with 400 GB space. Any ideas? a good way for an +individual on a TIGHT budget”. +36https://tinyurl.com/2p8nd2kw +37https://tinyurl.com/yc2eyvhs +38https://tinyurl.com/2bu5yu65 +39https://tinyurl.com/yra7d3y7 +40https://tinyurl.com/4pjp3ntj +19 + +Key Findings of RQ2 +Finding 2: Most of the SO users (71%, 687 out of 968 ARP questions) considered design contexts +when asking architecture related questions. +Finding 3: Application context is the most common (54%, 377 out of 687) category of design +contexts in ARP questions, whereas organizational context is the least significant design context +category (8%, 56 out of 687) in ARP questions. +4.3. Characteristics of architecture related questions that have more than one answer (RQ3) +Some architecture related questions are continuously getting more attention from SO users by an- +swering them. This motivated us to investigate why such architecture related questions get more than one +answer in SO by characterizing those questions. As mentioned in Section 3.2.2, we used two techniques +(i.e., open coding and constant comparison) from Grounded Theory [17], to examine the characteristics +of ARP questions that have more than one answer from 650 ARPs (a subset of 968 ARPs) (see Figure +1). As discussed in Section 3.2.2, we referred to the contents of the questions and comments attached to +questions to understand what factors (e.g., question formulation [35] or certain features in the question +(e.g., architectural diagram)) that contribute to such architecture related questions getting more than +one answer. The outputs of our data analysis generated four common characteristics of architecture +related questions that have more than one answer. We provide these four common characteristics and +their counts & percentages in Table 7, which shows that well-articulated architectural information is the +most (46%, 297 out of 650 ARP questions) frequent characteristic while upvoted architecture question +comes as the least (8%, 51 out of 650 ARP questions) frequent characteristic of architecture related +questions that have more than one answer. Moreover, we show the numbers of answers of the ARPs that +have more than one answer in Figure 3. +(1) Well-articulated architectural information in the question: The main reason an archi- +tecture related question would continuously be answered is that its architectural information is well- +articulated. This question provides an overview of the planned software system and its basic principles +(e.g., design contexts and architectural constraints) to help other SO users perceive what the question +is really about (the purpose of the question). For example, a comment: “+1 great question, I want to +say that this is a beautifully and very well-articulated question!)” was posted under this question 41 that +illustrates and explains well the encountered architecture concerns (i.e., architecting a system to meet +the scalability and visualization of data). +(2) Clear description together with architectural diagrams in the question: Another +reason an architecture related question would continuously be answered is that it is easy to read and +understand, for example, the question which clearly states necessary information about the components +and connectors together with interfaces and relationships to other components. Also, providing diagrams, +such as architectural component diagrams that depict and clarify the logical architecture view of software +systems, contributes to questions continuously being answered. For example, before one responder started +answering this question 42, this responder stated that: “Your question is clearly described. Thanks for +the little graph you drew to help clarify the overall architecture (...)”. +(3) Alternative architecture solutions to answer the question: Although different architec- +ture solutions act as alternative solutions to similar design problems, they differ in terms of their qualities +[34]. For example, two architecture solutions may both address the interoperability concern but may +differ into addressing performance concern. However, such alternative architecture solutions are of sig- +nificant importance as they provide a wide range of possibilities for choosing and making design decisions +on candidate architecture solutions for certain design issues. In this study, we noticed that questions +that ask about choosing between various architecture solutions, such as technologies (e.g., databases, +frameworks, and programming languages), are continuously getting new answers (alternative solutions). +For example, SO users were interested in choosing a right combination of message formats with message +transmission techniques in order to achieve quality attributes, including high performance, availability, +scalability, among others, for their Ruby and Java applications interaction, and they posted a question43: +41https://tinyurl.com/dfw27h38 +42https://tinyurl.com/74fr8mwe +43https://tinyurl.com/44sey2a2 +20 + +“We have cloud-hosted (RackSpace cloud) Ruby and Java apps that will interact as follows (....). We +are interested in evaluating both message formatting (such as JSON) as well as message transmission +techniques (RPC, REST, SOAP, etc.). Our criteria are high performance, availability, scalability (...). +What combination of message format and transmission method would you recommend? Why?”. +(4) Upvoted architecture question: Usually, in SO, if a question is consistently getting upvote +count, its likelihood of being answered or getting more answers increases [50], and architecture related +question is no exception. In our data analysis, we realized that this factor (upvoted architecture question) +also contributes to architecture related questions having more than one answer. For example, in this +architecture related post44, a responder stated in the title of the answer thread when s/he was answering +the question: “Since this question got upvoted several times I would like to share what I did in the end +(...)”. +Table 7: Characteristics of architecture related questions with more answers and their counts & percentages +Characteristic +Count +Well-articulated architectural information +297 (46%) +Clear description together with architectural diagrams in the question +174 (27%) +Alternative architecture solutions to answer the question +118 (18%) +Upvoted architecture question +51 (8%) +0 +5 +10 +15 +20 +25 +30 +35 +40 +1 +100 +199 +298 +397 +496 +595 +Number of answers +ARP that has more than one answer +Figure 3: Numbers of answers of the ARPs that have more than one answer +Key Findings of RQ3 +Finding 4: Well-articulated architectural information is the most (46%, 297 out of 650 ARP +questions) frequent characteristic of architecture related questions that have more than one an- +swer. +Finding 5: The presence of architectural diagrams (e.g., components diagrams) in the architec- +ture questions increases the chance of these questions to get more than one answer. +4.4. Characteristics of architecture related questions that only have one answer (RQ4) +We found out that some architecture related questions gain less attention from SO users to be con- +tinuously answered. Analogous to the two previous RQs (i.e., RQ1 and RQ3), we applied two techniques +(open coding and constant comparison) to study the characteristics of questions that only have one +answer from 318 ARPs (a subset of 968 ARPs) (see Figure 1). As detailed in Section 3.2.2, similar to +RQ3, we referred to the contents of the questions and the comments attached to questions to understand +44https://tinyurl.com/7bc8764a +21 + +what factors demotivate the responders to continue answering certain architecture related questions. We +identified five common characteristics of those questions with their counts & percentages in Table 8, +which shows that lacking information in the question (39%, 125 out of 318 ARP questions) and poorly +structured architecture question (22%, 69 out of 318 ARP questions) are the two major characteristics +of architecture related questions that only have one answer. Below, we elaborate these characteristics in +detail with examples from ARPs. +(1) Lacking information in the question: Architecture related questions that lack certain +significant information (e.g., missing information on components and connectors together with interfaces +and relationships to other components) fail to attract community members to provide their answers. +For example, one developer pointed out that some information is missing in this architecture related +question45: “Your question cannot be answered without doing (many) assumptions. More information is +needed about the module’s dependencies. Are they stateless? Can you draw a flow of the requests?”. +(2) Poorly structured architecture question: We found that architecture related questions that +are poorly structured (e.g., not well articulated) fail to clearly reveal their purposes to the community +members so that they can provide answers. These questions sound unclear, vague, or hard to follow. For +example, in this question46 a developer asked about how to implement the interactors in Android MVP +clean architecture but failed to clearly structure well his/her question, and another developer came and +commented: “So what exactly is your question?”. The asker came back and edited the question to make +it clear so that other community members could understand what the question is really about. +(3) Architecture considered as off-topic: Even though architecture related questions are being +asked in SO, SO is mainly designed for programming information seekers. Therefore, architecture being +not directly for programming related issues but mainly for high-level structure related concerns, this +leads to the situation that architecture related questions get less attention from the community and +consequently only get one answer or remain unanswered in SO. For example, a developer asked about +the overview of ZeroMQ architecture, and another developer commented under this question47 by saying +that: “I’m voting to close this question as off-topic because it’s not really about programming”. +(4) Proprietary technology in the question: We found a few questions, asking about propri- +etary technologies, such as databases and frameworks that are not widely used, get less attention in SO +and consequently get only one answer. For instance, in this architecture related post48, a community +member claimed to be one of the technology founders of RethinkDB in the title of the answer when s/he +was answering a question. Other community members kept asking more questions about that RethinkDB +(a not widely used database) in the comment thread, and those questions have remained unanswered. +For example, “Disclaimer: I’m one of the founders of RethinkDB. Sorry for the longish answer (...). +RethinkDB is designed with a very flexible architecture (...)”. +(5) Duplicate architecture question: Similar to other types of questions (e.g., programming +questions) in SO, some architecture related questions get few answers (i.e., one answer) or remain unan- +swered because they are duplicate architecture questions. Community members do not like to re-answer +questions that were answered before [50]. They would like the askers to review the site (i.e., to check if +their questions have not been posted and answered) before posting such new questions. For example, a +comment: “duplicate of stackoverflow.com/questions/15142386/. . . ”, was posted under this architecture +related question49. +Key Findings of RQ4 +Finding 6: Lacking information in the question (39%, 125 out of 318 ARP questions) and poorly +structured architecture question (22%, 69 out of 318 ARP questions) are the top two most frequent +characteristics of architecture related questions that only have one answer. +45https://tinyurl.com/btukhpzx +46https://tinyurl.com/3m6fzjbe +47https://tinyurl.com/rmpfarjc +48https://tinyurl.com/nrz66x3w +49https://tinyurl.com/2p9382hh +22 + +Table 8: Characteristics of architecture related questions with few answers and their counts & percentages +Characteristic +Count +Lacking information in the question +125 (39%) +Poorly structured architecture question +69 (22%) +Architecture considered as off-topic +51 (16%) +Proprietary technology in the question +32 (10%) +Duplicate architecture question +29 (9%) +4.5. Taxonomy of architecture solutions that are considered useful (RQ5) +This RQ aims to construct a taxonomy of architecture solutions that are considered useful in SO. +As discussed in Section 3.2.2, when answering this RQ, we first investigated how SO users discuss +the usefulness of architecture solutions attributed to their associated architecture related questions. We +needed to gain an understanding of the ways (e.g., terms) SO users may use to communicate the usefulness +of architecture solutions in SO. Understanding SO users’ discussions on the usefulness of these solutions +is important to direct Q&A platform owners in creating the mechanisms that can help SO users to +efficiently and effectively search and (re)use such useful architecture solutions. We found that SO users +frequently use two terms related to usefulness (i.e., “useful” and “helpful”) along with six other terms +(i.e., “definitely”, “very”, “really”, “super”, “extremely”, and “incredibly”) in the comment threads (see +Figure 2) to explicitly express their feedback about how useful they found certain architecture solutions +provided to their associated architecture related questions. Note that SO users may use other ways to +communicate the usefulness of architecture solutions in SO, and we cannot claim that we have identified +all usefulness terms. Secondly, as detailed in Section 3.2.2, we thoroughly and comprehensively examined +the contents of the solutions from 324 ARPs (a subset of 968 ARPs) with useful knowledge (see Figure 1) +to construct the taxonomy of these solutions. This examination yielded a taxonomy of 7 main categories, +20 subcategories of which 1 were encoded as “Others” (i.e., refer to codes that do not fit into the already +generated subcategories), and 85 types (see Figure 4). +23 + +Framework for embedded +system implementation +Drupal functionality +explanation +Architecture tactic +Architecture pattern +Explanation of architecture +Viewpoint for architecture +documentation (2) +Solution for architecture documentation +Explanation of CMS +architecture (16) +Architecture solution for deployment +Tactic for +availability (3) +Tactic for security (4) +Data replication +tactic +Solution for architecture configuration +Solution for architecture implementation +API for iOS application +implementation +API for architecture implementation +(18) +37 (11%) +30 (9%) +53 (16%) +59 (18%) +13 (4%) +127 (39%) +Taxonomy of architecture solutions that are considered useful in SO +5 (2%) +Configuration solution for +platform (47) +Deployment process +with AppHabor for .NET +application +Tactic for performance (21) +Solution for iOS app +configuration +Solution for Linux +application configuration +Solution for Android +app configuration +Solution for Window +application configuration +Solution for tracking system +configuration +Architecture pattern suggestion +to meet quality attribute (17) +Tactic for +maintainability (9) +Deployment process +with Azure App Service +Deployment process with +AWS elastic container +Deployment process with +Bluemix +Solution for hotel management +system configuration +Solution for hospital management +system configuration +Solution for embedded system +configuration +Solution for social network +system configuration +Solution for real-time system +configuration +Solution for image processing +system configuration +Solution for VxWorks +application configuration +Solution for MacOS +application configuration +Solution for cross-platform +configuration +Framework for architecture +implementation (29) +Solution for E-commerce +system configuration +Framework for machine learning +application implementation +Authentication tactic + +Solution for distributed system +configuration +Solution for game system +configuration +Solution for banking system +configuration +Scheduling resources +tactic +Use intermediary to +reduce coupling +Library for architecture +implementation (12) +Deployment process +with Kubernetes +Deployment process +with Docker Swarm +Others (5) +Communication link +encryption tactic +Configuration management +tool for deployment +Loading less data tactic for +computation +Pattern for modifiability, reusability, +and portability (Broker, MVC, +Layered, SOA) +Automatic class loading +tactic +Development viewpoint +for architecture +documentation +Explanation of database +architecture (6) +NoSQL database +functionality explanation +Functional +redundancy tactic +Pattern for scalability and +availability (Client-Server, +Broker, Microservice) +Encapsulation through +API introduction +In-memory caching tactic +EC2 functionality +explanation +Explanation of Cloud- +based architecture (28) +Less coupling tactic +WordPress functionality +explanation +TYPO 3 functionality +explanation +Graph database +functionality explanation +Tencent Kubernet +functionality explanation +Web server functionality +explanation +Explanation of Web server +architecture (3) +Application server +functionality explanation + Library for computer vision +application implementation +API for Web-based application +implementation +API for cryptocurrency +application implementation +API for facial recognition application +implementation +API for video game system +implementation +Library for vector graphics +application implementation +Library for game system +implementation +Library for Android application +implementation +Time stamp tactic +Proxy server functionality +explanation +Framework for Android +application implementation +Framework for iOS +application implementation +Framework for Windows +applications implementation +Framework for cross-platform +application implementation +Framework for MacOS +application implementation +Simulator tool for +architecture deployment +Pattern for Android application +(MVP, MVVM, MVC, Observer) +Pattern for performance +(Layered) +Tool for architecture +documentation (3) +Tool for database +architecture documentation +Tool for Web application +architecture documentation +AppHarbor functionality +explanation +Framework for Web-based +application implementation +Limit access +Azure App service +functionality explanation +Tool for desktop application +architecture documentation +Category +Taxonomy Legend +Subcategory +(Number of posts) +Type of solutions +Taxonomy +Number of posts +(Percentage) +Process for architecture +deployment (9) +Tool for architecture +deployment (3) + +Configuration solution for +domain (75) +Architecture configuration +model +Configuration model +generation from code +Tool for continuous +deployment +Library for Web-based +application +Library for linear algebra +application +Pattern for time-critical system + (Preemptive Multitasking) +Pattern for distributed +system (Broker, SOA) +Pattern for Web-based system +(MVC, Client-Server, SOA) +Usage of architecture +pattern (13) +Deployment viewpoint +for architecture +documentation +Figure 4: Taxonomy of architecture solutions that are considered useful +24 + +(1) Solution for architecture configuration: This is the largest category of architecture solu- +tions in our taxonomy (see Figure 4). The solutions in this category provide approaches and tools that +enable the configuration of components and connectors of the planned software systems. Among three +subcategories identified in this category, the configuration solution for domain subcategory collects more +than half of the solutions (75 out of 127 ARP solutions) discussed in this category (see Figure 4). +• Configuration solution for domain: This subcategory discusses approaches and tools for config- +uring applications of various domains, such as solution for distributed system configuration, solution +for banking system configuration, solution for E-commerce system configuration, and solution for +game system configuration (see Figure 4). Concerning the solution for distributed system configu- +ration type, a user asked about how to design and configure a 2/3 tier distributed application in +Java with certain components, including centrally shared database and multiple fat clients (Swing +based Graphical User Interface clients (GUIs)). S/he needed a simpler approach that could help +him or her to configure those clients so that each client can be informed about data changes com- +mitted to the database by another client. The first solution in this ARP50 is provided based on the +application domain (i.e., distributed application) described in the question. The solution suggests +to configure the application’s components (e.g., database and clients) by following Java EE dis- +tributed container/component-based architecture by stating that: “(...) Java EE is a distributed +container/component based architecture for the enterprise tier (...) You c/would design a messag- +ing domain with both topic/subscription based and straight up Queues. These can be declaratively +configured to be durable, or not, etc. (...)”. +• Configuration solution for platform provides approaches and tools that enable the configu- +ration of components and connectors of applications with regard to the platforms (e.g., Windows +OS) on which these applications will run in production. We identified seven commonly discussed +solution types in this category, such as solution for Android app configuration, solution for Win- +dows applications configuration, solution for iOS app configuration, and solution for cross-platform +configuration (see Figure 4). Regarding the solution for cross-platform configuration type, one SO +user needed to design and configure an application that sends data between two iOS devices (i.e., +iPad and iPhone) with iPad acting as an iBeacon. The first solution in this ARP51 explains how the +application’s components including the iPad and iPhone could be configured by using an approach +that could support Android as well by stating that: “(...) I was forced into an architecture that +would support Android as well, so I switched to BlueTooth. The iPad acting as an iBeacon also has +BlueTooth code that is looking for ‘peripherals’ with a certain signature. Once the iPhone detects +the iBeacon, the app then starts transmitting a BlueTooth peripheral signal with the appropriate +signature (...)”. +(2) Solution for architecture implementation: The ARP solutions in this category provide +technology solutions, such as frameworks and libraries (see Figure 4), for implementing diverse architec- +ture designs to address the system requirements (e.g., quality attributes). According to our dataset, we +classified these technology solutions into three subcategories, among which framework for architecture +implementation contains the majority of solutions (21 out of 59 ARP solutions) that SO users discuss in +this category (see Figure 4). +• Framework for architecture implementation: These solutions gather different types of frameworks +for implementing architecture design, for example, framework for Web-based application (such +as Laravel, Django, Express.js, and Play frameworks), framework for iOS application (such as +SwiftUI, Flutter, and React Native frameworks), and framework for Windows applications (such +as WinForm, WPF, and UWP frameworks) (see Figure 4). Regarding the framework for Web-based +application type, a user asked about (among other things) a framework that could facilitate the +implementation of REST APIs in a Web-based application which will serve the content to mobile +apps. The third answer in this ARP52 suggests the Play framework as the solution to that question +by stating that: “Use Play! to do it all. Writing REST services in Play is very very easy (...)”. +• API for architecture implementation accumulates different types of APIs as solutions to questions +that ask about APIs for implementing architecture design, for instance, API for video game system +50https://tinyurl.com/jfnuke2w +51https://tinyurl.com/ytw52k6p +52https://tinyurl.com/2p8pm9rs +25 + +(such as Pokeapi, Chicken Coop, Dota2, and Minecraft APIs), API for Web-based application +(such as REST, SOAP, RPC, and Geolocation APIs), and API for facial recognition application +(such as Lambda labs and Microsoft Computer Vision APIs) (see Figure 4). Concerning the API +for Web-based application type, one developer needed an API to implement a request-response +Web application in Service Orientated Architecture (SOA) in order to meet certain requirements +(including high performance). The second answer in this ARP53 suggests to use REST API over +SOAP API by noting that: “(...) consider also using REST API, it demands less overhead than +SOAP, and you can use JSON as document format which is also more compact than XML, lowering +network throughput requirements (...) SOAP has more fancy features that are not well supported +in all languages, if you use REST you will be more safe here (...)”. +• Library for architecture implementation: These solutions recommend various libraries for imple- +menting architecture, for example, library for linear algebra application (such as JBLSA, MTJ, +OjAlgo, and EJML), library for computer vision application (such as OpenCV library), library for +vector graphics application (such as DISLIN library), and library for Web-based application (such +as HPPC, Trove, and FastUtil) (see Figure 4). Regarding the library for vector graphic application +type, the first answer in this ARP54 suggests Raphaël library as the solutions to the question that +asks about vector graphics application by saying that: “(...) I chose RaphaëlJS and I have to say +it has been an absolute pleasure to use, and the help is fantastic too (...)”. +(3) Explanation of architecture: The ARP solutions in this category provide theoretical expla- +nations, purposes, or functionalities of architecture instead of providing concrete instructions on how to +do something (e.g., how to configure certain architectural components in the system). Explanation of +architecture category consists of four subcategories, among which explanation of cloud-based architecture +is the most (28 out 53 ARP solutions) discussed subcategory (see Figure 4). +• Explanation of cloud-based architecture provides theoretical explanations, differences, and func- +tionalities of the architecture of cloud computing services, such as Azure App service functionality +explanation, EC2 functionality explanation, and AppHarbor functionality explanation (see Figure +4). For instance, a user asked about the difference between Azure App Service and the AAzure +Service Fabric in terms of functionalities in software development. +The fourth answer in this +ARP55 provides a detailed explanation about the difference between those two Azure platforms in +terms of functionalities in software development as the solution to that question by stating that: +“(...) They’re two separate platforms, following different development paradigms. The App Service +will give you functionality that Service Fabric doesn’t provide out of the box. Stuff like auto-scale, +authentication, rate limiting, integration with SaaS applications, etc. (...)”. +• Explanation of CMS architecture describes the functionalities of the architecture of Content Man- +agement Systems (CMS), such as Drupal functionality explanation and TYPO 3 functionality ex- +planation (see Figure 4). For example, the first answer in this ARP56 provides the architecture +overview of Drupal (together with an architectural diagram) as the solution to the question that +asks about the functionality of Drupal (e.g., control flow and how a page gets generated) by saying +that: “(...) Although it’s procedural PHP, it’s purely event/listener driven in its architecture, and +there’s no simple ‘flow’ in the main PHP script for you to look though (...) Drupal’s index.php file +functions as a front-side controller (...)”. +• Explanation of database architecture groups the ARP solutions that explain or describe the ar- +chitecture of datab”ase systems, for instance, NoSQL database functionality explanation (such as +Apache Cassandra) and Graph database functionality explanation (such as Nebula Graph) (see +Figure 4). For example, the first answer in this ARP57 provides an explanation about Cassan- +dra in terms of data replication to deal with data failure scenario as the solution to the question +that asks about the way Cassandra handles such a scenario if one node goes down containing the +record (data) a user is querying by stating that: “Cassandra clusters do replicate data across the +nodes. The specific number of replicas is configurable, but generally production clusters will use a +replication factor of 3. This means that a given row will be stored on three different machines in +53https://tinyurl.com/pakzw2yk +54https://tinyurl.com/259h4f6e +55https://tinyurl.com/2p8hnkmm +56https://tinyurl.com/2a9hb3ek +57https://tinyurl.com/wpvw6j5h +26 + +the cluster (...) In terms of servicing requests, if a node receives a request for data that it does not +have it will forward that request to the nodes that do own the data”. +• Explanation of Web server architecture provides the functionalities or difference between the ar- +chitecture of Web servers, for example, Web server functionality explanation (such as XAMPP, +IIS, and WAMP servers) (see Figure 4). For instance, the first answer in this ARP58 provides +detailed difference between XAMPP, WAMP, and IIS servers as the solution to the question that +asks about the difference between those three types of Web severs by expressing that: “(...) Their +(XAMPP and WampServer) differences are in the format/structure of the package, the configura- +tions, and (...) IIS is a web-server application just like Apache is, except it’s made by Microsoft +and is Windows only (Apache runs on both Windows and Linux) (...)”. +(4) Architecture tactic: This category of ARP solutions provide and explain architecture tactics +that enable the realization of specific quality attribute (e.g., performance and security) of software +systems. Four subcategories of architecture tactics were identified in this category, among which tactic +for performance is the most (21 out of 37 ARP solutions) discussed subcategory (see Figure 4). +• Tactic for performance provides and explains architecture tactics that assist in the realization of +the system performance requirements. We identified four architecture tactics for performance, such +as scheduling resources tactic and in memory caching tactic (see Figure 4). Regarding in memory +caching tactic, a developer wanted to choose a suitable design technique between two data handling +design techniques (i.e., working directly with a database or working with objects and letting the +ORM handle the storage) in order to boost the performance of the inventory system that should +handle thousands of item types and quantities of each item stored in a database. According to the +scenario elaborated in the question, the first answer in this ARP59 suggests to apply in-memory +caching architecture tactic with ORM to have the system performance boosted by saying that +“(...) most of the time it is easier to do an SQL query, but an in-memory cache can really BOOST +performance. Yes, it uses memory. Who cares? Workstations can have 64GB memory these days +(...)”. +• Tactic for maintainability covers architecture tactics that enable the maintainability requirements +of systems. We identified three maintainability tactics, such as less coupling tactic and encapsula- +tion through API introduction tactic (see Figure 4). Concerning the less coupling tactic, a developer +needed to build a scalable, maintainable, and low-latency single sign-on for all web applications. +The first answer in this ARP60 suggests to apply less coupling tactic when designing the applica- +tions in order to make them maintainable by saying that: “I would not integrate the authentication +on the database level (...) This might become hard to maintain. I would prefer a loosely coupled +approach by exposing a simple service on your central server that lets the other app servers run +authentication requests (...)”. +• Tactic for security provides architecture tactics that help the realization of the system security +requirements. This subcategory includes three architecture tactics, authentication tactic, limiting +access tactic, and communication link encryption tactic (see Figure 4). Regarding authentication +tactic, the first answer in this ARP61 provides and explains authentication tactic to a question +that asked for how to set up two level authentication approaches of the ‘user JWT’ in microservice +based application by stating that: “(...) You can achieve the two levels of security you require by +using a single user token and claims based authorisation. If a call is made to the gateway with the +user token, the gateway authenticates the call based on the user token, retrieves the ‘userId’ claim +(...)”. +• Tactic for availability collects architecture tactics that enable the system availability requirements. +We collected three availability tactics, like data replication tactic and functional redundancy tactic +(see Figure 4). Concerning data replication tactic, a developer asked how to achieve the availability +requirement for an application that needs to use two Amazon EC2 instances each with Cassandra +database. The first answer in this ARP62 provides and explains the replication mechanism that +58https://tinyurl.com/ysacs8zd +59https://tinyurl.com/289ffurv +60https://tinyurl.com/22ckrdv4 +61https://tinyurl.com/bdhk4uud +62https://tinyurl.com/2p8db7mu +27 + +could be applied in his/her application (according to the design scenario described in the question) +by saying that: “(...) In your scenario (since you are in a single DC) you can use SimpleStrategy +for your replication strategy and a Replication Factor (RF) of 2. With this setup, you will have all +data replicated on both nodes. This will make the data available from either node with a covet”. +(5) Architecture pattern: The ARP solutions in this category provide architecture patterns for +addressing multiple system quality attributes, and also provide commonly used architecture patterns in +certain application domains (see Figure 4). Among the two subcategories identified in this category, the +architecture pattern suggestion to meet quality attribute subcategory contains the majority of solutions +(17 out of 30 ARP solutions of the architecture pattern category) that SO users discuss in this category +(see Table 4). +• Architecture pattern suggestion to meet quality attributes collects architecture patterns for address- +ing system quality attributes, such as patterns for modifiability, reusability, and portability (Broker, +MVC, and SOA) (see Figure 4). For example, a SO user asked about the best C# architecture +patterns enabling the communication between separate plugins of a multi-tenant website wherein +modifiability, reusability, and flexibility are the major concerns. The first answer in this ARP63 +recommends SOA pattern as a solution to that question by noting that: “(...) I might suggest +Service Oriented Architecture. Mostly because it can bend to a business in a very quick and agile +manner. This architecture provides many bonuses: Lightweight, Agile, Code Re-usability (...)”. +• Usage of architecture pattern gathers architecture patterns for questions that ask about the com- +monly used architecture patterns in certain application domains (see Figure 4), such as pattern for +time-critical system (Preemptive Multitasking), pattern for Android application (MVP, MVVM, +MVC, Observer), and pattern for distributed system (SOA, Broker). For example, a user asked +about architecture pattern for time-critical applications. The first answer in this ARP64 recom- +mends Preemptive Multitasking pattern to that question by saying that: “(...) This pattern is called +preemptive RTOS, which is capable of handling the events immediately (...)”. +(6) Architecture solution for deployment collects the ARP solutions that discuss the deploy- +ment of architecture of systems in the hosting devices (either on the Cloud or the local server) in order +to address the systems’ requirements. This category consists of two subcategories, among which process +for deployment is the most (9 out of 13 ARP solutions) discussed subcategory (see Figure 4). +• Process for architecture deployment collects the ARP solutions that discuss the processes for de- +ploying the architecture of applications for the purpose of achieving the applications’ requirements +(e.g., functional or nun-functional requirements). We identified several processes for architecture +deployment, for example, deployment process with Azure App service, deployment process with Ku- +bernetes, and deployment process with AppHabor for .NET applications (see Figure 4). Regarding +deployment process with Kubernetes, a responder provided and explained the deployment process +with Kubernetes to an asker who wanted to deploy a microservices architecture (which was built up +with 15 Spring Boot microservices) on five Kubernetes nodes with one cluster master. According +to the scenario described in the question, the first answer in this ARP65 suggested to use three +cluster masters at a minimum instead of one cluster master in order to avoid the data loss and +consequently address the system’s availability requirement by saying that: “(...) one master is not +enough. The loss of that VM, the underlying hardware, or a failure of the services on the master +will lead to an outage for all customers and potentially catastrophic data loss. Run 3 masters at +minimum”. +• Tool for architecture deployment collects the tools for deploying architecture of systems in order +to achieve the requirements of the systems. We collected several tools, such as simulator tool for +architecture deployment and tool for continuous deployment (see Figure 4). Regarding the tool for +continuous deployment, the first answer in this ARP66 recommends Argo CD tool as the solution +to the question that asks about a tool for microservices architecture continuous deployment on +Kubernetes by stating that: “(...) ArgoCD workflow provides that functionality (...)”. +63https://tinyurl.com/42m8ts56 +64https://tinyurl.com/25y4b9e6 +65https://tinyurl.com/483en2ts +66https://tinyurl.com/yc7j8z5j +28 + +(7) Solution for architecture documentation: The ARP solutions in this category provide +the approaches and tools that enable the documentation of architecture (see Figure 4). This category +consists of two subcategories, among which tool for architecture documentation is the most (3 out 5 ARP +solutions) discussed subcategory (see Figure 4). +• Tool for architecture documentation suggests the tools that can facilitate the documentation of +architecture, such as tool for Web application architecture documentation and tool for database +architecture documentation (see Figure 4). Concerning the tool for Web application architecture +documentation, the first answer in this ARP67 suggests NJsonSchema tool as the solution to the +question that asks about a tool for documenting a microservices-based application by saying that: +“(...) there is NJsonSchema tool https://github.com/NJsonSchema/NJsonSchema”. +• Viewpoint for architecture documentation provides the viewpoints for architecture documentation, +such as development viewpoint for architecture documentation, and deployment viewpoint for ar- +chitecture documentation. For example, the first answer in this ARP68 provides two viewpoints +for architecture documentation (i.e., development viewpoint for architecture documentation and +deployment viewpoint for architecture documentation) for documenting an architecture that is im- +plemented with Java. +Key Findings of RQ5 +Finding 7: SO users frequently use two terms related to usefulness (i.e., “useful” and “helpful”), +along with six other terms (i.e., “definitely”, “very”, “really”, “super”, “extremely”, and “incred- +ibly”) in the comment threads to explicitly express their feedback about how useful they found +certain architecture solutions provided to their associated architecture related questions. +Finding 8: We derived a taxonomy of useful architecture solutions consisting of 7 categories, 20 +subcategories, and 85 types, indicating the diversity of useful architecture solutions provided in +SO. +Finding 9: Solution for architecture configuration (39%, 127 out 324 ARP solutions), solution +for architecture implementation (18%, 59 out 324 ARP solutions), explanation of architecture +(16%, 53 out 324 ARP solutions), and architecture tactic (11%, 37 out 324 ARP solutions) are +the top four most frequently discussed categories of useful architecture solutions. +4.6. Characteristics of useful architecture solutions (RQ6) +As shown in Figure 2, SO users occasionally leave comments under an architecture solution to convey +that such solution is useful. Hence, this motivated us to study the characteristics of the architecture +solutions that are considered to be useful. Analogous to RQ5, we used the 324 ARPs (a subset of 968 +ARPs) with useful knowledge (see Figure 1) to analyze the architecture solutions and their attached +comments, and study the characteristics of those solutions. The qualitative data analysis (see Section +3.2.2) identified four common characteristics of architecture solutions that are considered useful by SO +users. Figure 5 depicts these characteristics along with their counts, in which complete and comprehensive +architecture solution appears to be the most (34%, 111 out of 324 ARP solutions) frequent characteristic +of architecture solutions that SO users consider to be useful. +(1) Complete and comprehensive architecture solution: A developer may ask more than +one question (sub-questions) in one single architecture related question in SO. A solution that addresses +all sub-questions asked in the question and provides comprehensive responses (e.g., providing rationale, +such as benefits and drawbacks of the provided architecture solution) to these sub-questions is considered +useful. For example, a developer posted this comment: “+1 for the most complete, comprehensive useful +response I’ve ever seen (...)” under the first answer in this ARP69 that comprehensively addresses all +sub-questions (e.g., difficult to visualize data in the system architecture implemented with Java and +Python) asked in the question. +(2) Concise explanation with architectural diagrams provides a brief explanation about +the key elements of the architecture solution. Some examples of these key elements could be the best +67https://tinyurl.com/4zm82snw +68https://tinyurl.com/2p8ezw7j +69https://tinyurl.com/dfw27h38 +29 + +architecture patterns, tactics, and technologies (e.g., databases) to be used in order to address the design +concerns described in the question. In addition, providing architectural diagrams, such as component +diagrams to represent and summarize the practical applicability of the solution also contributes to the +architecture solution being considered useful. +For example, one developer asked whether “command +handler” and “command bus” should belong to or be implemented in the application layer or domain +layer in the architecture. At first, in the first answer of this ARP70, a responder provided a concise and +relevant solution. But the asker was not satisfied with this solution and then s/he commented to request +a sequence diagram (which was provided later) to be associated with the solution for it to be useful: +“Thanks, David. It would be really useful if you could share a sequence diagram. Appreciate it”. +(3) Detailed architecture solution: These solutions provide and fully describe all necessary +architectural elements (such as patterns, components) and other various aspects to be considered (e.g., +solution trade-offs, constraints, and alternatives) when addressing the design concerns stated in the +question. For instance, a developer posted this comment: “Thank you for your detailed answer. This +is certainly very helpful” under the second answer in this ARP71 that lists and details all necessary +architectural elements (e.g., quality attributes) and other aspects (e.g., pros and cons of the solution, and +alternative solutions) that should be considered when addressing the design concerns (e.g., integrating +external modules (external Web applications) into Drupal or vice versa) stated in the question. +(4) Summarization of external but relevant content: Answer seekers do not like to have +external links (URLs) only as solutions posted to their questions since the links may die and the solutions +become not accessible and useless. During our data analysis, we observed that answer seekers prefer to +have the relevant content summary of URLs instead of the URLs only for the architecture solutions. +For example, the first answer in this ARP72 summarizes the content from three URLs to answer the +question which mainly asks about the design approach to follow in order to address system availability +with Cassandra database. A developer commented under the answer: “Thank you very much for your +information. Your Explanation is sufficient and the links you mentioned are very useful”. +111, 34% +87, 27% +80, 25% +44, 14% +Complete and comprehensive architecture solution +Concise explanation with architectural diagrams +Detailed architecture solution +Summarization of external but relevant content +Figure 5: The common identified characteristics of architecture solutions that are considered useful +Key Findings of RQ6 +Finding 10: Complete and comprehensive architecture solution is the most (34%, 111 out of +324 ARP solutions) frequent characteristic of architecture solutions that SO users consider to be +useful. +Finding 11: The presence of architectural diagrams (e.g., components diagrams) in the provided +architecture solutions increases the chance of these solutions to be considered useful. +5. Discussion +In this section, we revisit the findings of this study by interpreting the results in Section 5.1 and +discussing their implications for various stakeholders in Section 5.2. +70https://tinyurl.com/yh292pn8 +71https://tinyurl.com/pfezm7nn +72https://tinyurl.com/2p87vu99 +30 + +5.1. Analysis of the results +5.1.1. The delta between our results and the results from prior work +Similar to our study, several studies have analyzed ARPs from SO to mine architectural knowledge +discussed by SO users in order to support architecting activities. In this section, we discuss the relation- +ship and difference between our study results and the results in the prior studies (i.e., the three studies +by Soliman et al. [6][51][10]), which are closely related to our work. +Soliman et al. [6] identified and analyzed ARPs from SO that discuss architecture knowledge with +a focus on technology decisions (one type of architecture decision [46]). They classified these ARPs +based on two dimensions: the purpose of the question and the solution type of the question. +They +further classified the purpose dimension into three subtypes: solution synthesis, solution evaluation, and +multi-purposes, and the solution type dimension into three subtypes: technology feature, technology +bundle, and architecture configuration. In total, their analysis generated 6 types of ARPs. Our analysis +generated 9 categories and 21 subcategories of ARP questions (see the results of RQ1 in Table 5), such +as architecture configuration, architecture decision, architecture concept, architectural implementation, +architecture evolution, and architecture refactoring. Some of the types of ARPs (e.g., solution synthesis, +solution evaluation, architecture configuration) found by Soliman et al. in [6] are aligned with some +of our ARP types, and most of the types of ARPs presented in [6] can be subcategories of the main +categories reported in our work. For example, we have a main category encoded architecture decision, +and this category can cover three types of APRs (solution synthesis, solution evaluation, and multi- +purposes) reported in [6]. Moreover, our analysis generated new categories of ARPs (such as architecture +concept, architecture tool, architecture evolution, architecture refactoring, architecture deployment, and +architecture documentation). +Soliman et al. [51] used the same sample of ARPs that were used in their previous work (i.e., [6]) +and developed an ontology that covers architectural knowledge concepts in SO. The ontology consists of +three main ontology classes: simple ontology class, composite ontology class, and lexical trigger ontology +class. A simple ontology class is composed of subclasses, for example, technology solution, architecture +pattern, quality attribute, architecture component, and architecture connector. The composite ontology +class consists of several subclasses, such as architecture configuration, technology feature, technology +benefits and drawbacks, technology user-case, user request, and design rule. The lexical trigger ontology +class has subclasses, such as difficulty adjectives, advise verbs, value adjectives, wish verbs, support verbs, +versus prepositions. Some subclasses found by the analysis in [51], such as architecture configuration and +architecture pattern, are aligned with our results of RQ5 (see Figure 4). However, the analysis in [51] is +based on a sample of ARPs that mainly discuss technology information (e.g., requirements and constraints +on technology solutions, technology benefits and drawbacks, and technology features). +Our analysis +complements the work in [51] by adding several new categories, such as architecture tactic, explanation of +architecture, solution for architecture documentation, and solution for architecture deployment, leading +to more comprehensive categories and subcategories of ARP solutions provided in SO. +Soliman et al. [10] developed a search approach that relies on the classification approach to provide +suitable types of ARPs for each design step proposed by Kazman and Cervantes [12]. +The analysis +conducted by Soliman et al. [10] is also based on the sample of ARPs from their previous work [6], +and some other posts extracted from SO. Specifically, their search approach classifies SO posts into four +types: technology identification, technology evaluation, features and configuration, and programming +posts. The first three types of posts are ARP types that were reported in their previous study (i.e., +[6]), and in the first paragraph of this section, we have already described the difference and similarities +between these types of ARPs in [6] and the types of ARPs in our study (see results of RQ1 in Table 5). +In addition to the abovementioned difference between our study results and the results reported in +[6][51][10], in our study, we investigated a new set of research questions (RQ2, RQ3, RQ4, RQ5, and +RQ6). We explored other types of architecture knowledge, such as design contexts (RQ2) discussed +in architecture related questions, characteristics of ARPs (questions and solutions) (RQ3, RQ4, and +RQ6), and the usefulness of the ARP solutions (RQ5), which was not the concern of the abovementioned +studies (i.e., [6][51][10]). Moreover, our analysis covered the entire post, including the question and its +associated comments (RQ3, RQ4), the answers to the question and their associated comments (RQ5, +RQ6). The analysis in the abovementioned studies by Soliman et al. only focused on questions and +answers. Thus, our study results add new information to the state of the art, and practitioners and +researchers can benefit from our study results and findings (e.g., the taxonomy of architecture solutions +that are considered useful). +31 + +5.1.2. Identified categories of ARPs in SO could support architecting activities +The significant results of this study are categories of ARPs (questions (i.e., RQ1) and solutions +(i.e., RQ5)). +This study reveals that SO users ask a broad range (nine categories) of architecture +related questions, such as questions about architecture configuration, architecture decision, architecture +concept, architecture implementation, and architecture tool (see Table 5 in Section 4.1). In addition, +we classified the architecture solutions that are considered useful into seven categories, such as solution +for architecture configuration, solution for architecture implementation, explanation of architecture, and +architecture tactic (see Figure 4 in Section 4.5). One observation is that our identified categories of +these ARPs (questions and solutions) cover almost all the architecting activities that span from the +initial stages (e.g., architectural analysis, synthesis, and evaluation [18]) of architecture creation to +the later stages (e.g., architectural implementation, and maintenance and evolution [19]) in a system +lifecycle. Thus the identified categories of architecture related questions and solutions can support the +mentioned architecting activities during the architecture lifecycle. These results also support the findings +by Soliman et al. in [6] that SO should be considered as one of the important sources of architectural +knowledge. Moreover, practitioners reported Q&A sites (e.g., Stack Overflow) as the most useful when +searching architectural information according to our recent industrial survey [4]. Thus, practitioners +could rely well on SO to identify, such as, the benefits and drawbacks of architecture solutions in certain +application domains, for example, the benefits and drawbacks about the framework for iOS applications +in our taxonomy (see Figure 4) for architecture implementation. +5.1.3. Importance of design context in architecture design +The results of RQ2 reveal that in most (71%, 687 out of 968) of the studied ARP questions, SO +users considered the design contexts (i.e., knowledge about the environments in which the systems are +expected to operate [27]) when describing the design concerns in their architecture related questions +(Finding 2 in Section 4.2). One reason could be that SO users prefer to provide a brief description of +their project backgrounds and then expect responders to suggest potential architecture solutions with +their rationale based on the given design concerns and design contexts. Moreover, the results of RQ2 +show that most of the SO users do consider design context as one of the indispensable ingredients that +can drive the architecture design of a system [27]. +5.1.4. Identified characteristics of ARPs to improve their quality +From Table 7, Table 8, and Figure 5, we found that there are various characteristics of ARPs (ques- +tions and answers) in SO. For example, we observed that architecture related questions that articulate +well architectural information are likely to get more than one answer (see Table 7), while architecture +related questions that lack certain significant information and poorly structured (see Table 8) tend to +only get one answer. The reason is the following: well-articulated architecture questions provide an +overview of the planned system and describe well their design concerns, which helps potential responders +to fully understand the purposes of these questions so that they can provide answers. We also found +that answer seekers highly appreciated architecture solutions that are complete and comprehensive and +considered them to be useful. One reason is that these architecture solutions address design concerns +raised in all sub-questions (in the case that there are sub-questions in one question) by providing com- +prehensive solutions, for example, design contexts, pros and cons of the provided architecture solutions, +which helps the answer seekers understand why such architecture solutions are the way they are. The +identified characteristics of ARPs (questions and answers) in SO show that SO users have varying needs +in the description of ARPs (questions and answers) and the level of details. These findings could assist +in improving the quality of the posted architecture related questions and answers at SO. +5.2. Implications +5.2.1. For Stack Overflow +Increase the awareness of SO towards its users: SO introduces itself as a community Q&A +platform for asking and collecting programming related knowledge during software development. Thus, +the majority of the SO users use the platform as the place for sharing and learning coding related knowl- +edge only. However, ever since this site started growing and being popular, architects have begun to +share their competencies, experience, and architecture problems by asking architecture related questions +or providing architecture solutions, such as architecture patterns [52]. Akin to searching and (re)using +existing code examples provided in SO to solve programming related problems, SO users also search and +(re)use existing architecture solutions, such as architecture tactics [5] in SO for solving their architec- +ture design concerns (e.g., architecture design to meet quality attributes). Hence, SO not only curates +32 + +programming related knowledge, but also accumulates architecture solutions provided to a wide range of +architecture problems or design problems [6, 5]. However, during our study, we found that architecture +related questions were being seen as off-topics in SO and should not be asked at the site (see Table 8 in +Section 4.4) due to SO users’ perception or awareness of what SO is used for (i.e., a site for programming +related issues). Given this situation, there is a possibility that interesting architecture related questions +asked might remain unanswered or even be deleted by the site moderators. Although some SO users +see architecture related questions as off-topic, we think that architecture related questions will sustain +and continue to thrive in SO. According to our studied dataset (318 architecture related questions) rel- +evant for answering RQ4 (characteristics of architecture related questions that only have one answer), +we observed that architecture related questions that were commented to be off-topic are not many +(16%, 51 out 318 architecture related questions). This finding is promising for the long-term prospect +of architecture related questions in SO. Moreover, we argue that architecture related questions which +communicate architectural knowledge [23] are an important type of questions and have a system-wide +impact on software development. Many architecture related questions arise during development when +addressing specific design concerns (e.g., quality attributes) and their trade-offs. Therefore, architecture +related discussions (e.g., through architecture related questions) should not be seen as off-topic in SO, +and SO should consider increasing its awareness (i.e., to be a site for development related issues instead +of a site for programming related issues only) towards its users and welcome architecture questions to +be discussed on the site. +Adjust the current answers and comments organization mechanisms to improve the +search and (re)use of architecture solutions: SO attracts a large number of users with different +backgrounds, skills, expertise, and viewpoints. Thus during our data analysis, we have observed that an +architecture related question like any other questions (e.g., programming related question) in SO may +receive multiple (or alternative) answers. The study by Wang et al. [53] reported that nearly 6.5 million +questions (37% of all questions at SO) had more than one answer, and the average length of an answer is +789 characters. With the current SO answers organization mechanism, when there are multiple answers +to a question in a single post, at most one answer per question can be accepted/marked by the asker to +indicate that the answer is the most useful one [3]. This asker should be a registered user with at least 15 +reputation on SO [54]. The registered users without required reputation (i.e., less than 15 reputation) on +the site are restricted from accepting or voting (upvoting or downvoting) answers to indicate that such +answers are useful [54]. Consequently, leaving a large number of answers in SO that are not accepted or +marked as useful answers yet being useful, just because the users (askers) do not possess the required +minimum reputation to do so. During our data analysis, we observed that not all useful architecture +solutions are explicitly marked (i.e., accepted as useful) in SO to facilitate the search and (re)use of +those solutions (for example, see Figure 2). Also, we found that SO users may use terms related to +usefulness, such as “useful” and “helpful”, in the comment threads to explicitly express their feedback +about how useful they found certain architecture solutions provided to their associated questions in SO +(Finding 7 in Section 4.5). Our finding is in line with the findings by Zhang et al. in [42] and [41] that +comments provide additional information to support the answers, such as improvement of answers [42] +and obsoleted answers [41]. Prior studies criticized the comment organization mechanism at SO (e.g., +[55]). In order to keep each answer thread compact, SO implements a comment organization mechanism +to only show the top 5 comments [55]. Aiming at showing the most informative comments and hiding +less informative ones, the mechanism first ranks these comments based on their scores. When multiple +comments have the same score, they are then ranked by their creation time [55]. Hidden comments are +not indexed by Google73. Thus, due to this current comment ranking mechanism, informative comments +might be hidden in turn reducing the chances of someone retrieving or voting on them. Regardless of +its success and popularity, navigating SO remains a challenge, and it is insufficient how SO directs its +users to retrieve informative comments [56]. Comments that state the usefulness of answers (including +architecture solutions) are one of the most important informative comments. Thus, we provide SO the +following suggestion: +• Instead of simply ranking comments by their score then their creation time [55], the comments +organization mechanism needs to introduce a higher priority for more informative comments. SO +may consider adjusting its comments organization mechanism by, for instance, developing special +analytical techniques (e.g., machine learning approaches) that could filter and rank comments +73https://tinyurl.com/2p87yyfr +33 + +stating, for example, improvement of answers [42], usefulness of answers (e.g., useful architecture +solutions). +• SO may also refer to and extend our proposed taxonomy of useful architecture solutions (solutions +commented to be useful) to develop an automated tool that could assist the SO users in identifying +existing architecture solutions with, for example, useful knowledge. +5.2.2. For SO users +Throughout the qualitative analysis of RQ3, RQ4, and RQ5, we identified various characteristics of +ARPs (questions and solutions) in SO. Among these characteristics, we found that questions that provide +clear description together with architectural diagrams increase their likelihood of getting more than one +answer (see Table 7), while poorly structured architecture questions (see Table 8) tend to only get one +answer. Also, we found that architecture solutions that provide concise explanation with architectural +diagram is the second most common characteristic of architecture solutions that are considered useful (see +Figure 5). One observation is that SO users would like to see architectural diagrams, such as components +diagrams, in both questions and solutions as these diagrams can benefit both parts. Concerning questions, +providing architectural diagrams increases their chance of getting more responses (e.g., more than one +answer) (Finding 5 in Section 4.3). On the other hand, architectural diagrams in solutions boost their +chances of being considered useful (Finding 11 in Section 4.6). Therefore, both askers and responders +should better provide diagrams in their ARPs (questions and answers). One reason is that architecture +is at a high abstraction level, and it would be hard to describe an architecture problem and much harder +to explain an architecture solution with text only. Architectural diagrams make architecture to be more +understandable [57], and stakeholders can communicate about architectural problems and solutions more +easily using architectural diagrams. Moreover, various identified characteristics of ARPs in this study +(see Table 7, Table 8, and Figure 5) are indicators that SO users have varying needs in the formation of +both architecture related questions and architecture solutions and the level of details. Therefore, there +is a need to provide guidelines to SO users to follow when posting their architecture related questions +and solutions. +For SO askers: In the following, we provide the guidelines for SO askers to follow when posting +their architectural related questions with more likelihood of being answered by other SO users and get +more than one answer from SO users: +• Include architectural diagrams with clear description in the questions: We recommend askers to +add architectural diagrams (e.g., component diagrams) and specifically clarify the design concerns +in their architecture related questions to help other SO users better understand the purposes of +their questions. +• Write well-articulated architecture questions with descriptive details about the context: We suggest +that askers could describe well architectural information in their questions. This can be done, for +example, by providing an overall understanding of the system, as well as detailed information on +components in their scope together with interfaces and relationships to other components. Also, we +recommend askers to add information about the design contexts, since design contexts are critical +for other SO users to correctly understand your architecture related questions. +For SO responders: As stated throughout this study, we not only analyzed architecture related +questions, but also examined the characteristics of architecture solutions that are considered useful by +SO users. Thus, in the following, we provide guidelines to SO users to follow when posting their solutions +with more likelihood of being considered useful by other SO users: +• Write concise architecture solutions with architectural diagrams: Responders are recommended to +write concise architecture solutions by stating key points only in the solutions and add architectural +diagrams (if necessary) that depict and clarify, for example, the architecture implementation view +in their posted solutions. +• Include URLs in architecture solutions with sufficient and relevant architectural knowledge: Answer +seekers do not like to have external links (URLs) only as solutions posted to their questions [58]. +In the case when a responder wants to make the architecture solution short, s/he can provide links +to external websites that contain more explanations or complex examples, and his/her solution +should be self-contained. In other words, this solution should provide certain important and relevant +architectural knowledge which can make it explainable, such as design decisions and their rationale, +34 + +contexts, assumption, and other factors that together determine why a particular solution is the +way it is. +5.2.3. For researchers +Towards innovative tools to search and (re)use architectural knowledge in SO: The +results of our study (e.g., categories of architecture related questions in Section 4.1 and their useful +solutions in Section 4.5) provide insights into the nature of SO users’ discussions on architecture design +in SO. In addition, the results of this study re-emphasize the conclusion by Soliman et al. [51] that SO +should be considered as one of the important sources of architectural knowledge. However, SO captures +large amounts of information in its posts and this information is mainly represented as unstructured text. +Furthermore, the abstract nature of architectural concepts makes it difficult for keyword-based searches to +find architecture relevant information, and this might not be easy for SO users to capture and (re)use the +architectural knowledge (e.g., benefits, drawbacks, and trade-offs of using specific architecture patterns +in certain application domains) from SO. Therefore, researchers can contribute to improving the search +and (re)use of the architectural knowledge in SO by focusing on innovative techniques and tools that +could efficiently and effectively guide the capturing and usage of this knowledge to support architecting +activities (e.g., architectural analysis and synthesis [18]). +For example, researchers can refer to our +proposed taxonomy of useful architecture solutions in SO as a guidance to develop automated approaches +and tools that could mine and locate architecture solutions (e.g., solution for architecture configuration, +the most common category of useful architecture solutions in SO, see Figure 4) for addressing similar +design concerns (e.g., questions that ask about architecture configuration, see Table 5). This could help +SO users to check the questions and solutions that are relevant to their design concerns (e.g., banking +system configuration). +Furthermore, we observed that architecture configuration (27%), architecture +decision (19%), and architecture concept (15%) are the top three categories of most frequently asked +architecture related questions (Finding 1 in Section 4.1), and researchers may explore the challenges +(that are being faced by SO users) related to these most frequently asked categories of architecture +related questions. +Investigation of design contexts in Q&A sites to support architecture knowledge man- +agement: From Table 6 in Section 4.2, we found that SO users discuss about design contexts along with +design concerns when asking architecture related questions in SO. Three categories (application, plat- +form, and organization contexts) and eight subcategories (application domain, external service, software, +hardware, development schedule, stakeholders, and resources contexts) of design contexts were identified +from our studied sample of ARPs (see Table 6). Whilst we know that SO users discuss design contexts +along with design concerns when asking architecture related questions, there have been very few studies +on mining design contexts in the Q&A community sites, such as SO, to support architecture knowledge +management [59], which is an interesting area to be explored in future studies. +6. Threats to validity +In this section, we discuss the threats to the validity of the study results by following the guidelines +proposed by Wohlin et al. [60] and how these threats were mitigated in our research. +Internal validity concerns with the selection of search terms used to mine ARPs in SO. We used +search terms, such as “architecture” and “architectural”, to identify the related posts in SO (see Section +3.2), and this is a threat to the internal validity in our study because we might have missed other +terms, such as “design”, that SO users use to express architecture concepts. Hence, the search terms +we used in this study may not be able to identify the complete set of ARPs in SO. To reduce this +threat, we first conducted a pilot search and observed that SO users use the term “design” mostly in the +programming context (e.g., “singleton design pattern”74). In addition, as mentioned in Section 3.2, using +the search terms (e.g., “architecture” and “architectural”) to only search exclusively through tags can +be ineffective, because tags can be sometimes less informative [36] (see the example provided in Section +3.2.1). Thus, we decided to add the titles and bodies of the questions into the search. In this way, we +sought to minimize the risk of missing ARPs that use incorrect or irrelevant tags. Finally, we gathered +10,423 ARPs through the search which is quite a large dataset, and it may not be realistic to thoroughly +analyze this size of dataset with human effort in order to get accurate and comprehensive results from +74https://tinyurl.com/8yks7nhm +35 + +this dataset. Hence, we computed a statistically representative sample [16] of these 10,423 ARPs and +randomly selected 968 ARPs as the dataset to be analyzed in this study. However, to further mitigate +this threat, we downloaded and utilized the current SO data dump (i.e., Stack Exchange data dump on +October 5, 202275). This data dump is a snapshot of the underlying database used by SO and it stores +all the information for the questions, answers, tags, comments, votes, and user histories in XML files +(e.g., Posts.xml). We used Posts.xml file, which stores the questions and answers of all the SO posts, +as the basic to estimate how many ARPs we missed due to limiting the search to the “architect*” terms +in our study. According to the SO data dump of October 5, 2022, there are 23 million questions (posts) +and 34 million answers. We then used the power statistics and calculated a representative sample size of +these 23 million posts. With a 95% confidence level and 3% margin of error, the representative sample +size calculated is 1069 posts. Afterwards, we randomly selected 1069 posts from the 23 million posts +and manually checked them for calculating how many ARPs we might have missed due to limiting the +search to the “architect*” terms during the search of ARPs. Specifically, the first author labelled the +1069 posts to determine which of the posts are ARPs. The second author checked and validated the +labeling results. The disagreements were resolved in a meeting to improve the reliability of the labeling +results. Based on our manual labelling, we found that out of the 1069 posts, only 21 were ARPs (i.e., +the true positives), wherein 14.3% (i.e., 3 out of 21 ARPs) do not contain “architect*” terms and 85.7% +(i.e., 18 out of 21 APRs) contain “architect*” terms. Therefore, we admit that we might have missed +certain number of ARPs (i.e., 14.3%) that do not contain “architect*” terms. We added in our replication +package [39] the randomly selected posts (i.e., 1069 posts) and the labeling results (i.e., 18 ARPs which +contain “architect*” terms and 3 ARPs which do not contain “architect*” terms) for replication purpose. +Construct validity refers to the degree to which a study measures what it claims to be measuring +[60]. One threat to the construct validity in this study is concerning the manual analysis of the selected +SO posts. This is because manually analysis could bring personal bias due to multiple interpretations +and/or oversight. To mitigate this threat, we used two qualitative techniques (open coding and constant +comparison) from Grounded Theory [17] to analyze the extracted data and answer the RQs. Moreover, +we tried to minimize this threat by performing a pilot data coding before the formal data coding. As +discussed in Section 4, during the pilot data coding, the first author selected a random set of 100 ARPs +and encoded the extracted data (see Table 4) with respect to the purpose of each RQ (see Table 1). +Several physical meetings with the second author were scheduled to solve any confusion faced by the +first author during this pilot data coding. Moreover, the final results (i.e., concepts, categories, and +subcategories) from the pilot data analysis were checked and validated by other three authors (the +second, third, and fourth authors) of this study. The disagreements were resolved in a meeting using the +negotiated agreement approach [45] to improve the reliability of the pilot data analysis results. Another +threat is related to the identification of ARPs (solutions) with useful knowledge by checking the comments +attached to these posts in order to answer RQ5 and RQ6 (see Phase II, in Section 3). To mitigate this +threat, as we explained in Section 3, we did not count on the occurrence of the terms, e.g., “useful” (and +the similar) stated in comments to measure the usefulness of an architecture solution given to certain +architecture related question in our study. We rather referred to the usefulness related information in the +comments attached to the solutions to investigate SO users’ discussions on the usefulness of architecture +solutions. In other words, we judged the usefulness using the reaction of the SO users after seeing and +using the architecture solutions (see a comment in Figure 2). In addition, we (four authors of this study) +first read the solutions (from our studied representative sample) commented to be useful to see whether +there are really useful to address the questions [15] before we decided to include such posts (solutions) +with useful knowledge for analysis. Thus, we believe that we have adequately mitigated this threat. +External validity refers to the extent to which the findings of the study can be generalized in other +settings [60]. In this research, we only used SO as the source to investigate ARPs and their usefulness. +Even though SO is a widely used and popular developer Q&A site, this unique source still poses a threat +to the diversity of the study results. To mitigate this threat, our research could be further enhanced +by including more sources (e.g., GitHub) and look at architecture related questions to understand the +architecture design issues that are being faced by architects and developers. Also, researchers might +consider going to the fields and asking for feedback directly from architects and developers to better +understand the problems they are facing about architecture design and what architecture solutions can +be regarded as useful. +75https://archive.org/details/stackexchange_20221005 +36 + +Reliability refers to whether the study will provide the same results and findings when it is repli- +cated by other researchers [60]. In this study, this threat is largely related to the process of manual data +collection and analysis. To mitigate this threat, we (the authors of this study) followed a rigorous pro- +cedure that is consisted of data collection and analysis activities (see Section 3.2). Moreover, the results +from the classification and characterization stages were cross-checked by involving the four authors of +the study. To guarantee the reliability of our results and findings, a replication package, containing the +dataset used and the encoded data produced in this work, has been made available [39], allowing other +researchers to evaluate the rigor of the design and replicate the study. With these measures, this threat +has been partially reduced. +7. Related work +The research related to our work comes from studies that investigate software development knowl- +edge in Q&A sites, such as SO. In this section, we summarize relevant work in two categories: (1) +investigation of architectural knowledge in Q&A sites and (2) quality assessment of the knowledge in +Q&A sites. +7.1. Architectural knowledge in Q&A Sites +A few number of existing studies have studied architectural information provided in ARPs in SO +from different perspectives. Bi et al. [5] used a semi-automatic dictionary-based mining approach to ex- +tract Quality Attribute (QA) and Architecture Tactic (AT) related discussions in SO posts. Specifically, +they applied the dictionary-based classifier Support Vector Machine (SVM) to automatically identify +QA-AT related discussions from SO posts. Moreover, the authors went on to manually structure the +design relationships between Architectural Tactics (ATs) and Quality Attributes (QAs) used in prac- +tice and build a knowledge base of how developers use ATs with respect to QA concerns from related +discussions. Such knowledge can help architects better make ATs design decisions. Chinnappan et al. +[61] extracted data from five open sources of software repositories, including Stack Overflow and qualita- +tively mined architectural tactics for energy-efficiency robotics software applied by practitioners in real +robotics projects. To foster the applicability of the identified tactics (even beyond the robotics software +community), they describe them in a generic, implementation independent manner by means of diagrams +inspired by the UML component and sequence diagram notation. The presented energy-aware tactics can +serve as guidance for roboticists, as well as other developers interested in architecting and implementing +energy-aware software. Soliman et al. [11] conducted an empirical study with 50 software engineers, +who used Google to make design decisions using the Attribute Driven Design steps [12]. Based on the +relevance and Architecture Knowledge (AK) concepts specified by software engineers, they determined +how effective web search engines are to find relevant architectural information from various sources (in- +cluding Stack Overflow) and to capture AK. In another work, Soliman et al. [51] developed an ontology +that covers AK concepts in SO. The ontology provides a description of architecture related information +to represent and structure AK in SO. +Soliman et al. [10] also leveraged SO with the goal of improving how architects search for architec- +turally relevant information in online developer communities. They developed a new search approach +(i.e., a domain specific-search approach) for searching architecturally relevant information using SO. +They found that the new search approach outperforms the conventional keyword-based search approach +(searching through the search engines, such as Google). Tian et al. [14] conducted an empirical study of +SO users’ conception of architectural smells by analyzing the discussions from architecture smell related +posts in SO. They found that SO users often describe architectural smells with some general terms, such +as “bad”, “wrong”, “brittle” or violation of architecture patterns. Li et al.[62] extracted data from eight +most popular online developer communities, including Stack Overflow, to investigate how developers +perceive the notion of architecture erosion, its causes and consequences, as well as tools and practices +to identify and control architecture erosion. Among other major findings reported in their study, Li et +al. found that developers either focus on the structural manifestation of architecture erosion or on its +effect on run-time qualities, maintenance and evolution; alongside technical factors, architecture erosion +is caused to a large extent by non-technical factors. Zou et al. [63] used the topic model technique to +study non-functional requirements related to textual content in SO posts in order to understand the +actual requirements of developers. Our study complements the abovementioned work since it focuses +on the investigation of architectural knowledge in SO through the characterization and categorization of +architecture related posts. In addition, we examine the usefulness (i.e., are the answers useful to address +the questions? [15]) of these posts from the point of view of SO users. +37 + +The work closely related to ours is the study by Soliman et al. [6], which leveraged SO to categorize +ARPs based on technology related information provided in those posts. The main difference between +our study and their work is the fact that we look at the problems from a wider scope. In other words, +our study aims to categorize ARPs in SO by looking at various architectural information, such as ar- +chitecture tactics, provided in those posts, rather than limiting ourselves to one particular information +(i.e., technology information). Therefore, our work complements the work in [6] by digging deeper into +architecture related posts, for example, identifying additional categories of ARPs and exploring design +contexts of architecture related questions. Moreover, we characterize and analyze the usefulness of these +posts for practitioners and researchers. +7.2. Quality assessment of knowledge in Q&A Sites +The Q&A platforms, such as SO, play a significant role in knowledge sharing; however, they still face +significant challenges to ensure the quality of their knowledge. This is evident in the growing number of +studies that focus on analyzing the quality of the content in the programming related posts in SO from +different views, such as code and text. +Dagenais et al. [64] conducted an empirical study on the traceability links between an API and its +learning resources in SO. They found that the majority of API names (89%) in code snippets from online +forums are vague and cannot be easily solved due to the deficiency of code snippets. An et al. [65] studied +399 Android applications and revealed 1,279 cases of copyright violations of code reuse between GitHub +and SO. Fischer et al. [66] assessed the security-related matters of code snippets in SO and discovered +that 29% are insecure. Zhang et al. [41] investigated obsolescence of answers in SO and found that 31% +may have potential API usage violations that could yield unexpected behavior, such as system crashes +and resource leaks. Ragkhitwetsagul et al. [67] investigated Java code snippets in SO and identified +that 153 clones were copied to SO wherein 66% were obsolete. Zagalsky et al. [68] presented Example- +Overflow, a code search and recommendation tool to suggest high-quality code by using the knowledge +in SO. Zhang et al. [8] conducted an exploratory study on the prevalence and severity of API misuse in +SO. Treude et al. [69] surveyed GitHub users to comprehensively study the difficulty of code snippets in +SO. They found that less than half of the SO snippets are self-explanatory. Ragkhitwetsagul et al. [70] +conducted an online survey to investigate the answer obsolescence matter in SO. Their survey results +indicated that half of the top answerers are aware of obsolete code examples. However, users rarely and +even never fix obsolete code examples. Treude et al. [71] developed a tool to improve API documentation +with the use of “insight sentences” extracted from SO. Wong et al. [72] proposed an AutoComment tool +to automatically generate comments for Java and Android tagged Q&A posts in Q&A sites. The results +indicate that accurate, adequate, concise, and useful generated comments help users understand the +code. Gao et al. [73] investigated questions (with similar crash traces) to automatically fix recurring +crash bugs in Q&A sites. McDonnell et al. [74] investigated APIs evolution in Android ecosystem using +the version history data found in GitHub. Their results revealed that Android is evolving fast at a rate +of 115 API updates per month on average. Dalip et al. [75] suggested a method to rank answers with +regard to the feedback provided to answers. They witnessed that both user and review features are +essential to assess the quality of answers. Xu et al. [76] proposed an approach called AnswerBot to +automatically summarize answer posts relevant to a technical question in SO. Zhu et al. [15] proposed +a multi-dimensional model for assessing the quality of answers in social Q&A sites, such as Answerbag +and Yahoo! Answers, in the context of eLearning. Calefato et al. [77] conducted an empirical study +aimed at assessing 26 best-answer prediction models in SO. +These studies are related to our work since they investigated the quality of code examples provided in +SO posts, while our work investigates the quality (i.e., usefulness) of the architecture solutions provided +in SO posts. Nevertheless, our work differs from the aforementioned work in that they focused on low- +level source code (e.g., API), while our study focuses on high-level concepts (e.g., proposed architecture +patterns as solutions to address design concerns) to investigate their usefulness. We believe that our +study complements the existing work on the quality of SO posts by analyzing architecture related posts. +To the best of our knowledge, there has been no investigation of the architectural information +provided in ARPs with regard to their categories, characteristics, and usefulness (i.e., are the answers +useful to address the questions?) from the point of view of SO users, which is the focus of this study. +38 + +8. Conclusions and future work +Investigating architecture solutions (e.g., architecture tactics and patterns) as an important type +of architectural knowledge provided in online developer communities, such as SO, is crucial since this +knowledge is one of the most important development knowledge [22]. Architectural knowledge plays a +significant role for architects and developers in making informed architectural design decisions during +development [24]. Architecture solutions are the fundamental building blocks in modern software design +[22]. Contrarily to changing implementation (e.g., low-level source code), once an architecture solution +(e.g., an architecture pattern) is adopted and implemented, it is quite difficult and costly to change it +[22]. By analyzing and understanding how SO users deal with architectural problems or issues in online +developer communities, such as SO, brings three benefits: (1) it provides key insights about the types of +design problems SO users face during their architecture designs and the types of architecture solutions +discussed as well as their usefulness, (2) it can help to know the design contexts in which architecture +problems are raised, and (3) it can help to know the characteristics of architecture problems and solutions +discussed. These benefits provide an opportunity to develop new approaches and tools that can assist So +users search and (re)use architectural knowledge shared in online developer communities. To this end, in +this study, we investigated architecture related questions and their associated architecture solutions in +SO. Specifically, we used qualitative analysis approach to analyze a statistically representative random +sample of 968 ARPs from 10,423 ARPs manually identified. We intended to identify both the categories +and characteristics of architecture related questions and their solutions. We also explored the design +contexts in which those questions were raised. Finally, we studied SO users’ discussions on the usefulness +of the architecture solutions. We summarize our main results and findings as follows: +• SO users ask a broad spectrum of architecture related questions ranging from architecture tool to +architecture configuration, architecture implementation to architecture deployment. In addition, SO +users mostly discuss solution for architecture configuration (39%), followed by solution for architec- +ture implementation (18%), explanation of architecture (16%), and architecture tactic (11%). We +observed that ARPs (questions and answers) cover almost all architecting activities. +• SO users ask the most (27%, 261 out of 968) ARP questions about architecture configuration. +• Most of the SO users (71%, 687 out of 968 ARP questions) considered design contexts when asking +architecture related questions. +• Architecture related questions that provide clear description together with architectural diagrams +increase their likelihood of getting more than one answer, while poorly structured architecture +questions tend to only get one answer. +• Architecture solution for configuration from our proposed taxonomy is the most provided type of +architecture solutions that are considered useful in SO. +• SO users mainly consider architecture solutions that are complete and comprehensive and have +concise explanation with architectural diagrams to be helpful. +Our results and findings can help researchers and practitioners by knowing what types of architec- +tural knowledge, such as categories of architecture related questions and solutions, are provided in SO, +and what are the characteristics of good architecture related questions and useful architecture solutions. +Also, our results can motivate researchers and practitioners to consider SO as a valuable source of archi- +tectural knowledge (e.g., architecture patterns and tactics) and develop novel approaches and tools for +mining useful architecture knowledge from SO to support architecting activities and development. +In the next step: (1) We plan to conduct a comparative study of architecture solutions provided at +SO and other platforms (e.g., developer mailing lists and issue tracking systems), which may help reveal +insights into the current focus of architecture solutions utilization, and their advantages and deficiencies. +(2) We aim for validating and extending the proposed taxonomy of useful architecture solutions provided +at SO (see Figure 4) using an industrial survey from the practitioners’ perspective. (3) We also plan to +design and employ (semi-)automatic approaches to extract and summarize architectural information, and +establish the architecture issue-solution pairs from the retrieved architectural information, for example, +benefits and drawbacks of certain architecture solutions (e.g., patterns and tactics) for task-specific +architecture problems from multiple sources of architectural information (e.g., Q&A sites, GitHub, issue +tracking systems, technical blogs), which can facilitate the decision-making of architects by utilizing +architectural knowledge from peers and communities. +39 + +Acknowledgements +This work is partially sponsored by the Natural Science Foundation of China (NSFC) under Grant +No. 62172311. The authors would also like to acknowledge the financial support from the China Schol- +arship Council. +References +[1] C. Sadowski, K. T. Stolee, S. 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Calado, Exploiting user feedback to learn to rank answers in Q&A +forums: A case study with Stack Overflow, in: Proceedings of the 36th International ACM SIGIR +Conference on Research and Development in Information Retrieval (SIGIR), Dublin, Ireland, 2013, +pp. 543–552. +[76] B. Xu, Z. Xing, X. Xia, D. Lo, Answerbot: Automated generation of answer summary to developers’ +technical questions, in: Proceedings of the 32nd IEEE/ACM International Conference on Automated +Software Engineering (ASE), Urbana, IL, USA, 2017, pp. 706–716. +[77] F. Calefato, F. Lanubile, N. Novielli, An empirical assessment of best-answer prediction models in +technical Q&A sites, Empirical Software Engineering 24 (2) (2019) 854–901. +44 + diff --git a/tNAzT4oBgHgl3EQfBfpr/content/tmp_files/load_file.txt b/tNAzT4oBgHgl3EQfBfpr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4943a235fc5b57257156bc42f02e0b9541b7367 --- /dev/null +++ b/tNAzT4oBgHgl3EQfBfpr/content/tmp_files/load_file.txt @@ -0,0 +1,2618 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf,len=2617 +page_content='Characterizing Architecture Related Posts and Their Usefulness in Stack Overflow Musengamana Jean de Dieua, Peng Lianga,∗, Mojtaba Shahinb, Arif Ali Khanc aSchool of Computer Science, Wuhan University, 430072 Wuhan, China bSchool of Computing Technologies, RMIT University, 3000 Melbourne, Australia cM3S Empirical Software Engineering Research Unit, University of Oulu, 90014 Oulu, Finland Abstract Context: Stack Overflow (SO) has won the intention from software engineers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architects) to learn, practice, and utilize development knowledge, such as Architectural Knowledge (AK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' But little is known about AK communicated in SO, which is a type of high-level but important knowledge in development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Objective: This study aims to investigate the AK in SO posts in terms of their categories and charac- teristics as well as their usefulness from the point of view of SO users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Method: We conducted an exploratory study by qualitatively analyzing a statistically representative sample of 968 Architecture Related Posts (ARPs) from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Results: The main findings are: (1) architecture related questions can be classified into 9 core cate- gories, in which “architecture configuration” is the most common category, followed by the “architecture decision” category, and (2) architecture related questions that provide clear descriptions together with architectural diagrams increase their likelihood of getting more than one answer, while poorly structured architecture questions tend to only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Conclusions: Our findings suggest that future research can focus on enabling automated approaches and tools that could facilitate the search and (re)use of AK in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO users can refer to our proposed guidelines to compose architecture related questions with the likelihood of getting more responses in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Keywords: Architectural Knowledge, Architectural Level Element, Architecture Solution, Stack Overflow, Usefulness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Introduction Technical Questions and Answers (Q&A) sites, such as Stack Overflow (SO), have revolutionized how users seek knowledge on the Internet [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO has shown to be the most prominent community Q&A site for knowledge sharing and learning in software development, and SO leverages the knowledge and skills of its users, such as developers, to share their thoughts and experience by asking various types of technical questions related to development and providing answers to these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, SO users can learn novel techniques and tools from SO [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO is predominately being used to solve coding problems [3], and these problems are often not relevant or less interesting to architects because they focus on lower-level implementation details [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, ever since this site started growing and being popular, architects have begun to share their competencies, experience, and design problems by asking architecture related questions or providing architecture solutions, such as architecture tactics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In our recent industrial survey on how developers search for architectural information [4], practitioners acknowledged Q&A sites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', SO) as the most useful source of architectural information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', benefits and drawbacks of architecture solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hence, similar to searching and (re)using existing coding related answers provided in SO to solve programming related problems, software engineers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architects and developers) also search and (re)use existing architecture solutions in SO for addressing their design concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, SO not only ∗Corresponding author at: School of Computer Science, Wuhan University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' : +86 27 68776137;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' fax: +86 27 68776027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Email addresses: mjados@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com (Musengamana Jean de Dieu), liangp@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='cn (Peng Liang), mojtaba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='shahin@rmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='au (Mojtaba Shahin), arif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='khan@oulu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='fi (Arif Ali Khan) Preprint submitted to Journal of Systems and Software January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='00943v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='SE] 3 Jan 2023 accumulates code examples, but also curates a large number of architecture solutions provided to a wide range of architecture related questions or design problems [5] [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Although SO users discuss high-level knowledge in SO, for instance, architecture tactics and qual- ity attributes knowledge [5], architecture knowledge for technology decisions [6], to date the majority of the existing studies mainly focus on analyzing programming related knowledge in SO posts from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, Diamantopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [7] employed source code information to improve question-answering in SO, and Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [8] investigated the quality of code examples in SO programming related posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Little work has focused on analyzing architectural knowledge provided in Architecture Related Posts (ARPs) in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [5] mined posts from SO and structured the design relationships between architectural tactics and quality attributes used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [9] extracted SO posts and mined the design pattern use scenarios and related design pattern pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [10] developed a search approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a domain specific search approach) for searching architecture knowledge in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In another work, Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [11] conducted an empirical study with 50 software engineers, who used Google to make design decisions using Attribute Driven Design [12], and they determined how effective web search engines are to find relevant architectural information from various sources (including SO) and to capture AK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Malavolta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [13] extracted data from five open source software repositories (including SO), and mined architectural tactics for energy-efficiency applied by practitioners in real robotics projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [14] studied SO users’ conception of architectural smells using SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The abovementioned studies extracted ARPs from SO and investigated archi- tecture knowledge (high-level concepts) from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, prior work is only based on architecture related questions and their associated answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In contrast, our work covers the entire ARP, including its question, all comments under the question, all answers associated with the question, and all comments under the answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, no prior study has specifically investigated architectural knowledge provided in ARPs (answers) with regard to their usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, there has been no comprehensive research on exploring architectural knowledge communicated by SO users in terms of their types, design contexts, characteristics, and usefulness, which is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Analyzing and understanding how SO users deal with architecture design concerns in online developer communities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' such as SO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' brings three benefits: (1) it provides key insights about the types of design problems SO users face during their architecture design and the types of architecture solutions discussed as well as their usefulness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) it can help to know the design contexts in which architecture problems are raised,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and (3) it can help to know the characteristics of architecture problems and solutions discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These benefits provide an opportunity to develop new techniques and tools that can help SO users search and (re)use architectural knowledge shared in online developer communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, this study aims to complement prior works by analyzing the characteristics and categories of ARPs in SO as well as their usefulness from the point of view of SO users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, we treated usefulness (one quality criterion of posts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', answers, in Q&A sites [15]) using the definition in [15] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', are the answers useful to address the questions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To achieve the goal of this study (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1), we conducted an exploratory study to investigate various aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', categories and characteristics) of ARPs in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' More specifically, we extracted 32,182 posts from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We went on to manually filter out irrelevant posts and got 10,423 candidate ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Since 10,423 candidate ARPs were a quite large dataset, and it was not easy to manually analyze this size of dataset with human effort and get accurate and comprehensive results, we used the power statistics and calculated a representative sample size [16] of these 10,423 ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With a 95% confidence level and 3% margin of error, the final representative sample size calculated was 968 ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Then, we randomly selected 968 ARPs from the 10,423 ARPs and analyzed them for answering a set of research questions (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, we manually analyzed the 968 ARPs using open coding and constant comparison from Grounded Theory (GT) [17] to answer those research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The main results and findings of this study are that: (1) SO users ask a broad spectrum of architecture related questions ranging from architecture tool to architecture configuration, architecture implementation to architecture deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) The useful architecture solutions are classified into seven categories as a taxonomy (see Figure 4), such as solution for architecture configuration, solution for architecture implementation, architecture tactic, and architecture pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One observation is that the identified categories of these posts (questions and answers) cover almost all the architecting activities that span from the initial stages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architectural analysis and synthesis [18]) of architectural creation as well as the later stages (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architectural implementation and maintenance & evolution [19]) in a system lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus our identified categories of ARPs can support the mentioned architecting activities during the architecture lifecycle, and SO can be considered as one of the sources of architectural knowledge [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We found 2 that architecture related questions that provide clear descriptions together with architectural diagrams increase their likelihood of getting more than one answer, while poorly structured architecture questions tend to only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) SO users frequently use two terms related to usefulness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', useful and helpful) to explicitly communicate about the usefulness of certain architecture solutions provided to their associated architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This study makes the following three contributions: (1) a classification and characterization of ar- chitecture related questions that SO users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', developers) asked in SO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) a list of identified design contexts in which architecture related questions were raised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and (3) a classification (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a proposed taxonomy) and characterization of useful architecture solutions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our study findings can be ben- eficial to various stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, researchers can refer to our proposed taxonomy of useful architecture solutions in SO as a guidance to develop new automated approaches and tools that could mine and locate architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', solution for architecture configuration, see Figure 4) for addressing similar design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', questions that ask about architecture configuration, see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This can facilitate SO users to check the questions and solutions that are relevant to their design concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO can use our results to better adjust its answers and comments organization mechanisms and enhance the search and (re)use of useful architecture solutions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The rest of this paper is structured as follows: Section 2 presents the background of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Section 3 describes the research methodology, and Section 4 elaborates the study results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Section 5 analyzes the results and discusses their implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Section 6 presents the threats to the validity of the study results, and Section 7 summarizes the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finally, Section 8 concludes this work with potential areas of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Background In this section, we introduce the background concepts used in this study, including Stack Overflow, architecture knowledge, architecture problem, design context, and architecture solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Stack Overflow Stack Overflow is one of the websites that make Stack Exchange1 network, which provides a Q&A platform for its users to share knowledge across various domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', programming, design, statistics, mathematics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO users exchange knowledge related to software development by asking questions or providing answers to existing questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among other development knowledge, architecture knowledge, such as drawbacks and benefits of architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', patterns and tactics) in certain application domains, has been shared at SO to support architecting activities [20][21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Mining architecture knowledge in Q&A websites, specifically in SO, has been the subject of the architecture research community in recent years, such as architectural knowledge for technology decisions [6] and architecture tactics and quality attributes [5], in order to support the architecting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture knowledge Software Architecture (SA) is a set of structures comprising software elements, the relationships among them, and the properties of the elements and relationships [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Building an architecture of a software system often requires knowledge, especially architecture knowledge [23], and skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture knowledge, such as architecture decisions and their rationale [24], benefits and drawbacks of architecture solutions [6], is one of the most important types of knowledge in software development [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architectural knowledge is often described in various formats, such as textual and graphical representation [25] and this knowledge is recorded in various sources, such as books [22], technical blogs and tutorials [11], developer mailing lists (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', ArgoUML [26]), Q&A sites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', SO [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, we investigated architecture knowledge discussed in SO from various aspects, such as categories and characteristics of ARPs in SO, SO users’ discussions on the usefulness of architecture solutions provided in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture problem Architecture problems (such as “any testable architecture or design pattern for an MFC applica- tion?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2) arise during development when addressing specific architecture design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', quality 1https://stackexchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/sites 2https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2z69uzs5 3 attributes) and their trade-offs [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' There are various problems related to architecture design that are asked in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, we investigated the categories of architecture problems/questions, specifically, the categories of architecture related questions asked in SO (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Design context Design context of a software system comprises the knowledge that an architect needs to have about the environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a hardware platform) in which a system is expected to operate [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Design contexts can be seen as “conditions that influence design decisions but are not specified explicitly as requirements” [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Harper and Zheng suggested that design contexts are forces that influence stakeholders’ concerns [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' There are some works that categorize design contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Bedjeti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' identified four context categories of an architecture viewpoint (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', platform context, user context, application context, and organizational context) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Petersen and Wohlin provided a checklist for documenting design contexts from six perspectives: product, processes, practices and techniques, people, organization, and market [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Groher and Weinreich studied environmental factors that influence architecture decision making [31], and they identified eight categories: company size, business factors, organizational factors, technical factors, cultural factors, individual factors, project factors, and decision scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In our study, we investigated the design contexts that were discussed in architectural related posts in SO, and we referred to the classification of design contexts proposed in two existing studies [27][30] (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture solution Architecture solutions are the fundamental building blocks in modern software design and they are used to address architecture design concerns [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' There are various architecture solutions, such as patterns, tactics, and frameworks for addressing different design concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Model–View–Controller, Client-Server, Publish-Subscribe patterns) are reusable solutions to commonly occurring problems in architecture design within given contexts [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Contrarily to changing implemen- tation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', low-level code), once an architecture solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', an architecture pattern) is adopted and implemented, it is quite difficult and costly to change it [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture patterns determine the overall structure and behavior of a software system [32] and are typically selected early during development for achieving multiple system requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', quality attributes) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, we studied SO users’ discussions on the usefulness of architecture solutions, for example, patterns, tactics, frameworks (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5), as well as the categories and characteristics (in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='6) of architecture solutions that were considered useful in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Research design We carried out an exploratory study on various aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', categories and characteristics) of ARPs in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the following subsections, we describe the details of the research design of this study, including the goal and Research Questions (RQs) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1 and the execution of this study in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Goal and research questions The overall goal of this study based on Goal-Question-Metric approach [33] is “to analyze the ARPs (questions and answers) in SO for the purpose of investigating their categories, characteristics, and usefulness from the point of view of SO users in the context of software development in practice”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Following the goal of this research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' we derived six research questions (see Table 1) that aim to examine four aspects that highlight the question and answer threads of SO posts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' including (1) categorization of architecture related questions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) the design contexts in which architecture related questions were raised,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) characterization of architecture related questions that have more than one answer and characteristics of architecture related questions that only have one answer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and (4) categorization and characterization of architecture solutions that are considered useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Study Execution In this subsection, we describe the process of data collection and analysis of ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Figure 1 shows an overview of the two processes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', data collection and analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 4 Table 1: Research questions and their rationale Research Question Rationale RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What architecture related questions are asked in SO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture related questions (design problems) are mainly asked to ad- dress certain design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', quality attributes of a system) during architecting activities, for example, architectural analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO curates dif- ferent types of architecture related questions that are raised with various design issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The answer to this RQ can help researchers to be aware of the areas of interest of SO users in architecture design and help practition- ers to get an insight into the architecture related questions asked in SO so that they can provide practical contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What are the design con- texts in which architecture related questions were raised?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Design contexts comprise the knowledge about the environments in which systems are expected to operate [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Design contexts are indispensable ingredients that can drive the architecture design of a system [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' A system of similar functionalities can operate differently in different contexts [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Although the importance of considering design contexts during architecture design has been recognized, there is limited understanding on what design contexts are considered in architecture design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The answer to this RQ can help researchers and practitioners be aware of typical design contexts in which architecture related questions are raised in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What are the characteristics of architecture related questions in SO that have more than one an- swer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One major challenge during architecture design is choosing the right archi- tecture solutions to address the requirements of the systems [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Although different architecture solutions act as alternative solutions to similar ar- chitecture problems, they differ in terms of their qualities [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, providing more than one answer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', alternative solutions) to architecture problems/questions is important as they provide a wide range of possibili- ties for making architecture design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With this RQ, we identify and examine the characteristics of architecture related questions that get more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' By characteristics of an architecture related question, we mean certain features, such as architectural diagrams, in the content of the question or question formulation [35], that distinguish the architecture related question to another or make the architecture related question get attraction from SO users and get more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The answer to this RQ can help researchers and practitioners know what motivates SO users to provide more solutions to these questions, and consequently improve or prevent unanswered architecture related questions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What are the characteristics of architecture related questions in SO that only have one answer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Some architecture related questions fail to continuously get attention from SO users by answering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Similar to RQ3, we want to examine the factors behind this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We study the characteristics of architecture related questions that only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The answer of this RQ can help researchers and practitioners know what demotivates SO users to continue answering these questions and design general guidelines for SO users to compose architecture related questions with the likelihood of getting more responses in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What are the types of ar- chitecture solutions provided in SO that are considered useful by SO users?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' There are many architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', tactics and patterns) to ad- dress architecture related questions provided in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, the quality of solutions/answers provided in SO has been a major concern for researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As elaborated in the related work (see Section 7), this is evident in the growing number of studies, in which the focus is on an- alyzing the quality of the content in SO posts from different perspectives, for example, code and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The results of this RQ can help researchers and practitioners be aware of types (a taxonomy) of architecture solutions considered useful in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What are the characteristics of architecture solutions in SO that are considered useful by SO users?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [8] argued that accepted, highly voted, and frequently viewed SO posts are not always reliable or useful in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Identifying the features of architecture solutions that are considered useful helps to better under- stand what SO users consider when accepting architecture solutions as useful ones, thus providing insights for improving the current answering mechanism of architecture related questions and helping SO users retrieve their desired architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5 Phase I: Gathering Architecture Related Posts (ARPs) Stack Overflow Query with keywords Returned posts Returned posts Query with keywords 32,182 retrieved posts Filter ARPs from others posts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', programming and hardware related posts) Is it an ARP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Stack Exchange API Pilot filtering Disagreements & Discussions Formal filtering Apply inclusion & exclusion criteria 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='423 valid ARPs Consensus on inclusion & exclusion criteria Phase II: Determining the representative sample size 968 ARPs (RQ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ2) Identifying questions with more than one answer Identifying ARPs with useful information Data extraction and analysis 650 questions (RQ3) ARP categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' characteristics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and design contexts Identifying questions with only one answer 318 questions (RQ4) 324 ARPs (RQ5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ6) Figure 1: An overview of data collection and analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Data collection Our data collection is divided into two phases, namely Phase I: Gathering architecture related posts and Phase II: Determining the representative sample size, as detailed below: Phase I: Gathering architecture related posts a) Search terms: Before we decided the most suitable terms for capturing posts relevant to architec- ture design, we first performed a pilot search with several terms, namely “architect*” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “architect”, “architecture”, “architectural”, and “architecting”) and “design*” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “design” and “designing”), within SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The process was carried out by using a SQL query through the query interface provided by StackEx- change Data Explorer3, which is a web interface that allows the execution of SQL queries on data from Q&A sites, including SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' After the pilot search with the mentioned terms, we saw that SO users mostly use the terms “design*” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “design” and “designing”) in the programming context in SO, for instance, “singleton design pattern”4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, we were aware that Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [6] identified distinctive terms between ARPs and pure programming posts from their studied sample of SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, in our study, we did not use those distinctive terms to search ARPs in SO due to the following two reasons: (1) The purposes of our work and Soliman et al.’s work in [6] are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The purpose of the work in [6] is technology related architecture knowledge extraction from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, the authors in [6] identified and analyzed ARPs that mainly discuss architectural knowledge for technology decisions, such as the pros and cons of a technology solution in a certain application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, the authors in [6] claimed that they did not find many pure architectural concepts (such as architectural pattern or tactic) in their dataset of ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In contrast, our study takes the problems from a wider scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, our study aims to identify and analyze ARPs from SO by looking at various architectural information, including architecture patterns, tactics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, using the specific distinctive terms, such as versus, alternative, pros, cons, xmpp, that were found in [6] may lead to missing other relevant ARPs, which may affect the completeness of the retrieved ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Using the distinctive terms found in Soliman et al.’s work [6] may lead to bias in the search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The relevancy and completeness of extracted ARPs may affect the correctness of the answers to our six RQs (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, including the specific distinctive terms that were found in [6] in the search queries may lead to the situation that the search results are biased to those terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 3https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='stackexchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/stackoverflow/query/new 4https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/8yks7nhm 6 Therefore, we selected the general terms “architect*” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “architect”, “architecture”, “architec- tural”, and “architecting”) to be used in our search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It is worth mentioning that we did not use the search terms to search exclusively through tags only because tags can sometimes be less informative and ineffective [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' There are several disadvantages of using tags as the only approach to determine whether a post is related to a topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This is due to the reason that a user who created a post could be unsure about the title of the most appropriate tag for their discussion, which can lead to the use of incorrect or irrelevant tags [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this architecture related post5 that asked for an architecture pattern that can be used in the design of a single webform application, a developer used tags (“jc#”, “asp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='net”, and “web”), and these tags cannot immediately tell in which contexts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture or programming context) they are really used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Another problem with user-defined tags is that users may try to add as many tags as possible (SO allows up to 5 tags) to raise the number of views and probably increase the probability of getting responses quickly [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, while tags can be helpful to capture posts related to architecture design, using tags exclusively may miss important posts on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hence, we decided to add the title and body of the questions into the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, by following the criteria of the query interface provided by Stack Exchange6, for the term “architect”, we searched in the title, tags, and body of posts by using this query: SELECT p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Id, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Tags, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Title, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Body as “Questions Body”, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Score as “Questions Score”, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Answercount as “Answer Count” FROM Posts p WHERE (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Body like ‘%architect%’ or p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Title like ‘%architect%’ or p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Tags like ‘%architect%’) AND p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Score >0 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='AnswerCount <>0 ORDER BY p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Score DESC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In our replication package [39], we provided the complete SQL query used to search ARPs in SO, such as how the title, tags, and body of a post were combined during the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The searching process resulted in 32,182 posts (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Note that we used the mentioned search terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “architect”, “architecture”, “architectural”, and “architecting”), not for the purpose of accumulating all ARPs in SO, but for gathering sufficient data for a relatively comprehensive analysis to achieve the goal of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' b) Filtering ARPs from other posts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', programming and hardware related posts): We found that SO users use the term “architecture” not only in the context of software architecture, but also in other contexts, such as hardware architecture context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', ARM6 CPU architecture7) and programming con- text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', array architecture8), when describing their concerns in the SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, we need to filter the retrieved 32,182 posts and exclude those posts related to programming and hardware architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To do so, we performed context analysis and applied our defined inclusion and exclusion criteria (see Table 2) to accurately filter and separate software ARPs from other types of posts mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Before the formal post filtering (manual inspection), to reach an agreement about the inclusion and exclusion criteria (see Table 2), a pilot filtering was performed whereby the first author took a random sample of 1,000 posts from the 32,182 posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' He manually checked them with our defined criteria (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The other three authors checked and examined the results so that all the authors (four authors) of this study could get a consensus on the understanding of the defined inclusion and exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thereafter, we got controversy and misunderstanding on 51 posts from the filtered results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Such controversy and misunderstanding were discussed between the four authors of this study till a consensus was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first author carried on with the formal post filtering based on the inclusion and exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The process continued till all the 32,182 posts were manually checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This step resulted in 10,423 candidate ARPs (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The results from this round were checked and verified by the other three authors of this study, and it took us twenty one full days to identify and separate ARPs from programming and hardware architecture related posts in these 32,182 posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Phase II: Determining the representative sample size The 10,423 candidate ARPs (filtered from the previous phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Phase I)) are a quite large dataset, and it is not realistic to analyze this size of dataset with human effort and get accurate and comprehensive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, in order to get statistically significant results, we used the power statistics and calculated a representative sample size [16] of these 10,423 ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' At a confidence level of 95%, we set a margin of error (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', how much we can expect our analysis results to reflect the view of the overall dataset) to 3% for the whole 10,423 ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The final representative sample size calculated is 968 5https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/mb9y37z4 6https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='stackexchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/stackoverflow/query/new 7https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/f8sjvzz2 8https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/3ps7b3ek 7 Table 2: Inclusion and exclusion criteria for filtering ARPs from programming and hardware related posts Inclusion criteria I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' An ARP should contain a discussion on software architecture, for example, architecture design and architecture tactics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' An ARP should contain at least one answer attached to its architecture related question as we aim to study the factors that make these questions have more than one answer or only have one answer and the usefulness of their answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' An ARP should contain at least one data item that can be extracted according to the data items defined in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Exclusion criteria E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' An ARP that has a score (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', medium number of down/upvote) that is less than 1 is excluded since we want to make sure that all studied posts have attracted enough attention from the community [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Then, we randomly selected 968 ARPs from the 10,423 ARPs and analyzed them for answering the six RQs (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To be more specific, except for RQ1 and RQ2 on which we used 968 (a representative sample size of ARPs) to answer them, we used subsets of the 968 ARPs that satisfy our defined criteria (in the following steps) with respect to the purposes of the remaining RQs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', RQ3, RQ4, RQ5, and RQ6) (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We followed the following steps to further divide our calculated representative sample size of ARPs (968) into subsets of the ARPs that are relevant to answering the remaining RQs: Step 1: Identification of questions that have more than one answer (RQ3) and questions that only have one answer (RQ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As stated in the rationale of these two RQs (see Table 1), we want to examine the factors that make such architecture questions get more than one answer or only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus we considered comments posted on architecture related questions by referring to these two studies [41][42], in which the authors argued that the quality of an answer is a combination of both the answer and its associated comments as comments may provide additional information about the answer, for example, improvement of answers [42] and obsoleted answers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, in our study, we included comments posted on architecture related questions and studied what makes these posts get more than one answer (RQ3) or only get one answer (RQ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' More specifically, the first author manually checked 968 ARPs and their comments, and got 650 architecture related questions (wherein each question has more than one answer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These questions were used to answer RQ3 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' On the other hand, the first author followed the same procedure (manual inspection of the 968 ARPs for RQ3) and got 318 architecture related questions (wherein each question has only one answer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These questions were used to answer RQ4 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Step 2: Identification of ARPs with useful knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For answering RQ5 and RQ6, we need to identify ARPs with useful knowledge from the representative random sample of ARPs (968 ARPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We referred to these two studies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', [41, 42]) to identify ARPs with usefulness knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These studies by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [41, 42] argued that the quality of an answer is a combination of both the answer and its associated comments as comments may provide additional information to support the answer, such as improvement of answers [42] and obsoleted answers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, in this study, we included the information in comments to gain a deep understanding of how SO users discuss the usefulness of architecture solutions provided to their architecture related questions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, we did not consider vote score for answering RQ5 and RQ6 since even highly voted SO posts are not always reliable or useful as argued by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We observed that SO users occasionally use terms related to usefulness, such as “helpful”, in the comments to indicate that certain architecture solutions provided to their architecture related questions are useful (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, based on this observation along with the aid of our defined selection criteria in Table 3, we manually checked and filtered the 968 ARPs to identify solution threads with useful knowledge (each solution thread includes all solutions to a question (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', accepted & not-accepted solutions) and all the comments that are associated with the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, to filter out ARPs that do not discuss the usefulness of architecture solutions and reach an agreement about the criteria defined in Table 3, two authors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', the first and second authors) did a pilot ARPs filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They independently and manually examined a random sample of 20 ARPs from the 968 8 ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Similar procedure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', selecting a random sample of data from a large dataset and subsequent manual filtering) has also been employed in recent studies, such as [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To measure the inter-rater agreement between the first two authors, we calculated the Cohen’s Kappa coefficient [44] and got an agreement of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Note that before a solution was finally included as a relevant one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a useful solution), the first and second authors first read the solution that was commented to be useful/helpful in order to verify if it is really useful to address the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Disagreements on the ARPs were discussed between the two authors till a consensus was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Then the first author carried on to check and filter the remaining ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The number of resulting ARPs (with useful knowledge) that were used to answer the two RQs (RQ5 and RQ6) is 324 ARPs (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Figure 2: An example answer that was commented to be helpful Table 3: Inclusion and exclusion criteria for identifying ARPs with useful knowledge Inclusion criterion I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' A comment in an answer thread must contain one of the keywords related to usefulness, such as “useful”, “helpful”, “beneficial”, “handy”, and “effective”, and this comment is used to signify the usefulness of the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Exclusion criteria E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The keyword related to usefulness, for example, “useful”, “helpful”, “beneficial”, is used to talk about something else (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a question or answer itself is related to a “usefulness” topic rather than being a sign that the answer is likely useful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' An ARP with controversy discussions on the answer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', if there are two comments in the same answer thread, and one states the usefulness of the answer while another states its uselessness) is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Data extraction and analysis (1) Data extraction: We performed the data extraction process by identifying the relevant informa- tion to be extracted from 968 ARPs to answer our defined RQs (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In Table 4, we present the data items for which the relevant information was extracted from the candidate ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It also shows the RQs that are supposed to be answered using the extracted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The data extraction was subse- quently followed by data analysis, and these two processes were conducted and recorded with the aid of MAXQDA (a qualitative data analysis tool)9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 9https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='maxqda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/ 9 Okay so as an electrical engineer who works with software involving C# and serial ports almost every day, here is some advice that I can offer you from an architectural perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Use this architecture in the situation where you need to control the motors across several application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Have some process/service running in the background that controls your port 1oo% of the time (via your APl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' You can launch an application then talk to the background service (responsible for controlling the motor) via TCP sockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This way you can launch as many applications as you want and everybody will get access to the APl without having to worry about serial port access issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Use this architecture in the situation where you need to control the motors in a single application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=" This one's similar to what you're already proposing in your question, which by the way I think is a pretty good way of doing things." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Instantiate the class to control the motors from your APl and then use constructor/property injection, or some kind of Dl to pass a reference to the controller to everybody who needs it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=" Share Edit Follow answered Feb 24 '17 at 17:22 Snoop 95311127 Thanks, Snoopy!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=" I've been working on this project for a week now, and I've learned a ton so far." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I decided to go with a static service for controlling all of the different devices from my APl, and just running any commands I need to send to them in background threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It seems to be working really well, and the result is even cooler (l can control motors over the web, so cool).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thanks for the tips, definitely helpful!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=" Brian CorbinFeb 27 '17 at 0:13 Table 4: Data items to be extracted from the ARPs with their description," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' analysis approaches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and relevant RQs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Data item ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Data analysis approach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='RQs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Content ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='The main content of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='question in the ARP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Open coding & constant com- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='parison ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='RQ1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Design context ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='The design context elabo- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='rated in the content of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='question in the ARP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Predefined classifications in [27] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='and [30] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='RQ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='D3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Content ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='its ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='comments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='The main content of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='question and a summary of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='the question’s comments in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='the ARP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Open coding & constant com- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='parison ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='RQ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RQ4 D4 Content of the an- swer The main content of the an- swer in the ARP Open coding & constant com- parison RQ5 D5 Content of the an- swer and its com- ments The main content of the an- swer and a summary of the answer’s comments in the ARP Open coding & constant com- parison RQ6 (2) Data analysis: Similarly to several existing studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', [4]), we used open coding & constant comparison to answer RQ1 and RQ3-RQ6 in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Open coding & constant comparison are two widely used techniques from Grounded Theory [17] during qualitative data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Grounded Theory (GT) is a bottom-up approach and focuses on theory generation, rather than extending or verifying existing theories [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Open coding generates codes for incidents that can be further classified into concepts and categories [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Constant comparison is a continuous process for verifying the generated concepts and categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Both concepts and categories evolve and saturate until they fit the data [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, in this study, we employed open coding & constant comparison techniques from GT to generate the concepts and categories for answering RQ1 and RQ3-RQ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, we used open coding to encode the extracted data items for RQ1 and RQ3-RQ6 (see Table 4) to generate codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Afterwards, we applied constant comparison to compare the codes identified in one piece of data with the codes that emerge from other data to identify the codes which have similar semantic meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We proceeded to group similar codes into high-level concepts and categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' On the other hand, we employed predefined classifications of design contexts in [27] and [30] to answer RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We followed the same procedure (encoding and grouping similar codes into high-level categories) to answer RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Before the formal data analysis, the first author conducted a pilot data analysis for each RQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, this analysis process involved the following steps: (1) The first author selected a random set of 100 ARPs from the representative sample size calculated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 968 ARPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) The first author coded the extracted data (see Table 4) for each RQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' When such posts were unclear and the first author got confused while coding the extracted data, physical meetings with the second author were scheduled to solve such confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) The first author applied constant comparison and grouped all the codes into higher-level concepts and turned them into categories and subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The grouping process was iterative, in which the first author continuously went back and forth between the concepts, categories, subcategories, and contents of the questions, answers, and comments to revise and refine the concepts, categories, and subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (4) Thereafter, other three authors (the second, third, and fourth authors) checked and validated the results from the pilot data analysis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', concepts, categories, and subcategories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The disagreements were resolved in a meeting using a negotiated agreement approach [45] to improve the reliability of the pilot data analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first author carried on with the formal data analysis and followed similar steps used during the pilot data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the following paragraphs, we provide details of the formal data analysis process: a) For analyzing RQ1, RQ3, and RQ4 As abovementioned, we used open coding and constant comparison [17] to manually analyze the extracted data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', content of the question for RQ1 and content of the question and its comments for 10 RQ3 and RQ4) as shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With these RQs, we investigated architecture related questions from two aspects, namely categorization (RQ1) and characterization (RQ3 and RQ4) of these questions (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, regarding the categorization of the questions, the first author studied the content of each architecture related question (from the ARPs that are relevant to answer RQ1 (see Figure 1)) by exploring and identifying their main purposes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', design concerns), such as asking for help on how to refactor the architecture of a system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', refactoring of circular dependencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thereafter, the first author summarized each question’s purpose in a short sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Firstly, the first author went on to encode the summarized sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This process was iterative, in which he continuously applied this technique till all ARPs in the dataset were encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Secondly, the first author applied constant comparison to compare the codes identified in one summarized sentence with the codes that emerged from other summarized sentences to check the codes which have similar semantic meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first author proceeded to group similar codes into high-level concepts, categories and subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The grouping process was iterative, in which the first author continuously went back and forth between the concepts, categories, subcategories, and contents of the questions to revise and refine both the concepts, categories and their subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To mitigate the personal bias during the formal data analysis, the other authors (second, third, and fourth authors) of this study participated in the validation of the generated codes, concepts, categories, and subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The disagreements were resolved in a meeting using the negotiated agreement approach [45] to improve the reliability of the analysis results for RQ1 as during the pilot data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We finally got 9 high-level categories and 21 subcategories as the results of RQ1, and these results are fully elaborated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1, for answering RQ3 and RQ4, we manually checked the 968 ARPs to identify ARPs with more than one answer (RQ3) and ARPs with only one answer (RQ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This led to two subsets of the 968 ARPs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 650 ARPs and 318 ARPs) relevant to answering RQ3 and RQ4 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, in the formal data analysis, the first author analyzed the questions and their attached comments (from the ARPs of the two mentioned subsets, 650 ARPs and 318 ARPs, of the 968 ARPs) by encoding the extracted data for the two RQs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', content of the question and comments as shown in Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' He wanted to study if there might be such factors, for example, question formulation [35] or certain features in the question (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture diagram) that contribute to such architecture related questions having more than one answer or only having one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, when investigating RQ3 (questions with more than one answer), one community member posted a comment under an architecture related question saying that “+1 great question, very well-articulated”10, for this comment, he picked a phrase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a summary of that comment) “well-articulated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Subsequently, he went on to study the content of the question (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', how architectural information is stated in the question) under which this comment was commented, and then he came up with one code that fits this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' He followed the same processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', grouping similar codes into high-level concepts, categories) that the used when analyzing RQ1 to analyze these two RQs (RQ3 and RQ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To mitigate the personal bias, the results from this analysis were checked and validated by other three authors of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As in the analysis of RQ1, we held a meeting and followed the negotiated agreement approach [45] to discuss and resolve any disagreement, therefore improving the reliability of the analysis results for RQ3 and RQ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the final analysis, we generated four characteristics of architecture related questions that have more than one answer and five characteristics of architecture related questions that are that only have one answer as the results of RQ3 and RQ4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The details about these four characteristics for RQ3 and five characteristics for RQ4 are provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' b) For analyzing RQ2 We employed pre-defined classifications in [27] and [30] to answer RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, the first author manually analyzed the extracted data for RQ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', design contexts, see Table 4) from the questions in the ARPs relevant to answering RQ2 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first author then examined the extracted data to investigate the design contexts in which architecture related questions were raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' By referring to the categories of design contexts presented in the abovementioned studies, three main categories and eight subcategories were generated from the analyzed ARP questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The personal bias was mitigated through the validation of the generated categories and subcategories with the other three authors of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As in the analysis of the previous RQs, the disagreements were discussed and resolved in a meeting using the negotiated agreement approach [45] to improve the reliability of the analysis results for RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The results of RQ2 are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 10https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/dfw27h38 11 c) For analyzing RQ5 and RQ6 These two RQs aim to investigate the usefulness of architecture solutions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1, in order to answer these two RQs, we defined a set of criteria (see Table 3) and filtered ARPs with useful knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For clarity, we examined these two RQs from two aspects: We investigated how SO users discuss the usefulness of architecture solutions attributed to their associated architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We needed to gain insights into ways (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', terms) SO users may use to communicate the usefulness of architecture solutions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, understanding SO users’ discussions on the usefulness of these solutions is important to direct Q&A platform owners in creating the mechanisms that can help their users to efficiently and effectively search and (re)use such useful architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To achieve this, we manually checked comments attached to the solutions in the 968 ARPs with the aid of our defined criteria (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We found that SO users occasionally use terms related to usefulness, such as “helpful”, in the comments along with other terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “very”, “super”) to explicitly convey how useful they found certain architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1, the number of resulting ARPs (with useful knowledge) that were used to answer the two RQs (RQ5 and RQ6) is 324 ARPs (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It is worth noting that we did not count on the occurrence of the terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “useful” (and similar) stated in comments to measure the usefulness of such architecture solution given to an architecture related question in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As explained above, we referred to information in the comments attacked to the solutions to examine SO users’ discussions on the usefulness of architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We mean the reaction of SO users after seeing and using architecture solutions given to their associated architecture related questions, for example, see a comment in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, before such ARPs (solutions) were finally included for analysis, we first read the solutions commented to be helpful and their associated questions to check if they are really useful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', are the solutions/answers useful to address the questions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5, we provide more details about the identified terms related to usefulness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “helpful”) along with other terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “extremely”, “very”) (from 324 ARPs) that SO users use in comments to explicitly signify how useful they found architecture solutions provided to their associated questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' During the data analysis for these two RQs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', the taxonomy of architecture solutions considered useful (RQ5) and their characteristics (RQ6)), the first author followed the same procedures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', coding and grouping similar codes into high-level categories) that were used when analyzing RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One thing to elucidate when analyzing RQ5 to construct a taxonomy is that the first round of grouping yielded seven main categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the second round, the all the authors of this study proceeded to further generate subcategories and types from these seven main categories, ensuring that these main categories, their subcategories, and types follow an “is a” relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The grouping process was iterative, in which the authors continuously went back and forth between categories, subcategories, types, and solutions to refine the taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The final results of this analysis yielded a taxonomy of 7 main categories, 20 subcategories of which 1 were encoded as “Others” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', refer to codes that did not fit into the already generated subcategories), and 85 types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Note that the negotiated agreement approach [45] was used to discuss and resolve any disagreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The final taxonomy as the result of RQ5 is elaborated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Four characteristics were distilled as the results of RQ6 and are detailed in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Note that while categorizing and characterizing ARPs in SO by reading through those posts, we observed that a single ARP may contain multiple types of architecture knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this ARP11 from our dataset, an SO user asked about alternative architecture patterns for Model View Controller (MVC) pattern (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', alternative architecture solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the question body, the user asked the reasons that could drive someone to decide to use those alternative architecture patterns over MVC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture decisions and their rationale), the types of systems that the alternative architecture patterns are typical used for (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', design context), and the pros and cons that come along with using those alternative architecture patterns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', benefits and drawbacks of architecture solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We encoded such a post with multiple types of architecture knowledge accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, while analyzing ARPs in our dataset, we noted down and then discussed the results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', categories of architecture related questions and taxonomy of useful architecture solutions) during the qualitative data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This has led to several interesting findings and actionable implications for various stakeholders, which are presented in Section 4 and Section 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The dataset collected and used in this study and the details of data analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', coding in MAXQDA) are available online for replication and validation purposes [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 11https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2d8r6w8m 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Results In this section, we present the results to our RQs that we got from data analysis (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The result of each RQ is presented in a dedicated subsection, ending with the key findings of the corresponding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Categories of architecture related questions (RQ1) Categories of architecture related questions that SO users ask in SO were determined using the open coding and constant comparison techniques described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We examined questions in the 968 ARPs to answer this RQ (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our data analysis yielded 9 main categories and 21 subcategories of architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Table 5 shows the mentioned categories, their subcategories, their percentages of occurrence (out of 968 ARP questions), and count information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As shown in Table 5, architecture configuration (27%, 261 out of 968 ARP questions), architecture decision (19%, 181 out of 968 ARP questions), and architecture concept (15%, 142 out of 968 ARP questions) are the top three categories of most frequently asked architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the following, we report those categories and subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Where required, we provide an SO question example to support the understanding of the categories and their subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Table 5: Categories of architecture related questions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' their subcategories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and their counts & percentages Category Subcategory Count Architecture configuration (27%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 261) Architecture configuration with technologies support 144 Architecture pattern configuration 117 Architecture decision (19%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 181) Technology decision 104 Behavioral decision 77 Architecture concept (15%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 142) Architecture overview 62 Basic architectural concept 32 Architecture component functionality 26 Specific architecture pattern 22 Architecture implementation (12%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 119) Architecture component implementation 79 Architecture pattern implementation 40 Architecture tool (10%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 99) Architecture modeling tool 34 Model-based code generation tool 30 Usage of architecture tool 21 Code-based model generation tool 14 Architecture evolution (6%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 55) Architecture extension to meet new requirements 42 Component extension to meet new requirements 13 Architecture refactoring (5%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 45) Refactoring of circular dependencies 21 Refactoring of large components 13 Refactoring of big ball of mud 11 Architecture deployment (4%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 34) Application deployment to meet quality attributes 24 Application deployment to meet functional requirements 10 Architecture documentation (3%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 32) 32 (1) Architecture configuration questions in this category ask about how to configure compo- nents and connectors in software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The types of components and connectors could either belong to certain technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Windows Communication Foundation (WCF) and Windows Presentation Foundation (WPF)) or other architectural concepts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This category is the most common (27%, 261 out of 968 ARP questions) category of architecture related questions in SO (see 13 Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We further classified this category into two subcategories, in which architecture configuration with technologies support surpasses half of the questions (144 out of 261 ARP questions of the architecture configuration category) that SO users ask in this category (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture configuration with technologies support is concerned about how to configure an ar- chitecture of an application with specific technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', WPF, WCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question12, a developer asked about how to configure or build a scalable Web-based application by using WCF: “Does anyone have any experience with how well web services build with Microsoft’s WCF will scale to a large number of users?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The level I’m thinking of is in the region of 1000+ client users connecting to a collection of WCF services providing the business logic for our application (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture pattern configuration seeks practical guidance on how to configure a specific architec- ture pattern (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Model View Controller and hexagonal architecture patterns) when designing an application to achieve certain requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', functional requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question13, a developer asked about how to configure an application that conforms to a Hexagonal architecture pattern by stating that: “I’m looking for some guidance or best practices for how to configure and structure an application which conforms to Hexagonal architecture that supports multiple (driver) adapters simultaneously (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Architecture decision: SO users ask this type of questions mostly when they want to decide between two or more alternative architecture solutions when deigning their software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among two subcategories identified in this category, the technology decision subcategory contains the majority of questions (104 out of 181 ARP questions of the architecture decision category) that SO users ask in this category (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Technology decision is mainly concerned about choosing between two or more technology solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', frameworks, databases) to meet certain requirements at the architecture level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, various aspects can be considered during this choice, such as technology features, benefits, and drawbacks [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question14, a developer asked about the reasons that could drive him or her to decide to use Cassandra over HBase for his/her application by stating that: “HBase is known for being a key-value store and random reads with .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='get and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='put functions based on the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Is Cassandra a better choice for suiting a requirement of key-value store?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Can it support random reads based on key?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' If so, in which conditions should I choose Cassandra over HBase in a Spark Streaming application?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Behavioral decision is concerned with deciding how certain elements in a system would interact together to provide some functionality or to satisfy certain quality attributes [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a developer wanted to decide on either to let clients connect directly to the database or let the connection go through the web service by asking this question15: “Recently I have been developing a system to run a high secured database (using vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='net and SQL Server 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I want to increase the security of the database so no connection will be made directly to the database but instead a HttpWebRequest is sent to a web service which then connects to the database and returns the requested data table in XML format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' My concern is just about the performance, I cannot decide either to let clients connect directly to the database or let the connection go through the web service”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) Architectural concept includes theoretical related questions about software architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We divided this category into four subcategories, among which architecture overview contains the majority of questions (62 out of 142 ARP questions of the architecture concept category) that SO users ask in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture overview questions are concerned with the information about the general working mechanism or overview of certain existing architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question16, a developer asked about architectural overview of Drupal version 7: “Could someone provide an architectural overview of the Drupal 7 control flow?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Perhaps in the sense of a flowchart about how a page gets generated (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 12https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/yuxjp2su 13https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/4kn6t27e 14https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/k9xzkman 15https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/4pjh3ufk 16https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2a9hb3ek 14 Basic architectural concept refers to questions that seek explanations about basic concepts in soft- ware architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question17, a developer was seeking explanations about several architecture concepts, such as a architecture pattern: “Is MVC a pattern or architecture or framework?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What is a pattern?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What is an architecture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture component functionality is concerned with the use, purpose, or functionality of certain components in the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question18, a developer was asking about the use or purpose of the lifecycle aware component in Android based application: “We already have a Lifecycle in our Activity/Fragment then why will we use Lifecycle aware component & kindly guide me the main purpose of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' And if we use lifecycle aware then why we use lifecycle that we knew already”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specific architecture pattern questions ask about particular architecture patterns that are commonly used in the design of certain applications to address functional or non-functional requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question19, a developer asked about commonly used architecture patterns for three- dimensional (3D) video game applications: “What are some of the more common design patterns used when developing 3D games?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Are there any high-level architectural design patterns that are commonly used?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (4) Architecture implementation questions ask about how to implement a certain software system according to its architecture design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The architecture design is refined in detailed design, and then implemented in code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture implementation category has two subcategories, among which architecture component implementation occupies the majority of questions (79 out 119 ARP questions of the architecture implementation category) that SO users ask in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture component implementation is concerned with how components should be implemented in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question20: “How to implement a single component sharing in different modules in Angular 7 while using lazy loading?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture pattern implementation questions are about the ways certain architecture patterns are implemented with regard to the fundamental design principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question21: “How to implement MVC in Swift?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I’ve been building Swift apps where basically all the functionality is in the ViewController.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I know this isn’t the optimal way to do it because design patterns help you expand the app but I don’t really understand them (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' How do I go about turning this into a Model-View-Controller design?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (5) Architecture tool: There are various architecture tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Enterprise Architect, Archi, Cloudcraft) that can be used to assist in the architecture design of a software system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With our dataset, we found architecture related questions in which SO users ask about these tools and classified them in the architecture tool category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We further classified this category into four subcategories, among which architecture modeling tool contains the majority of questions (34 out of 99 ARP questions of the architecture tool category) that SO users ask in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture modeling tool questions ask about tools that can enable the creation or drawing of architectural diagrams to model or represent an architecture of a software system during the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question22: “I am a newbie in TOGAF and I need to start a first trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I am trying to model my architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Which tool do you advise me to use in order to model my architecture using TOGAF?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Model-based code generation tool refers to questions that ask about architecture tools that can enable the generation of code from architectural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question23: “Please suggest me any open source tool to generate C# code from UML designer (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' My requirement is to have a code generation tool for C#”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Usage of architecture tool questions look for instructions on how to use certain architecture tools 17https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/5burxuca 18https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/3x35jzu6 19https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2u2df8z5 20https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/kp73y3wk 21https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/mpmvwb5c 22https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/ndf7mrnc 23https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/jwkvtzwc 15 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Archi, Microsoft Visio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question24: “How to add UML/layer diagram to an existing solution in VS 2015 community?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' There is no architecture menu there?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Code-based model generation tool is concerned with tools that assist in architectural models gen- eration or recovery from the codebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question25: “I need to make a UML class diagram for a project (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') I do not really want to write all the classes/functions manually, so I was trying to generate the diagram from the source code but can’t seem to find a way or tool to do it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (6) Architecture evolution: SO users ask this type of architecture related questions when seeking help on how they can re-architect and expand their existing architecture for the purposes of achieving certain new requirements (functional or non-functional requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among two subcategories iden- tified in this category, the architecture extension to meet new requirements subcategory contains the majority of questions (42 out of 55 ARP questions of the architecture evolution category) that SO users ask in this category (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture extension to meet new requirements is concerned with practical guidance for expanding an existing architecture of a system to address certain new functional or non-functional require- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The changes do not only happen in one component, but they may happen in almost the whole architecture of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question26: “I am expanding/converting a legacy Web Forms application into a totally new MVC application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The expansion is both in terms of technology as well as business use case (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The new project has two primary goals: Extensibil- ity (for currently and future pipeline requirements) and Performance (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Is there a way in DDD to achieve both, Extensibility that DDD provides and performance that DBDD provides?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Component extension to meet new requirements includes questions that ask about the extension of certain architectural components to meet some new functional or non-functional requirements in existing and running software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This is different from the above subcategory, as here the change or extension happens in local to a specific component or layer, rather than affecting the whole architecture of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question27: “We are currently evaluating CQRS and Event Sourcing architecture (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What happens if, after an application has been up and running for a while, there is a new requirement to add an additional field to a ViewModel on the ReadModel database?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Say, the Customer Zip Code is required on the CustomerList ViewModel, where it was not previously”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (7) Architecture refactoring: SO users ask this type of architecture related questions when they want to restructure architecture of systems aiming at improving non-functional attributes of those systems without modifying their external behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This category includes three subcategories, where most of the questions (21 out of 45 ARP questions) are related to the subcategory refactoring of circular dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Refactoring of circular dependencies is concerned with techniques and tools that can help remove undesirable circular or cyclic dependency issues among modules so that layering violations can be addressed and dependency structure can be improved in the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question28: “I am working on the MVC project where I am following the layered architecture (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Now, my Business Logic Layer(BLL) is depending on the Data Access Layer (DAL) which is depending on BLL because domain objects are inside BLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' So, both are having reference to each other (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' How can I overcome the circular dependency?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Refactoring of large components is concerned with approaches that can help refactor large compo- nents in software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question29: “I have a pretty large table component and I want to separate its body section into new component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Each time I am trying to do this, the styling of table gets broken (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I would like to have exactly this same page after this refactoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Does anyone know how to pass styling to this new child component, or how to make thing styling work again ?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 24https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/52zffbw3 25https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/n7rz24mu 26https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/ymbyzcvz 27https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/3rh9xmhs 28https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/475dvbp5 29https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/475dvbp5 16 Refactoring of big ball of mud: Big ball of mud occurs when a software system lacks a perceivable, flexible, and appropriate architecture [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This subcategory includes questions that ask about approaches and tools for big ball of mud refactoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question30: “What step would you take to refactor a ball of mud CF app into something modern and maintainable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (8) Architecture deployment collects architecture related questions that ask about how certain software systems should be deployed in the hosting environments to meet requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', functional and non-functional requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' According to our studied dataset, we divided this category into two subcategories, among which application deployment to meet quality attributes contains the majority of questions (24 out of 34 ARP questions of the architecture deployment category) that SO users ask in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Application deployment to meet quality attributes includes architecture related questions that ask about methods and tools that assist in the deployment of applications in the hosting environments to meet quality attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', availability and performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question31, a developer asked how to deploy a microservice based system with zero downtime: “At the moment I’m working on an application which will be based on the Microservice architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As main technologies, we planned to use Spring Boot and Docker for each Micro Service development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One of the goals/requirements is to provide a Zero Downtime Deployment feature for the users (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Any suggestions on the Zero Downtime Deployment process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' If you have any great ideas for a different architecture or maybe you’ve used tools which can help us here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Application deployment to meet functional requirements covers architecture related questions that ask about methods and tools that assist in the deployment of software systems to meet functional requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question32, a developer asked about the method s/he can follow in order to deploy his/her microservices based application in the production environment so that each service of the application can call each other: “I am trying to deploy my microservices architecture to production env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Now I have 15 services, 1 Facade Layer, Facade Layer calls services, gets data, aggregates them, and generates the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, services call each other(rarely but yes, they call each other) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' So I have decided that I will have 5 Boxes (5 high-end servers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' A, B, C, D, E A will be LVS (for Load Balancing) B & C will host the Facade layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' So when the request came for Facade, it will come from A and load balanced to B & C (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' So B & C box will contain each one haproxy instance also since when Facade Layer calls services, it will be load balanced (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' But my question is how should I allow my services to call each other?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (9) Architecture documentation: This is the only category of architecture related questions with no subcategories in our studies dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The architecture documentation category includes ques- tions that ask about methods and tools that assist in the documentation of architecture of software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question33, a developer asked about the best practices and tools for documenting architecture of different types of systems: “What are the best practices and software tools for documenting software design and architecture for PC based applications based on Java or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='NET?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Embedded Applications based on VxWorks or Embedded Linux or Windows CE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Key Findings of RQ1 Finding 1: SO users ask a broad range (9 categories) of architecture related questions, among which architecture configuration (27%, 261 out of 968 ARP questions), architecture decision (19%, 181 out of 968 ARP questions), and architecture concept (15%, 142 out of 968 ARP questions) are the top three categories of most frequently asked architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Categories of design contexts (RQ2) This RQ aims to investigate the categories of design contexts in which architecture related questions were raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, to answer this RQ, we used a predefined classifications of design contexts from [27] and [30]) when analyzing the extracted data for RQ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', design contexts) 30https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/j6rdefeb 31https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/4tyd4yt6 32https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/37dmd6av 33https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/msnbc7xb 17 from the 968 ARP questions (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We found that most (71%, 687 out of 968) of our analyzed ARP questions describe their design contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', the knowledge about the environments in which the systems are expected to operate [27]), and then the responders provided potential solutions with rationale based on the given design issues and design contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, we identified three main categories and eight subcategories of design contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We report the mentioned categories, their subcategories, their percentages of occurrence (out of 687 ARP questions), and count information in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It is also evident from Table 6 that application context is the most common (54%, 377 out of 687 ARP questions) category of design contexts, and organizational context is the least significant category (8%, 56 out of 687 ARP questions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Table 6: Categories of design contexts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' their subcategories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and their counts & percentages Design Context Subcategory Count Application context (55%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 377) Application domain context 313 External service context 64 Platform context (37%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 254) Software context 139 Hardware context 115 Organizational context (8%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 56) Development schedule context 36 Stakeholders context 13 Resources context 7 (1) Application context refers to the software system or product that is to be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It is accessed through a device (platform entity) to deliver services to end-users [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This category includes two subcategories, in which the application domain context subcategory is the most (313 out of 377 ARP questions) common one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Application domain context describes the domain/type of the application that is being developed (such as E-commerce system, banking system, distributed system) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Some SO users like to reveal in their architecture related questions what kind of application domains they are about to design in order to get potential and relevant architecture solutions that fit their application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question34, a developer mentioned that s/he was designing an E- commerce system: “I am designing an E-commerce using microservices architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Suppose that I have two contexts: a product catalog, inventory and pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It’s seems clear to me that they have a clear responsibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' But to serve the show case (the product list) I need to make a request for the product catalog, get a list of ID’s and then use it to query the Inventory micro services to check inventory status (in stock or stock out).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Besides that I need to make a request to Pricing to get the price of each product (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I have been reading about microservices architecture and when you are dealing with many ‘joins’ it’s possible that the these contexts should be a single one (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We can use a domain event to notify ‘search’ microsecond that something has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' So we can resolve show case with a single request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This look like a CQRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Is there a correct approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Which one is better ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Trade-offs?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' External service context refers to specifications of external software services that the application uses [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question35, a developer mentioned that s/he was designing a system that will require to use Azure or Amazon cloud services: “Basically my question is on the application architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Designing for hosting is easy but cloud computing adds new challenges (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I am not certain what I should do in designing an application for safety engineers, so a high uptime is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' So, if my application is written in ASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='NET, using SQL Server, it would seem that my best bet is to design for Azure, but would Amazon’s solution be a good choice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' How would I decide if I should just have everything on the same system or have the data on Amazon’s cloud and the ASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='NET on Azure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I decide on the language, does that lock me into a cloud solution?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Platform context comprises the hardware technology a user employs to access an application, 34https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p93nesu 35https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p8phmr6 18 the software it runs, and the network capabilities of such technology [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In our dataset, we identified two platform contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', software and hardware context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Software context comprises information about the software elements of the device, such as the Op- erating System (OS) or other installed applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This subcategory collects the ARP questions that describe the software elements of the device (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', OS) on which the planned software system will need to run in production [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We found that some SO users provide this kind of information when asking architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question36: “I need to build one mobile application starts with windows phone 7 and then need to convert the application to other platforms like Android, iOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The application contains many screens with data capture and all the data stores it in local storage and finally, it is passed to a central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I would like to know how the architecture needs to be designed (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hardware context comprises the platform entity which defines the device through which the user accesses and uses the application, and can be of different types, such as desktop, laptop computers, and wearable mobile devices [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The hardware context category gathers ARP questions that mention hardware technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', desktop computers) through which the users access and utilize the planned applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question37, a developer mentioned that s/he was developing a desktop application: “We want to start develop an intermediate desktop software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We decided to use the WPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We don’t want to use the MVVM pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Because we are not familiar with MVVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Is it true to develop WPF application without MVVM pattern (using 3 layer architecture but without MVVM) although does it have better performance than win forms yet?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) Organizational context refers to the development schedule (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', time-to-market), the people (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', stakeholders), or the resources that could influence the development of software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This category includes two subcategories, in which the development schedule context subcategory is the most (36 out of 56 ARP questions) common one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Development schedule describes the time put on software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this ques- tion38 a developer mentioned that the development time for a software project was restricted to only three months: “For personal and university research reasons I am thinking of building a simple CRM using a service-oriented architecture (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The architecture that I’m designing defines: - We- bGUI (a client of the other services) - AnalyticsService (a service that receives data, analyzes, and collects it) - CustomerCareService (a service that uses RESTful APIs to apply CRUD operations (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What sort of authentication is more suitable for a client (user token vs OAuth or similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I’ve about 3 months to do it (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Stakeholders context describes the people who are involved in the development of a software system, for example, project managers, owners, architects, developers, users, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this question39 an asker mentioned a number of developers that was involved in the development of 3D Map application by stating that: “I’m trying to develop 3D Map, and I found 3 solutions: Use game engine (like unity) or Use 3D graphic API (OpenGL, etc) or Web app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Is there another way to do it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' And which one of those three solutions (design decision) is better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (with reason) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.) Developers: 3 programmers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Resources context denotes the lack (or availability) of resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', financial or technological competencies) at disposal to develop an application [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, one developer needed to update an application with a tight budget and stated in this question40 that: “I have to build a database/image-rich application that’s only going to increase in size (scalability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I am on a budget, but do have a rather good 3Ghz Xeon server with 400 GB space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Any ideas?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' a good way for an individual on a TIGHT budget”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 36https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p8nd2kw 37https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/yc2eyvhs 38https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2bu5yu65 39https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/yra7d3y7 40https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/4pjp3ntj 19 Key Findings of RQ2 Finding 2: Most of the SO users (71%, 687 out of 968 ARP questions) considered design contexts when asking architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finding 3: Application context is the most common (54%, 377 out of 687) category of design contexts in ARP questions, whereas organizational context is the least significant design context category (8%, 56 out of 687) in ARP questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Characteristics of architecture related questions that have more than one answer (RQ3) Some architecture related questions are continuously getting more attention from SO users by an- swering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This motivated us to investigate why such architecture related questions get more than one answer in SO by characterizing those questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, we used two techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', open coding and constant comparison) from Grounded Theory [17], to examine the characteristics of ARP questions that have more than one answer from 650 ARPs (a subset of 968 ARPs) (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, we referred to the contents of the questions and comments attached to questions to understand what factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', question formulation [35] or certain features in the question (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architectural diagram)) that contribute to such architecture related questions getting more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The outputs of our data analysis generated four common characteristics of architecture related questions that have more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We provide these four common characteristics and their counts & percentages in Table 7, which shows that well-articulated architectural information is the most (46%, 297 out of 650 ARP questions) frequent characteristic while upvoted architecture question comes as the least (8%, 51 out of 650 ARP questions) frequent characteristic of architecture related questions that have more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, we show the numbers of answers of the ARPs that have more than one answer in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (1) Well-articulated architectural information in the question: The main reason an archi- tecture related question would continuously be answered is that its architectural information is well- articulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This question provides an overview of the planned software system and its basic principles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', design contexts and architectural constraints) to help other SO users perceive what the question is really about (the purpose of the question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a comment: “+1 great question, I want to say that this is a beautifully and very well-articulated question!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' )” was posted under this question 41 that illustrates and explains well the encountered architecture concerns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecting a system to meet the scalability and visualization of data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Clear description together with architectural diagrams in the question: Another reason an architecture related question would continuously be answered is that it is easy to read and understand, for example, the question which clearly states necessary information about the components and connectors together with interfaces and relationships to other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, providing diagrams, such as architectural component diagrams that depict and clarify the logical architecture view of software systems, contributes to questions continuously being answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, before one responder started answering this question 42, this responder stated that: “Your question is clearly described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thanks for the little graph you drew to help clarify the overall architecture (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) Alternative architecture solutions to answer the question: Although different architec- ture solutions act as alternative solutions to similar design problems, they differ in terms of their qualities [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, two architecture solutions may both address the interoperability concern but may differ into addressing performance concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, such alternative architecture solutions are of sig- nificant importance as they provide a wide range of possibilities for choosing and making design decisions on candidate architecture solutions for certain design issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, we noticed that questions that ask about choosing between various architecture solutions, such as technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', databases, frameworks, and programming languages), are continuously getting new answers (alternative solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, SO users were interested in choosing a right combination of message formats with message transmission techniques in order to achieve quality attributes, including high performance, availability, scalability, among others, for their Ruby and Java applications interaction, and they posted a question43: 41https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/dfw27h38 42https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/74fr8mwe 43https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/44sey2a2 20 “We have cloud-hosted (RackSpace cloud) Ruby and Java apps that will interact as follows (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We are interested in evaluating both message formatting (such as JSON) as well as message transmission techniques (RPC, REST, SOAP, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our criteria are high performance, availability, scalability (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' What combination of message format and transmission method would you recommend?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Why?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (4) Upvoted architecture question: Usually, in SO, if a question is consistently getting upvote count, its likelihood of being answered or getting more answers increases [50], and architecture related question is no exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In our data analysis, we realized that this factor (upvoted architecture question) also contributes to architecture related questions having more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this architecture related post44, a responder stated in the title of the answer thread when s/he was answering the question: “Since this question got upvoted several times I would like to share what I did in the end (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Table 7: Characteristics of architecture related questions with more answers and their counts & percentages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Characteristic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Count ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Well-articulated architectural information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='297 (46%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Clear description together with architectural diagrams in the question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='174 (27%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Alternative architecture solutions to answer the question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='118 (18%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Upvoted architecture question ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='51 (8%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='199 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='298 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='397 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='496 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='595 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Number of answers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='ARP that has more than one answer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Figure 3: Numbers of answers of the ARPs that have more than one answer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Key Findings of RQ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Finding 4: Well-articulated architectural information is the most (46%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 297 out of 650 ARP questions) frequent characteristic of architecture related questions that have more than one an- swer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finding 5: The presence of architectural diagrams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', components diagrams) in the architec- ture questions increases the chance of these questions to get more than one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Characteristics of architecture related questions that only have one answer (RQ4) We found out that some architecture related questions gain less attention from SO users to be con- tinuously answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Analogous to the two previous RQs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', RQ1 and RQ3), we applied two techniques (open coding and constant comparison) to study the characteristics of questions that only have one answer from 318 ARPs (a subset of 968 ARPs) (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, similar to RQ3, we referred to the contents of the questions and the comments attached to questions to understand 44https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/7bc8764a 21 what factors demotivate the responders to continue answering certain architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We identified five common characteristics of those questions with their counts & percentages in Table 8, which shows that lacking information in the question (39%, 125 out of 318 ARP questions) and poorly structured architecture question (22%, 69 out of 318 ARP questions) are the two major characteristics of architecture related questions that only have one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Below, we elaborate these characteristics in detail with examples from ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (1) Lacking information in the question: Architecture related questions that lack certain significant information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', missing information on components and connectors together with interfaces and relationships to other components) fail to attract community members to provide their answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, one developer pointed out that some information is missing in this architecture related question45: “Your question cannot be answered without doing (many) assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' More information is needed about the module’s dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Are they stateless?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Can you draw a flow of the requests?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Poorly structured architecture question: We found that architecture related questions that are poorly structured (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', not well articulated) fail to clearly reveal their purposes to the community members so that they can provide answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These questions sound unclear, vague, or hard to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, in this question46 a developer asked about how to implement the interactors in Android MVP clean architecture but failed to clearly structure well his/her question, and another developer came and commented: “So what exactly is your question?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The asker came back and edited the question to make it clear so that other community members could understand what the question is really about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) Architecture considered as off-topic: Even though architecture related questions are being asked in SO, SO is mainly designed for programming information seekers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, architecture being not directly for programming related issues but mainly for high-level structure related concerns, this leads to the situation that architecture related questions get less attention from the community and consequently only get one answer or remain unanswered in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a developer asked about the overview of ZeroMQ architecture, and another developer commented under this question47 by saying that: “I’m voting to close this question as off-topic because it’s not really about programming”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (4) Proprietary technology in the question: We found a few questions, asking about propri- etary technologies, such as databases and frameworks that are not widely used, get less attention in SO and consequently get only one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, in this architecture related post48, a community member claimed to be one of the technology founders of RethinkDB in the title of the answer when s/he was answering a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Other community members kept asking more questions about that RethinkDB (a not widely used database) in the comment thread, and those questions have remained unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, “Disclaimer: I’m one of the founders of RethinkDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Sorry for the longish answer (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' RethinkDB is designed with a very flexible architecture (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (5) Duplicate architecture question: Similar to other types of questions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', programming questions) in SO, some architecture related questions get few answers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', one answer) or remain unan- swered because they are duplicate architecture questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Community members do not like to re-answer questions that were answered before [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They would like the askers to review the site (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', to check if their questions have not been posted and answered) before posting such new questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a comment: “duplicate of stackoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/questions/15142386/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' ”, was posted under this architecture related question49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Key Findings of RQ4 Finding 6: Lacking information in the question (39%, 125 out of 318 ARP questions) and poorly structured architecture question (22%, 69 out of 318 ARP questions) are the top two most frequent characteristics of architecture related questions that only have one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 45https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/btukhpzx 46https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/3m6fzjbe 47https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/rmpfarjc 48https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/nrz66x3w 49https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p9382hh 22 Table 8: Characteristics of architecture related questions with few answers and their counts & percentages Characteristic Count Lacking information in the question 125 (39%) Poorly structured architecture question 69 (22%) Architecture considered as off-topic 51 (16%) Proprietary technology in the question 32 (10%) Duplicate architecture question 29 (9%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Taxonomy of architecture solutions that are considered useful (RQ5) This RQ aims to construct a taxonomy of architecture solutions that are considered useful in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, when answering this RQ, we first investigated how SO users discuss the usefulness of architecture solutions attributed to their associated architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We needed to gain an understanding of the ways (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', terms) SO users may use to communicate the usefulness of architecture solutions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Understanding SO users’ discussions on the usefulness of these solutions is important to direct Q&A platform owners in creating the mechanisms that can help SO users to efficiently and effectively search and (re)use such useful architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We found that SO users frequently use two terms related to usefulness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “useful” and “helpful”) along with six other terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “definitely”, “very”, “really”, “super”, “extremely”, and “incredibly”) in the comment threads (see Figure 2) to explicitly express their feedback about how useful they found certain architecture solutions provided to their associated architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Note that SO users may use other ways to communicate the usefulness of architecture solutions in SO, and we cannot claim that we have identified all usefulness terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Secondly, as detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, we thoroughly and comprehensively examined the contents of the solutions from 324 ARPs (a subset of 968 ARPs) with useful knowledge (see Figure 1) to construct the taxonomy of these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This examination yielded a taxonomy of 7 main categories, 20 subcategories of which 1 were encoded as “Others” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', refer to codes that do not fit into the already generated subcategories), and 85 types (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Framework for embedded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='system implementation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Drupal functionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='explanation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Architecture tactic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Architecture pattern ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Explanation of architecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Viewpoint for architecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='documentation (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for architecture documentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Explanation of CMS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='architecture (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Architecture solution for deployment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Tactic for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='availability (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Tactic for security (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Data replication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='tactic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for architecture configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for architecture implementation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='API for iOS application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='implementation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='API for architecture implementation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='37 (11%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='30 (9%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='53 (16%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='59 (18%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='13 (4%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='127 (39%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Taxonomy of architecture solutions that are considered useful in SO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5 (2%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Configuration solution for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='platform (47) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Deployment process ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='with AppHabor for .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='NET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Tactic for performance (21) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for iOS app ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for Linux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='application configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for Android ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='app configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for Window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='application configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Solution for tracking system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Architecture pattern suggestion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='to meet quality attribute (17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Tactic for ' metadata={'source': 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+page_content='application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Library for linear algebra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Pattern for time-critical system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='(Preemptive Multitasking) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='Pattern for distributed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='system (Broker,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SOA) Pattern for Web-based system (MVC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Client-Server,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SOA) Usage of architecture pattern (13) Deployment viewpoint for architecture documentation Figure 4: Taxonomy of architecture solutions that are considered useful 24 (1) Solution for architecture configuration: This is the largest category of architecture solu- tions in our taxonomy (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The solutions in this category provide approaches and tools that enable the configuration of components and connectors of the planned software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among three subcategories identified in this category, the configuration solution for domain subcategory collects more than half of the solutions (75 out of 127 ARP solutions) discussed in this category (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Configuration solution for domain: This subcategory discusses approaches and tools for config- uring applications of various domains, such as solution for distributed system configuration, solution for banking system configuration, solution for E-commerce system configuration, and solution for game system configuration (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Concerning the solution for distributed system configu- ration type, a user asked about how to design and configure a 2/3 tier distributed application in Java with certain components, including centrally shared database and multiple fat clients (Swing based Graphical User Interface clients (GUIs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' S/he needed a simpler approach that could help him or her to configure those clients so that each client can be informed about data changes com- mitted to the database by another client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first solution in this ARP50 is provided based on the application domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', distributed application) described in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The solution suggests to configure the application’s components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', database and clients) by following Java EE dis- tributed container/component-based architecture by stating that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') Java EE is a distributed container/component based architecture for the enterprise tier (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') You c/would design a messag- ing domain with both topic/subscription based and straight up Queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These can be declaratively configured to be durable, or not, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Configuration solution for platform provides approaches and tools that enable the configu- ration of components and connectors of applications with regard to the platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Windows OS) on which these applications will run in production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We identified seven commonly discussed solution types in this category, such as solution for Android app configuration, solution for Win- dows applications configuration, solution for iOS app configuration, and solution for cross-platform configuration (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding the solution for cross-platform configuration type, one SO user needed to design and configure an application that sends data between two iOS devices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', iPad and iPhone) with iPad acting as an iBeacon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first solution in this ARP51 explains how the application’s components including the iPad and iPhone could be configured by using an approach that could support Android as well by stating that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') I was forced into an architecture that would support Android as well, so I switched to BlueTooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The iPad acting as an iBeacon also has BlueTooth code that is looking for ‘peripherals’ with a certain signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Once the iPhone detects the iBeacon, the app then starts transmitting a BlueTooth peripheral signal with the appropriate signature (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Solution for architecture implementation: The ARP solutions in this category provide technology solutions, such as frameworks and libraries (see Figure 4), for implementing diverse architec- ture designs to address the system requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', quality attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' According to our dataset, we classified these technology solutions into three subcategories, among which framework for architecture implementation contains the majority of solutions (21 out of 59 ARP solutions) that SO users discuss in this category (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Framework for architecture implementation: These solutions gather different types of frameworks for implementing architecture design, for example, framework for Web-based application (such as Laravel, Django, Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='js, and Play frameworks), framework for iOS application (such as SwiftUI, Flutter, and React Native frameworks), and framework for Windows applications (such as WinForm, WPF, and UWP frameworks) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding the framework for Web-based application type, a user asked about (among other things) a framework that could facilitate the implementation of REST APIs in a Web-based application which will serve the content to mobile apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The third answer in this ARP52 suggests the Play framework as the solution to that question by stating that: “Use Play!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' to do it all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Writing REST services in Play is very very easy (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' API for architecture implementation accumulates different types of APIs as solutions to questions that ask about APIs for implementing architecture design, for instance, API for video game system 50https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/jfnuke2w 51https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/ytw52k6p 52https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p8pm9rs 25 (such as Pokeapi, Chicken Coop, Dota2, and Minecraft APIs), API for Web-based application (such as REST, SOAP, RPC, and Geolocation APIs), and API for facial recognition application (such as Lambda labs and Microsoft Computer Vision APIs) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Concerning the API for Web-based application type, one developer needed an API to implement a request-response Web application in Service Orientated Architecture (SOA) in order to meet certain requirements (including high performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The second answer in this ARP53 suggests to use REST API over SOAP API by noting that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') consider also using REST API, it demands less overhead than SOAP, and you can use JSON as document format which is also more compact than XML, lowering network throughput requirements (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') SOAP has more fancy features that are not well supported in all languages, if you use REST you will be more safe here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Library for architecture implementation: These solutions recommend various libraries for imple- menting architecture, for example, library for linear algebra application (such as JBLSA, MTJ, OjAlgo, and EJML), library for computer vision application (such as OpenCV library), library for vector graphics application (such as DISLIN library), and library for Web-based application (such as HPPC, Trove, and FastUtil) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding the library for vector graphic application type, the first answer in this ARP54 suggests Raphaël library as the solutions to the question that asks about vector graphics application by saying that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') I chose RaphaëlJS and I have to say it has been an absolute pleasure to use, and the help is fantastic too (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) Explanation of architecture: The ARP solutions in this category provide theoretical expla- nations, purposes, or functionalities of architecture instead of providing concrete instructions on how to do something (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', how to configure certain architectural components in the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Explanation of architecture category consists of four subcategories, among which explanation of cloud-based architecture is the most (28 out 53 ARP solutions) discussed subcategory (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Explanation of cloud-based architecture provides theoretical explanations, differences, and func- tionalities of the architecture of cloud computing services, such as Azure App service functionality explanation, EC2 functionality explanation, and AppHarbor functionality explanation (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, a user asked about the difference between Azure App Service and the AAzure Service Fabric in terms of functionalities in software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The fourth answer in this ARP55 provides a detailed explanation about the difference between those two Azure platforms in terms of functionalities in software development as the solution to that question by stating that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') They’re two separate platforms, following different development paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The App Service will give you functionality that Service Fabric doesn’t provide out of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Stuff like auto-scale, authentication, rate limiting, integration with SaaS applications, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Explanation of CMS architecture describes the functionalities of the architecture of Content Man- agement Systems (CMS), such as Drupal functionality explanation and TYPO 3 functionality ex- planation (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, the first answer in this ARP56 provides the architecture overview of Drupal (together with an architectural diagram) as the solution to the question that asks about the functionality of Drupal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', control flow and how a page gets generated) by saying that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') Although it’s procedural PHP, it’s purely event/listener driven in its architecture, and there’s no simple ‘flow’ in the main PHP script for you to look though (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') Drupal’s index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='php file functions as a front-side controller (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Explanation of database architecture groups the ARP solutions that explain or describe the ar- chitecture of datab”ase systems, for instance, NoSQL database functionality explanation (such as Apache Cassandra) and Graph database functionality explanation (such as Nebula Graph) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, the first answer in this ARP57 provides an explanation about Cassan- dra in terms of data replication to deal with data failure scenario as the solution to the question that asks about the way Cassandra handles such a scenario if one node goes down containing the record (data) a user is querying by stating that: “Cassandra clusters do replicate data across the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The specific number of replicas is configurable, but generally production clusters will use a replication factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This means that a given row will be stored on three different machines in 53https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/pakzw2yk 54https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/259h4f6e 55https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p8hnkmm 56https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2a9hb3ek 57https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/wpvw6j5h 26 the cluster (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') In terms of servicing requests, if a node receives a request for data that it does not have it will forward that request to the nodes that do own the data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Explanation of Web server architecture provides the functionalities or difference between the ar- chitecture of Web servers, for example, Web server functionality explanation (such as XAMPP, IIS, and WAMP servers) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, the first answer in this ARP58 provides detailed difference between XAMPP, WAMP, and IIS servers as the solution to the question that asks about the difference between those three types of Web severs by expressing that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') Their (XAMPP and WampServer) differences are in the format/structure of the package, the configura- tions, and (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') IIS is a web-server application just like Apache is, except it’s made by Microsoft and is Windows only (Apache runs on both Windows and Linux) (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (4) Architecture tactic: This category of ARP solutions provide and explain architecture tactics that enable the realization of specific quality attribute (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', performance and security) of software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Four subcategories of architecture tactics were identified in this category, among which tactic for performance is the most (21 out of 37 ARP solutions) discussed subcategory (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tactic for performance provides and explains architecture tactics that assist in the realization of the system performance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We identified four architecture tactics for performance, such as scheduling resources tactic and in memory caching tactic (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding in memory caching tactic, a developer wanted to choose a suitable design technique between two data handling design techniques (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', working directly with a database or working with objects and letting the ORM handle the storage) in order to boost the performance of the inventory system that should handle thousands of item types and quantities of each item stored in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' According to the scenario elaborated in the question, the first answer in this ARP59 suggests to apply in-memory caching architecture tactic with ORM to have the system performance boosted by saying that “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') most of the time it is easier to do an SQL query, but an in-memory cache can really BOOST performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Yes, it uses memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Who cares?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Workstations can have 64GB memory these days (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tactic for maintainability covers architecture tactics that enable the maintainability requirements of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We identified three maintainability tactics, such as less coupling tactic and encapsula- tion through API introduction tactic (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Concerning the less coupling tactic, a developer needed to build a scalable, maintainable, and low-latency single sign-on for all web applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first answer in this ARP60 suggests to apply less coupling tactic when designing the applica- tions in order to make them maintainable by saying that: “I would not integrate the authentication on the database level (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') This might become hard to maintain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' I would prefer a loosely coupled approach by exposing a simple service on your central server that lets the other app servers run authentication requests (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tactic for security provides architecture tactics that help the realization of the system security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This subcategory includes three architecture tactics, authentication tactic, limiting access tactic, and communication link encryption tactic (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding authentication tactic, the first answer in this ARP61 provides and explains authentication tactic to a question that asked for how to set up two level authentication approaches of the ‘user JWT’ in microservice based application by stating that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') You can achieve the two levels of security you require by using a single user token and claims based authorisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' If a call is made to the gateway with the user token, the gateway authenticates the call based on the user token, retrieves the ‘userId’ claim (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tactic for availability collects architecture tactics that enable the system availability requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We collected three availability tactics, like data replication tactic and functional redundancy tactic (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Concerning data replication tactic, a developer asked how to achieve the availability requirement for an application that needs to use two Amazon EC2 instances each with Cassandra database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first answer in this ARP62 provides and explains the replication mechanism that 58https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/ysacs8zd 59https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/289ffurv 60https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/22ckrdv4 61https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/bdhk4uud 62https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p8db7mu 27 could be applied in his/her application (according to the design scenario described in the question) by saying that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') In your scenario (since you are in a single DC) you can use SimpleStrategy for your replication strategy and a Replication Factor (RF) of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With this setup, you will have all data replicated on both nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This will make the data available from either node with a covet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (5) Architecture pattern: The ARP solutions in this category provide architecture patterns for addressing multiple system quality attributes, and also provide commonly used architecture patterns in certain application domains (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among the two subcategories identified in this category, the architecture pattern suggestion to meet quality attribute subcategory contains the majority of solutions (17 out of 30 ARP solutions of the architecture pattern category) that SO users discuss in this category (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture pattern suggestion to meet quality attributes collects architecture patterns for address- ing system quality attributes, such as patterns for modifiability, reusability, and portability (Broker, MVC, and SOA) (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a SO user asked about the best C# architecture patterns enabling the communication between separate plugins of a multi-tenant website wherein modifiability, reusability, and flexibility are the major concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first answer in this ARP63 recommends SOA pattern as a solution to that question by noting that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') I might suggest Service Oriented Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Mostly because it can bend to a business in a very quick and agile manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This architecture provides many bonuses: Lightweight, Agile, Code Re-usability (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Usage of architecture pattern gathers architecture patterns for questions that ask about the com- monly used architecture patterns in certain application domains (see Figure 4), such as pattern for time-critical system (Preemptive Multitasking), pattern for Android application (MVP, MVVM, MVC, Observer), and pattern for distributed system (SOA, Broker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a user asked about architecture pattern for time-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first answer in this ARP64 recom- mends Preemptive Multitasking pattern to that question by saying that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') This pattern is called preemptive RTOS, which is capable of handling the events immediately (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (6) Architecture solution for deployment collects the ARP solutions that discuss the deploy- ment of architecture of systems in the hosting devices (either on the Cloud or the local server) in order to address the systems’ requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This category consists of two subcategories, among which process for deployment is the most (9 out of 13 ARP solutions) discussed subcategory (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Process for architecture deployment collects the ARP solutions that discuss the processes for de- ploying the architecture of applications for the purpose of achieving the applications’ requirements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', functional or nun-functional requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We identified several processes for architecture deployment, for example, deployment process with Azure App service, deployment process with Ku- bernetes, and deployment process with AppHabor for .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='NET applications (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding deployment process with Kubernetes, a responder provided and explained the deployment process with Kubernetes to an asker who wanted to deploy a microservices architecture (which was built up with 15 Spring Boot microservices) on five Kubernetes nodes with one cluster master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' According to the scenario described in the question, the first answer in this ARP65 suggested to use three cluster masters at a minimum instead of one cluster master in order to avoid the data loss and consequently address the system’s availability requirement by saying that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') one master is not enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The loss of that VM, the underlying hardware, or a failure of the services on the master will lead to an outage for all customers and potentially catastrophic data loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Run 3 masters at minimum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tool for architecture deployment collects the tools for deploying architecture of systems in order to achieve the requirements of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We collected several tools, such as simulator tool for architecture deployment and tool for continuous deployment (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regarding the tool for continuous deployment, the first answer in this ARP66 recommends Argo CD tool as the solution to the question that asks about a tool for microservices architecture continuous deployment on Kubernetes by stating that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') ArgoCD workflow provides that functionality (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 63https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/42m8ts56 64https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/25y4b9e6 65https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/483en2ts 66https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/yc7j8z5j 28 (7) Solution for architecture documentation: The ARP solutions in this category provide the approaches and tools that enable the documentation of architecture (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This category consists of two subcategories, among which tool for architecture documentation is the most (3 out 5 ARP solutions) discussed subcategory (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tool for architecture documentation suggests the tools that can facilitate the documentation of architecture, such as tool for Web application architecture documentation and tool for database architecture documentation (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Concerning the tool for Web application architecture documentation, the first answer in this ARP67 suggests NJsonSchema tool as the solution to the question that asks about a tool for documenting a microservices-based application by saying that: “(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') there is NJsonSchema tool https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/NJsonSchema/NJsonSchema”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Viewpoint for architecture documentation provides the viewpoints for architecture documentation, such as development viewpoint for architecture documentation, and deployment viewpoint for ar- chitecture documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, the first answer in this ARP68 provides two viewpoints for architecture documentation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', development viewpoint for architecture documentation and deployment viewpoint for architecture documentation) for documenting an architecture that is im- plemented with Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Key Findings of RQ5 Finding 7: SO users frequently use two terms related to usefulness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “useful” and “helpful”), along with six other terms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “definitely”, “very”, “really”, “super”, “extremely”, and “incred- ibly”) in the comment threads to explicitly express their feedback about how useful they found certain architecture solutions provided to their associated architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finding 8: We derived a taxonomy of useful architecture solutions consisting of 7 categories, 20 subcategories, and 85 types, indicating the diversity of useful architecture solutions provided in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finding 9: Solution for architecture configuration (39%, 127 out 324 ARP solutions), solution for architecture implementation (18%, 59 out 324 ARP solutions), explanation of architecture (16%, 53 out 324 ARP solutions), and architecture tactic (11%, 37 out 324 ARP solutions) are the top four most frequently discussed categories of useful architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Characteristics of useful architecture solutions (RQ6) As shown in Figure 2, SO users occasionally leave comments under an architecture solution to convey that such solution is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hence, this motivated us to study the characteristics of the architecture solutions that are considered to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Analogous to RQ5, we used the 324 ARPs (a subset of 968 ARPs) with useful knowledge (see Figure 1) to analyze the architecture solutions and their attached comments, and study the characteristics of those solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The qualitative data analysis (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2) identified four common characteristics of architecture solutions that are considered useful by SO users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Figure 5 depicts these characteristics along with their counts, in which complete and comprehensive architecture solution appears to be the most (34%, 111 out of 324 ARP solutions) frequent characteristic of architecture solutions that SO users consider to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (1) Complete and comprehensive architecture solution: A developer may ask more than one question (sub-questions) in one single architecture related question in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' A solution that addresses all sub-questions asked in the question and provides comprehensive responses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', providing rationale, such as benefits and drawbacks of the provided architecture solution) to these sub-questions is considered useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, a developer posted this comment: “+1 for the most complete, comprehensive useful response I’ve ever seen (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=')” under the first answer in this ARP69 that comprehensively addresses all sub-questions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', difficult to visualize data in the system architecture implemented with Java and Python) asked in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) Concise explanation with architectural diagrams provides a brief explanation about the key elements of the architecture solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Some examples of these key elements could be the best 67https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/4zm82snw 68https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p8ezw7j 69https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/dfw27h38 29 architecture patterns, tactics, and technologies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', databases) to be used in order to address the design concerns described in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, providing architectural diagrams, such as component diagrams to represent and summarize the practical applicability of the solution also contributes to the architecture solution being considered useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, one developer asked whether “command handler” and “command bus” should belong to or be implemented in the application layer or domain layer in the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' At first, in the first answer of this ARP70, a responder provided a concise and relevant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' But the asker was not satisfied with this solution and then s/he commented to request a sequence diagram (which was provided later) to be associated with the solution for it to be useful: “Thanks, David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' It would be really useful if you could share a sequence diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Appreciate it”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) Detailed architecture solution: These solutions provide and fully describe all necessary architectural elements (such as patterns, components) and other various aspects to be considered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', solution trade-offs, constraints, and alternatives) when addressing the design concerns stated in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For instance, a developer posted this comment: “Thank you for your detailed answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This is certainly very helpful” under the second answer in this ARP71 that lists and details all necessary architectural elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', quality attributes) and other aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', pros and cons of the solution, and alternative solutions) that should be considered when addressing the design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', integrating external modules (external Web applications) into Drupal or vice versa) stated in the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (4) Summarization of external but relevant content: Answer seekers do not like to have external links (URLs) only as solutions posted to their questions since the links may die and the solutions become not accessible and useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' During our data analysis, we observed that answer seekers prefer to have the relevant content summary of URLs instead of the URLs only for the architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, the first answer in this ARP72 summarizes the content from three URLs to answer the question which mainly asks about the design approach to follow in order to address system availability with Cassandra database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' A developer commented under the answer: “Thank you very much for your information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Your Explanation is sufficient and the links you mentioned are very useful”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 34% 87,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 27% 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 25% 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 14% Complete and comprehensive architecture solution Concise explanation with architectural diagrams Detailed architecture solution Summarization of external but relevant content Figure 5: The common identified characteristics of architecture solutions that are considered useful Key Findings of RQ6 Finding 10: Complete and comprehensive architecture solution is the most (34%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 111 out of 324 ARP solutions) frequent characteristic of architecture solutions that SO users consider to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finding 11: The presence of architectural diagrams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', components diagrams) in the provided architecture solutions increases the chance of these solutions to be considered useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Discussion In this section, we revisit the findings of this study by interpreting the results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1 and discussing their implications for various stakeholders in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 70https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/yh292pn8 71https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/pfezm7nn 72https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p87vu99 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Analysis of the results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The delta between our results and the results from prior work Similar to our study, several studies have analyzed ARPs from SO to mine architectural knowledge discussed by SO users in order to support architecting activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this section, we discuss the relation- ship and difference between our study results and the results in the prior studies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', the three studies by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [6][51][10]), which are closely related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [6] identified and analyzed ARPs from SO that discuss architecture knowledge with a focus on technology decisions (one type of architecture decision [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They classified these ARPs based on two dimensions: the purpose of the question and the solution type of the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They further classified the purpose dimension into three subtypes: solution synthesis, solution evaluation, and multi-purposes, and the solution type dimension into three subtypes: technology feature, technology bundle, and architecture configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In total, their analysis generated 6 types of ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our analysis generated 9 categories and 21 subcategories of ARP questions (see the results of RQ1 in Table 5), such as architecture configuration, architecture decision, architecture concept, architectural implementation, architecture evolution, and architecture refactoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Some of the types of ARPs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', solution synthesis, solution evaluation, architecture configuration) found by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' in [6] are aligned with some of our ARP types, and most of the types of ARPs presented in [6] can be subcategories of the main categories reported in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, we have a main category encoded architecture decision, and this category can cover three types of APRs (solution synthesis, solution evaluation, and multi- purposes) reported in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, our analysis generated new categories of ARPs (such as architecture concept, architecture tool, architecture evolution, architecture refactoring, architecture deployment, and architecture documentation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [51] used the same sample of ARPs that were used in their previous work (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', [6]) and developed an ontology that covers architectural knowledge concepts in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The ontology consists of three main ontology classes: simple ontology class, composite ontology class, and lexical trigger ontology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' A simple ontology class is composed of subclasses, for example, technology solution, architecture pattern, quality attribute, architecture component, and architecture connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The composite ontology class consists of several subclasses, such as architecture configuration, technology feature, technology benefits and drawbacks, technology user-case, user request, and design rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The lexical trigger ontology class has subclasses, such as difficulty adjectives, advise verbs, value adjectives, wish verbs, support verbs, versus prepositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Some subclasses found by the analysis in [51], such as architecture configuration and architecture pattern, are aligned with our results of RQ5 (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, the analysis in [51] is based on a sample of ARPs that mainly discuss technology information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', requirements and constraints on technology solutions, technology benefits and drawbacks, and technology features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our analysis complements the work in [51] by adding several new categories, such as architecture tactic, explanation of architecture, solution for architecture documentation, and solution for architecture deployment, leading to more comprehensive categories and subcategories of ARP solutions provided in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [10] developed a search approach that relies on the classification approach to provide suitable types of ARPs for each design step proposed by Kazman and Cervantes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The analysis conducted by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [10] is also based on the sample of ARPs from their previous work [6], and some other posts extracted from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, their search approach classifies SO posts into four types: technology identification, technology evaluation, features and configuration, and programming posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The first three types of posts are ARP types that were reported in their previous study (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', [6]), and in the first paragraph of this section, we have already described the difference and similarities between these types of ARPs in [6] and the types of ARPs in our study (see results of RQ1 in Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition to the abovementioned difference between our study results and the results reported in [6][51][10], in our study, we investigated a new set of research questions (RQ2, RQ3, RQ4, RQ5, and RQ6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We explored other types of architecture knowledge, such as design contexts (RQ2) discussed in architecture related questions, characteristics of ARPs (questions and solutions) (RQ3, RQ4, and RQ6), and the usefulness of the ARP solutions (RQ5), which was not the concern of the abovementioned studies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', [6][51][10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, our analysis covered the entire post, including the question and its associated comments (RQ3, RQ4), the answers to the question and their associated comments (RQ5, RQ6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The analysis in the abovementioned studies by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' only focused on questions and answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, our study results add new information to the state of the art, and practitioners and researchers can benefit from our study results and findings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', the taxonomy of architecture solutions that are considered useful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Identified categories of ARPs in SO could support architecting activities The significant results of this study are categories of ARPs (questions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', RQ1) and solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', RQ5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This study reveals that SO users ask a broad range (nine categories) of architecture related questions, such as questions about architecture configuration, architecture decision, architecture concept, architecture implementation, and architecture tool (see Table 5 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, we classified the architecture solutions that are considered useful into seven categories, such as solution for architecture configuration, solution for architecture implementation, explanation of architecture, and architecture tactic (see Figure 4 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One observation is that our identified categories of these ARPs (questions and solutions) cover almost all the architecting activities that span from the initial stages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architectural analysis, synthesis, and evaluation [18]) of architecture creation to the later stages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architectural implementation, and maintenance and evolution [19]) in a system lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus the identified categories of architecture related questions and solutions can support the mentioned architecting activities during the architecture lifecycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These results also support the findings by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' in [6] that SO should be considered as one of the important sources of architectural knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, practitioners reported Q&A sites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Stack Overflow) as the most useful when searching architectural information according to our recent industrial survey [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, practitioners could rely well on SO to identify, such as, the benefits and drawbacks of architecture solutions in certain application domains, for example, the benefits and drawbacks about the framework for iOS applications in our taxonomy (see Figure 4) for architecture implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Importance of design context in architecture design The results of RQ2 reveal that in most (71%, 687 out of 968) of the studied ARP questions, SO users considered the design contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', knowledge about the environments in which the systems are expected to operate [27]) when describing the design concerns in their architecture related questions (Finding 2 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One reason could be that SO users prefer to provide a brief description of their project backgrounds and then expect responders to suggest potential architecture solutions with their rationale based on the given design concerns and design contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, the results of RQ2 show that most of the SO users do consider design context as one of the indispensable ingredients that can drive the architecture design of a system [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Identified characteristics of ARPs to improve their quality From Table 7, Table 8, and Figure 5, we found that there are various characteristics of ARPs (ques- tions and answers) in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, we observed that architecture related questions that articulate well architectural information are likely to get more than one answer (see Table 7), while architecture related questions that lack certain significant information and poorly structured (see Table 8) tend to only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The reason is the following: well-articulated architecture questions provide an overview of the planned system and describe well their design concerns, which helps potential responders to fully understand the purposes of these questions so that they can provide answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We also found that answer seekers highly appreciated architecture solutions that are complete and comprehensive and considered them to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One reason is that these architecture solutions address design concerns raised in all sub-questions (in the case that there are sub-questions in one question) by providing com- prehensive solutions, for example, design contexts, pros and cons of the provided architecture solutions, which helps the answer seekers understand why such architecture solutions are the way they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The identified characteristics of ARPs (questions and answers) in SO show that SO users have varying needs in the description of ARPs (questions and answers) and the level of details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These findings could assist in improving the quality of the posted architecture related questions and answers at SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Implications 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For Stack Overflow Increase the awareness of SO towards its users: SO introduces itself as a community Q&A platform for asking and collecting programming related knowledge during software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, the majority of the SO users use the platform as the place for sharing and learning coding related knowl- edge only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, ever since this site started growing and being popular, architects have begun to share their competencies, experience, and architecture problems by asking architecture related questions or providing architecture solutions, such as architecture patterns [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Akin to searching and (re)using existing code examples provided in SO to solve programming related problems, SO users also search and (re)use existing architecture solutions, such as architecture tactics [5] in SO for solving their architec- ture design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture design to meet quality attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hence, SO not only curates 32 programming related knowledge, but also accumulates architecture solutions provided to a wide range of architecture problems or design problems [6, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, during our study, we found that architecture related questions were being seen as off-topics in SO and should not be asked at the site (see Table 8 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='4) due to SO users’ perception or awareness of what SO is used for (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a site for programming related issues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Given this situation, there is a possibility that interesting architecture related questions asked might remain unanswered or even be deleted by the site moderators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Although some SO users see architecture related questions as off-topic, we think that architecture related questions will sustain and continue to thrive in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' According to our studied dataset (318 architecture related questions) rel- evant for answering RQ4 (characteristics of architecture related questions that only have one answer), we observed that architecture related questions that were commented to be off-topic are not many (16%, 51 out 318 architecture related questions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This finding is promising for the long-term prospect of architecture related questions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, we argue that architecture related questions which communicate architectural knowledge [23] are an important type of questions and have a system-wide impact on software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Many architecture related questions arise during development when addressing specific design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', quality attributes) and their trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, architecture related discussions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', through architecture related questions) should not be seen as off-topic in SO, and SO should consider increasing its awareness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', to be a site for development related issues instead of a site for programming related issues only) towards its users and welcome architecture questions to be discussed on the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Adjust the current answers and comments organization mechanisms to improve the search and (re)use of architecture solutions: SO attracts a large number of users with different backgrounds, skills, expertise, and viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus during our data analysis, we have observed that an architecture related question like any other questions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', programming related question) in SO may receive multiple (or alternative) answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The study by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [53] reported that nearly 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5 million questions (37% of all questions at SO) had more than one answer, and the average length of an answer is 789 characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With the current SO answers organization mechanism, when there are multiple answers to a question in a single post, at most one answer per question can be accepted/marked by the asker to indicate that the answer is the most useful one [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This asker should be a registered user with at least 15 reputation on SO [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The registered users without required reputation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', less than 15 reputation) on the site are restricted from accepting or voting (upvoting or downvoting) answers to indicate that such answers are useful [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Consequently, leaving a large number of answers in SO that are not accepted or marked as useful answers yet being useful, just because the users (askers) do not possess the required minimum reputation to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' During our data analysis, we observed that not all useful architecture solutions are explicitly marked (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', accepted as useful) in SO to facilitate the search and (re)use of those solutions (for example, see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, we found that SO users may use terms related to usefulness, such as “useful” and “helpful”, in the comment threads to explicitly express their feedback about how useful they found certain architecture solutions provided to their associated questions in SO (Finding 7 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our finding is in line with the findings by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' in [42] and [41] that comments provide additional information to support the answers, such as improvement of answers [42] and obsoleted answers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Prior studies criticized the comment organization mechanism at SO (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In order to keep each answer thread compact, SO implements a comment organization mechanism to only show the top 5 comments [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Aiming at showing the most informative comments and hiding less informative ones, the mechanism first ranks these comments based on their scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' When multiple comments have the same score, they are then ranked by their creation time [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hidden comments are not indexed by Google73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, due to this current comment ranking mechanism, informative comments might be hidden in turn reducing the chances of someone retrieving or voting on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Regardless of its success and popularity, navigating SO remains a challenge, and it is insufficient how SO directs its users to retrieve informative comments [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Comments that state the usefulness of answers (including architecture solutions) are one of the most important informative comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, we provide SO the following suggestion: Instead of simply ranking comments by their score then their creation time [55], the comments organization mechanism needs to introduce a higher priority for more informative comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO may consider adjusting its comments organization mechanism by, for instance, developing special analytical techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', machine learning approaches) that could filter and rank comments 73https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/2p87yyfr 33 stating, for example, improvement of answers [42], usefulness of answers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', useful architecture solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO may also refer to and extend our proposed taxonomy of useful architecture solutions (solutions commented to be useful) to develop an automated tool that could assist the SO users in identifying existing architecture solutions with, for example, useful knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For SO users Throughout the qualitative analysis of RQ3, RQ4, and RQ5, we identified various characteristics of ARPs (questions and solutions) in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among these characteristics, we found that questions that provide clear description together with architectural diagrams increase their likelihood of getting more than one answer (see Table 7), while poorly structured architecture questions (see Table 8) tend to only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, we found that architecture solutions that provide concise explanation with architectural diagram is the second most common characteristic of architecture solutions that are considered useful (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One observation is that SO users would like to see architectural diagrams, such as components diagrams, in both questions and solutions as these diagrams can benefit both parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Concerning questions, providing architectural diagrams increases their chance of getting more responses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', more than one answer) (Finding 5 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' On the other hand, architectural diagrams in solutions boost their chances of being considered useful (Finding 11 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, both askers and responders should better provide diagrams in their ARPs (questions and answers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One reason is that architecture is at a high abstraction level, and it would be hard to describe an architecture problem and much harder to explain an architecture solution with text only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architectural diagrams make architecture to be more understandable [57], and stakeholders can communicate about architectural problems and solutions more easily using architectural diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, various identified characteristics of ARPs in this study (see Table 7, Table 8, and Figure 5) are indicators that SO users have varying needs in the formation of both architecture related questions and architecture solutions and the level of details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, there is a need to provide guidelines to SO users to follow when posting their architecture related questions and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For SO askers: In the following, we provide the guidelines for SO askers to follow when posting their architectural related questions with more likelihood of being answered by other SO users and get more than one answer from SO users: Include architectural diagrams with clear description in the questions: We recommend askers to add architectural diagrams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', component diagrams) and specifically clarify the design concerns in their architecture related questions to help other SO users better understand the purposes of their questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Write well-articulated architecture questions with descriptive details about the context: We suggest that askers could describe well architectural information in their questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This can be done, for example, by providing an overall understanding of the system, as well as detailed information on components in their scope together with interfaces and relationships to other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, we recommend askers to add information about the design contexts, since design contexts are critical for other SO users to correctly understand your architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For SO responders: As stated throughout this study, we not only analyzed architecture related questions, but also examined the characteristics of architecture solutions that are considered useful by SO users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, in the following, we provide guidelines to SO users to follow when posting their solutions with more likelihood of being considered useful by other SO users: Write concise architecture solutions with architectural diagrams: Responders are recommended to write concise architecture solutions by stating key points only in the solutions and add architectural diagrams (if necessary) that depict and clarify, for example, the architecture implementation view in their posted solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Include URLs in architecture solutions with sufficient and relevant architectural knowledge: Answer seekers do not like to have external links (URLs) only as solutions posted to their questions [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the case when a responder wants to make the architecture solution short, s/he can provide links to external websites that contain more explanations or complex examples, and his/her solution should be self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In other words, this solution should provide certain important and relevant architectural knowledge which can make it explainable, such as design decisions and their rationale, 34 contexts, assumption, and other factors that together determine why a particular solution is the way it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For researchers Towards innovative tools to search and (re)use architectural knowledge in SO: The results of our study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', categories of architecture related questions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1 and their useful solutions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='5) provide insights into the nature of SO users’ discussions on architecture design in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, the results of this study re-emphasize the conclusion by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [51] that SO should be considered as one of the important sources of architectural knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, SO captures large amounts of information in its posts and this information is mainly represented as unstructured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Furthermore, the abstract nature of architectural concepts makes it difficult for keyword-based searches to find architecture relevant information, and this might not be easy for SO users to capture and (re)use the architectural knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', benefits, drawbacks, and trade-offs of using specific architecture patterns in certain application domains) from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, researchers can contribute to improving the search and (re)use of the architectural knowledge in SO by focusing on innovative techniques and tools that could efficiently and effectively guide the capturing and usage of this knowledge to support architecting activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architectural analysis and synthesis [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' For example, researchers can refer to our proposed taxonomy of useful architecture solutions in SO as a guidance to develop automated approaches and tools that could mine and locate architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', solution for architecture configuration, the most common category of useful architecture solutions in SO, see Figure 4) for addressing similar design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', questions that ask about architecture configuration, see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This could help SO users to check the questions and solutions that are relevant to their design concerns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', banking system configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Furthermore, we observed that architecture configuration (27%), architecture decision (19%), and architecture concept (15%) are the top three categories of most frequently asked architecture related questions (Finding 1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1), and researchers may explore the challenges (that are being faced by SO users) related to these most frequently asked categories of architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Investigation of design contexts in Q&A sites to support architecture knowledge man- agement: From Table 6 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, we found that SO users discuss about design contexts along with design concerns when asking architecture related questions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Three categories (application, plat- form, and organization contexts) and eight subcategories (application domain, external service, software, hardware, development schedule, stakeholders, and resources contexts) of design contexts were identified from our studied sample of ARPs (see Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Whilst we know that SO users discuss design contexts along with design concerns when asking architecture related questions, there have been very few studies on mining design contexts in the Q&A community sites, such as SO, to support architecture knowledge management [59], which is an interesting area to be explored in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Threats to validity In this section, we discuss the threats to the validity of the study results by following the guidelines proposed by Wohlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [60] and how these threats were mitigated in our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Internal validity concerns with the selection of search terms used to mine ARPs in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We used search terms, such as “architecture” and “architectural”, to identify the related posts in SO (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2), and this is a threat to the internal validity in our study because we might have missed other terms, such as “design”, that SO users use to express architecture concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hence, the search terms we used in this study may not be able to identify the complete set of ARPs in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To reduce this threat, we first conducted a pilot search and observed that SO users use the term “design” mostly in the programming context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “singleton design pattern”74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, as mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2, using the search terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “architecture” and “architectural”) to only search exclusively through tags can be ineffective, because tags can be sometimes less informative [36] (see the example provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, we decided to add the titles and bodies of the questions into the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this way, we sought to minimize the risk of missing ARPs that use incorrect or irrelevant tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finally, we gathered 10,423 ARPs through the search which is quite a large dataset, and it may not be realistic to thoroughly analyze this size of dataset with human effort in order to get accurate and comprehensive results from 74https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='com/8yks7nhm 35 this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Hence, we computed a statistically representative sample [16] of these 10,423 ARPs and randomly selected 968 ARPs as the dataset to be analyzed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, to further mitigate this threat, we downloaded and utilized the current SO data dump (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Stack Exchange data dump on October 5, 202275).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This data dump is a snapshot of the underlying database used by SO and it stores all the information for the questions, answers, tags, comments, votes, and user histories in XML files (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='xml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We used Posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='xml file, which stores the questions and answers of all the SO posts, as the basic to estimate how many ARPs we missed due to limiting the search to the “architect*” terms in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' According to the SO data dump of October 5, 2022, there are 23 million questions (posts) and 34 million answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We then used the power statistics and calculated a representative sample size of these 23 million posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With a 95% confidence level and 3% margin of error, the representative sample size calculated is 1069 posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Afterwards, we randomly selected 1069 posts from the 23 million posts and manually checked them for calculating how many ARPs we might have missed due to limiting the search to the “architect*” terms during the search of ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, the first author labelled the 1069 posts to determine which of the posts are ARPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The second author checked and validated the labeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The disagreements were resolved in a meeting to improve the reliability of the labeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Based on our manual labelling, we found that out of the 1069 posts, only 21 were ARPs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', the true positives), wherein 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 3 out of 21 ARPs) do not contain “architect*” terms and 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='7% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 18 out of 21 APRs) contain “architect*” terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, we admit that we might have missed certain number of ARPs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='3%) that do not contain “architect*” terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We added in our replication package [39] the randomly selected posts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 1069 posts) and the labeling results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', 18 ARPs which contain “architect*” terms and 3 ARPs which do not contain “architect*” terms) for replication purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Construct validity refers to the degree to which a study measures what it claims to be measuring [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' One threat to the construct validity in this study is concerning the manual analysis of the selected SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This is because manually analysis could bring personal bias due to multiple interpretations and/or oversight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To mitigate this threat, we used two qualitative techniques (open coding and constant comparison) from Grounded Theory [17] to analyze the extracted data and answer the RQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, we tried to minimize this threat by performing a pilot data coding before the formal data coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' As discussed in Section 4, during the pilot data coding, the first author selected a random set of 100 ARPs and encoded the extracted data (see Table 4) with respect to the purpose of each RQ (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Several physical meetings with the second author were scheduled to solve any confusion faced by the first author during this pilot data coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, the final results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', concepts, categories, and subcategories) from the pilot data analysis were checked and validated by other three authors (the second, third, and fourth authors) of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The disagreements were resolved in a meeting using the negotiated agreement approach [45] to improve the reliability of the pilot data analysis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Another threat is related to the identification of ARPs (solutions) with useful knowledge by checking the comments attached to these posts in order to answer RQ5 and RQ6 (see Phase II, in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To mitigate this threat, as we explained in Section 3, we did not count on the occurrence of the terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', “useful” (and the similar) stated in comments to measure the usefulness of an architecture solution given to certain architecture related question in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We rather referred to the usefulness related information in the comments attached to the solutions to investigate SO users’ discussions on the usefulness of architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In other words, we judged the usefulness using the reaction of the SO users after seeing and using the architecture solutions (see a comment in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, we (four authors of this study) first read the solutions (from our studied representative sample) commented to be useful to see whether there are really useful to address the questions [15] before we decided to include such posts (solutions) with useful knowledge for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Thus, we believe that we have adequately mitigated this threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' External validity refers to the extent to which the findings of the study can be generalized in other settings [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this research, we only used SO as the source to investigate ARPs and their usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Even though SO is a widely used and popular developer Q&A site, this unique source still poses a threat to the diversity of the study results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To mitigate this threat, our research could be further enhanced by including more sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', GitHub) and look at architecture related questions to understand the architecture design issues that are being faced by architects and developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, researchers might consider going to the fields and asking for feedback directly from architects and developers to better understand the problems they are facing about architecture design and what architecture solutions can be regarded as useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 75https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='org/details/stackexchange_20221005 36 Reliability refers to whether the study will provide the same results and findings when it is repli- cated by other researchers [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this study, this threat is largely related to the process of manual data collection and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To mitigate this threat, we (the authors of this study) followed a rigorous pro- cedure that is consisted of data collection and analysis activities (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, the results from the classification and characterization stages were cross-checked by involving the four authors of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To guarantee the reliability of our results and findings, a replication package, containing the dataset used and the encoded data produced in this work, has been made available [39], allowing other researchers to evaluate the rigor of the design and replicate the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' With these measures, this threat has been partially reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Related work The research related to our work comes from studies that investigate software development knowl- edge in Q&A sites, such as SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In this section, we summarize relevant work in two categories: (1) investigation of architectural knowledge in Q&A sites and (2) quality assessment of the knowledge in Q&A sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architectural knowledge in Q&A Sites A few number of existing studies have studied architectural information provided in ARPs in SO from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [5] used a semi-automatic dictionary-based mining approach to ex- tract Quality Attribute (QA) and Architecture Tactic (AT) related discussions in SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, they applied the dictionary-based classifier Support Vector Machine (SVM) to automatically identify QA-AT related discussions from SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, the authors went on to manually structure the design relationships between Architectural Tactics (ATs) and Quality Attributes (QAs) used in prac- tice and build a knowledge base of how developers use ATs with respect to QA concerns from related discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Such knowledge can help architects better make ATs design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Chinnappan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [61] extracted data from five open sources of software repositories, including Stack Overflow and qualita- tively mined architectural tactics for energy-efficiency robotics software applied by practitioners in real robotics projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To foster the applicability of the identified tactics (even beyond the robotics software community), they describe them in a generic, implementation independent manner by means of diagrams inspired by the UML component and sequence diagram notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The presented energy-aware tactics can serve as guidance for roboticists, as well as other developers interested in architecting and implementing energy-aware software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [11] conducted an empirical study with 50 software engineers, who used Google to make design decisions using the Attribute Driven Design steps [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Based on the relevance and Architecture Knowledge (AK) concepts specified by software engineers, they determined how effective web search engines are to find relevant architectural information from various sources (in- cluding Stack Overflow) and to capture AK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In another work, Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [51] developed an ontology that covers AK concepts in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The ontology provides a description of architecture related information to represent and structure AK in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [10] also leveraged SO with the goal of improving how architects search for architec- turally relevant information in online developer communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They developed a new search approach (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', a domain specific-search approach) for searching architecturally relevant information using SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They found that the new search approach outperforms the conventional keyword-based search approach (searching through the search engines, such as Google).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [14] conducted an empirical study of SO users’ conception of architectural smells by analyzing the discussions from architecture smell related posts in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They found that SO users often describe architectural smells with some general terms, such as “bad”, “wrong”, “brittle” or violation of architecture patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [62] extracted data from eight most popular online developer communities, including Stack Overflow, to investigate how developers perceive the notion of architecture erosion, its causes and consequences, as well as tools and practices to identify and control architecture erosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Among other major findings reported in their study, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' found that developers either focus on the structural manifestation of architecture erosion or on its effect on run-time qualities, maintenance and evolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' alongside technical factors, architecture erosion is caused to a large extent by non-technical factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Zou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [63] used the topic model technique to study non-functional requirements related to textual content in SO posts in order to understand the actual requirements of developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our study complements the abovementioned work since it focuses on the investigation of architectural knowledge in SO through the characterization and categorization of architecture related posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, we examine the usefulness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', are the answers useful to address the questions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [15]) of these posts from the point of view of SO users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 37 The work closely related to ours is the study by Soliman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [6], which leveraged SO to categorize ARPs based on technology related information provided in those posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The main difference between our study and their work is the fact that we look at the problems from a wider scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In other words, our study aims to categorize ARPs in SO by looking at various architectural information, such as ar- chitecture tactics, provided in those posts, rather than limiting ourselves to one particular information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', technology information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Therefore, our work complements the work in [6] by digging deeper into architecture related posts, for example, identifying additional categories of ARPs and exploring design contexts of architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Moreover, we characterize and analyze the usefulness of these posts for practitioners and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Quality assessment of knowledge in Q&A Sites The Q&A platforms, such as SO, play a significant role in knowledge sharing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' however, they still face significant challenges to ensure the quality of their knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' This is evident in the growing number of studies that focus on analyzing the quality of the content in the programming related posts in SO from different views, such as code and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Dagenais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [64] conducted an empirical study on the traceability links between an API and its learning resources in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They found that the majority of API names (89%) in code snippets from online forums are vague and cannot be easily solved due to the deficiency of code snippets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [65] studied 399 Android applications and revealed 1,279 cases of copyright violations of code reuse between GitHub and SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [66] assessed the security-related matters of code snippets in SO and discovered that 29% are insecure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [41] investigated obsolescence of answers in SO and found that 31% may have potential API usage violations that could yield unexpected behavior, such as system crashes and resource leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Ragkhitwetsagul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [67] investigated Java code snippets in SO and identified that 153 clones were copied to SO wherein 66% were obsolete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Zagalsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [68] presented Example- Overflow, a code search and recommendation tool to suggest high-quality code by using the knowledge in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [8] conducted an exploratory study on the prevalence and severity of API misuse in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Treude et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [69] surveyed GitHub users to comprehensively study the difficulty of code snippets in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They found that less than half of the SO snippets are self-explanatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Ragkhitwetsagul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [70] conducted an online survey to investigate the answer obsolescence matter in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Their survey results indicated that half of the top answerers are aware of obsolete code examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' However, users rarely and even never fix obsolete code examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Treude et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [71] developed a tool to improve API documentation with the use of “insight sentences” extracted from SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [72] proposed an AutoComment tool to automatically generate comments for Java and Android tagged Q&A posts in Q&A sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' The results indicate that accurate, adequate, concise, and useful generated comments help users understand the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [73] investigated questions (with similar crash traces) to automatically fix recurring crash bugs in Q&A sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' McDonnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [74] investigated APIs evolution in Android ecosystem using the version history data found in GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Their results revealed that Android is evolving fast at a rate of 115 API updates per month on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Dalip et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [75] suggested a method to rank answers with regard to the feedback provided to answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' They witnessed that both user and review features are essential to assess the quality of answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [76] proposed an approach called AnswerBot to automatically summarize answer posts relevant to a technical question in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [15] proposed a multi-dimensional model for assessing the quality of answers in social Q&A sites, such as Answerbag and Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Answers, in the context of eLearning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Calefato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' [77] conducted an empirical study aimed at assessing 26 best-answer prediction models in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These studies are related to our work since they investigated the quality of code examples provided in SO posts, while our work investigates the quality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', usefulness) of the architecture solutions provided in SO posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Nevertheless, our work differs from the aforementioned work in that they focused on low- level source code (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', API), while our study focuses on high-level concepts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', proposed architecture patterns as solutions to address design concerns) to investigate their usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We believe that our study complements the existing work on the quality of SO posts by analyzing architecture related posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To the best of our knowledge, there has been no investigation of the architectural information provided in ARPs with regard to their categories, characteristics, and usefulness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', are the answers useful to address the questions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=') from the point of view of SO users, which is the focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 38 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Conclusions and future work Investigating architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture tactics and patterns) as an important type of architectural knowledge provided in online developer communities, such as SO, is crucial since this knowledge is one of the most important development knowledge [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architectural knowledge plays a significant role for architects and developers in making informed architectural design decisions during development [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture solutions are the fundamental building blocks in modern software design [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Contrarily to changing implementation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', low-level source code), once an architecture solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', an architecture pattern) is adopted and implemented, it is quite difficult and costly to change it [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' By analyzing and understanding how SO users deal with architectural problems or issues in online developer communities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' such as SO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' brings three benefits: (1) it provides key insights about the types of design problems SO users face during their architecture designs and the types of architecture solutions discussed as well as their usefulness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) it can help to know the design contexts in which architecture problems are raised,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' and (3) it can help to know the characteristics of architecture problems and solutions discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' These benefits provide an opportunity to develop new approaches and tools that can assist So users search and (re)use architectural knowledge shared in online developer communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' To this end, in this study, we investigated architecture related questions and their associated architecture solutions in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Specifically, we used qualitative analysis approach to analyze a statistically representative random sample of 968 ARPs from 10,423 ARPs manually identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We intended to identify both the categories and characteristics of architecture related questions and their solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We also explored the design contexts in which those questions were raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Finally, we studied SO users’ discussions on the usefulness of the architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We summarize our main results and findings as follows: SO users ask a broad spectrum of architecture related questions ranging from architecture tool to architecture configuration, architecture implementation to architecture deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In addition, SO users mostly discuss solution for architecture configuration (39%), followed by solution for architec- ture implementation (18%), explanation of architecture (16%), and architecture tactic (11%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' We observed that ARPs (questions and answers) cover almost all architecting activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO users ask the most (27%, 261 out of 968) ARP questions about architecture configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Most of the SO users (71%, 687 out of 968 ARP questions) considered design contexts when asking architecture related questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture related questions that provide clear description together with architectural diagrams increase their likelihood of getting more than one answer, while poorly structured architecture questions tend to only get one answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Architecture solution for configuration from our proposed taxonomy is the most provided type of architecture solutions that are considered useful in SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' SO users mainly consider architecture solutions that are complete and comprehensive and have concise explanation with architectural diagrams to be helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Our results and findings can help researchers and practitioners by knowing what types of architec- tural knowledge, such as categories of architecture related questions and solutions, are provided in SO, and what are the characteristics of good architecture related questions and useful architecture solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' Also, our results can motivate researchers and practitioners to consider SO as a valuable source of archi- tectural knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', architecture patterns and tactics) and develop novel approaches and tools for mining useful architecture knowledge from SO to support architecting activities and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' In the next step: (1) We plan to conduct a comparative study of architecture solutions provided at SO and other platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', developer mailing lists and issue tracking systems), which may help reveal insights into the current focus of architecture solutions utilization, and their advantages and deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (2) We aim for validating and extending the proposed taxonomy of useful architecture solutions provided at SO (see Figure 4) using an industrial survey from the practitioners’ perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' (3) We also plan to design and employ (semi-)automatic approaches to extract and summarize architectural information, and establish the architecture issue-solution pairs from the retrieved architectural information, for example, benefits and drawbacks of certain architecture solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', patterns and tactics) for task-specific architecture problems from multiple sources of architectural information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=', Q&A sites, GitHub, issue tracking systems, technical blogs), which can facilitate the decision-making of architects by utilizing architectural knowledge from peers and communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} +page_content=' 39 Acknowledgements This work is partially 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNAzT4oBgHgl3EQfBfpr/content/2301.00943v1.pdf'} diff --git a/u9AzT4oBgHgl3EQfB_ou/content/2301.00950v1.pdf b/u9AzT4oBgHgl3EQfB_ou/content/2301.00950v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..304b517452eec6cb385474bd8fb74c77d617d65a --- /dev/null +++ b/u9AzT4oBgHgl3EQfB_ou/content/2301.00950v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a44a567cbda61a8a4fa024540fe39016b53bea147a00241b61666e0c569e3162 +size 41060494 diff --git a/u9AzT4oBgHgl3EQfdPx0/content/tmp_files/2301.01417v1.pdf.txt b/u9AzT4oBgHgl3EQfdPx0/content/tmp_files/2301.01417v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3bdfa7f00f1bf20771f126ee26b930da12a021fd --- /dev/null +++ b/u9AzT4oBgHgl3EQfdPx0/content/tmp_files/2301.01417v1.pdf.txt @@ -0,0 +1,563 @@ +4EOR1A-01 +1 +On-sky performance of new 90 GHz detectors for +the Cosmology Large Angular Scale Surveyor +(CLASS) +Carolina N´u˜nez, John W. Appel, Michael K. Brewer, Sarah Marie Bruno, Rahul Datta, Charles L. Bennett, +Ricardo Bustos, David T. Chuss, Sumit Dahal, Kevin L. Denis, Joseph Eimer, Thomas Essinger-Hileman, Kyle +Helson, Tobias Marriage, Carolina Morales P´erez, Ivan L. Padilla, Matthew A. Petroff, Karwan Rostem, +Duncan J. Watts, Edward J. Wollack, and Zhilei Xu (徐智磊) +Abstract—The Cosmology Large Angular Scale Surveyor +(CLASS) is a polarization-sensitive telescope array located at an +altitude of 5,200 m in the Chilean Atacama Desert and designed +to measure the polarized Cosmic Microwave Background (CMB) +over large angular scales. The CLASS array is currently observ- +ing with three telescopes covering four frequency bands: one +at 40 GHz (Q); one at 90 GHz (W1); and one dichroic system +at 150/220 GHz (HF). During the austral winter of 2022, we +upgraded the first 90 GHz telescope (W1) by replacing four of the +seven focal plane modules. These new modules contain detector +wafers with an updated design, aimed at improving the optical +efficiency and detector stability. We present a description of the +design changes and measurements of on-sky optical efficiencies +derived from observations of Jupiter. +Index Terms—Transition-edge sensors (TES) devices, Mi- +crowave detectors, Millimeter wave detectors, Microwave antenna +arrays, Superconducting bolometers +I. INTRODUCTION +T +HE Cosmology Large Angular Scale Surveyor (CLASS) +is a four-telescope polarization-sensitive array located on +Cerro Toco at 5,200 m in the Chilean Atacama Desert. The +CLASS telescopes are designed to measure “E-mode” (even +parity) and “B-mode” (odd parity) polarization patterns in the +Cosmic Microwave Background (CMB) over large angular +scales (> 1°), with the goal of improving our understanding +of inflation, reionization, and dark matter [1], [2]. +The CLASS array design consists of four telescopes: one +at 40 GHz (Q), two at 90 GHz (W1 & W2), and one dichroic +Carolina N´u˜nez, John W. Appel, Charles L. Bennett, Michael K. Brewer, +Sarah Marie Bruno, Rahul Datta, Joseph Eimer, Tobias Marriage, Car- +olina Morales P´erez, and Ivan L. Padilla are with the Department of Physics +and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA. +Ricardo Bustos is with the Departamento de Ingenier´ıa El´ectrica, Univer- +sidad Cat´olica de la Sant´ısima Concepci´on, Concepci´on, Chile. +Sumit Dahal, Kevin L. Denis, Thomas Essinger-Hileman, Karwan Rosten, +and Edward J. Wollack are with the NASA Goddard Space Flight Center, +Greenbelt, MD 20771, USA. +Kyle Helson is with the Center for Space Sciences and Technology, +University of Maryland, Baltimore County, Baltimore, MD 21250, USA, and +also with the NASA Goddard Space Flight Center, Greenbelt, MD 20771, +USA. +Matthew A. Petroff is with the Center for Astrophysics, Harvard & +Smithsonian, Cambridge, MA 02138, USA. +Duncan J. Watts is with the Institute of Theoretical Astrophysics, University +of Oslo, Oslo, Norway. +Zhilei Xu (徐智磊) is with the MIT Kavli Institute, Massachusetts Institute +of Technology, Cambridge, MA 02139, USA. +system at 150/220 GHz (HF). The CLASS array is currently +observing with three telescopes across all four targeted fre- +quency bands. The Q-band instrument was deployed in 2016, +W1 was deployed in 2018, and HF was deployed in 2019. +CLASS focal planes consist of arrays of highly sensitive +feedhorn-coupled transition-edge sensor (TES) bolometers. +The CLASS TES bolometers provide the background-limited +sensitivity required to achieve the experiment’s science goals. +Characterization and on-sky performance of these focal planes +are presented in [3]–[6]. +During the austral winter of 2022, we upgraded the first +90 GHz telescope (W1) by replacing four (out of seven) of +the focal plane modules. These new modules contain detector +wafers with an upgraded design, aimed at improving the +optical efficiency and detector stability performance issues +described in [6]. In-lab testing and characterization of these +detectors, including electrothermal parameters, bandpass mea- +surements, and dark noise performance, are described in [7]. +As a supplement to this previous work, this paper presents +preliminary on-sky performance of these new detectors. In § II, +we describe the upgraded wafer design. In § III, we describe +optical efficiency measurements derived from dedicated obser- +vations of Jupiter. +II. DESIGN +The CLASS 90 GHz detector wafers, which integrate 37 +detector pixels, were fabricated at NASA Goddard Space +Flight Center. Each of the detector pixels consists of a sym- +metric planar ortho-mode transducer (OMT), which reads out +two orthogonal linear polarizations to two TES bolometers. +Signals from opposing antenna probes are routed through a +vialess crossover and a terminated vialess crossover, which +symmetrizes the response between both OMT signal paths, +and then coherently combined onto a single microstrip trans- +mission line using the difference output of a Magic Tee, which +transmits a single mode [8]. On-chip filtering and micro- +machined silicon packaging define the signal bandpass [9]. +Finally, the signal from each of the two orthogonal polariza- +tions is passed to a MoAu bilayer TES bolometer. For a full +description of the original CLASS 90 GHz wafer, see [10]– +[12]. +As summarized in [7], we present the design changes of the +new 90 GHz detector wafers. The updated wafer includes the +arXiv:2301.01417v1 [astro-ph.IM] 4 Jan 2023 + +4EOR1A-01 +2 +following main design changes to the original CLASS 90 GHz +detector design: +1) a simplified absorber that terminates power from the sky +(brought in via a Nb microstrip) onto the TES island, +with a resistive PdAu meander that consists of a stepped +impedance transition from Nb to PdAu; +2) the addition of a direct normal-metal connection be- +tween the TES and the heat capacity element formed by +the Pd, to effectively lump the electronic heat capacity +into a single element; +3) the revision of the choke filter circuit design to extend +onto the membrane’s diffusive bolometer legs. +4) the addition of a revised absorber at the Magic Tee and +at the terminated vialess crossover, realized as a lossy +stepped impedance transition between Nb to PdAu that +decreased the total meander length, device footprint, and +sensitivity to detailed implementation. +These design changes were introduced in order to improve +optical efficiency (1, 3) and stability of the TES transition +(2), which were described in [6]. The redesigned absorber +at the Magic Tee and at the terminated crossovers (4) may +have also improved the optical efficiency by achieving lower +reflectance. The use of discrete interfaces along the length of +absorber (rather than along the length of a taper) minimizes the +uncertainty in the microwave propagation parameters arising +from proximization [13] at the superconducting and normal +metal interfaces. The updated TES is shown in Fig. 1. The full +detector pixel and zoom-ins of the Magic Tee and terminated +vialess crossover are shown in Fig. 2. +III. ON-SKY PERFORMANCE: OPTICAL EFFICIENCIES +We measure the detector optical efficiencies via observations +of Jupiter. Specifically, we scan back and forth in azimuth at +a constant elevation, while Jupiter rises or sets through the +telescope beams. We can then measure the optical efficiency +by comparing the observed and expected amplitude of Jupiter +(i.e., the peak power received by the telescope when pointed +directly at Jupiter). In the Rayleigh-Jeans limit, the expected +amplitude of Jupiter in this microwave frequency range is +given by: +AJ = kBTJ∆νBdil , +(1) +where kB is the Boltzmann constant, TJ is the Jupiter bright- +ness temperature at 90 GHz (172.8 K) as reported by the +WMAP team [14], ∆ν is the observing bandwidth of 31 GHz, +and Bdil is the beam dilution factor given by the ratio of +the oblateness-corrected solid angle of Jupiter (ΩJ) and the +convolution of the detector beam with Jupiter (ΩB). We can +then obtain the absolute efficiency (η) by taking the ratio of +the measured amplitude Aobs with the expected amplitude AJ: +η = Aobs +AJ += +AobsΩB +kB∆νTJΩJ +. +(2) +To determine Aobs, we do the following: initially, raw data +from the Jupiter observations are converted from DAC units +to units of power using the I-V +bin calibration method +described in [15] and low-pass filtered to remove the emission +Fig. 1: Upgraded CLASS TES showing three primary design +changes described in § II: 1) a simplified absorber with a +resistive PdAu meander; 2) a direct normal-metal contact +between the MoAu TES and the Pd heat capacity element; 3) +the revised choke filter circuit implementation extending onto +the membrane’s diffusive bolometer legs. Power from the sky +is brought in via the tapered Nb microstrip and terminates +through the absorber onto the TES island. The short Si beam +sets the thermal conductance (G) and regulates the flow of +power between the TES island and the bath. The MoAu bilayer +determines the superconducting critical temperature (Tc) of the +TES. In-lab characterization of electrothermal parameters for +the TES can be found in [7]. +signal from the variable-delay polarization modulator (VPM) +[16], [17]. The data are de-projected from the sky into the +instrument frame, a 10° radius map centered on Jupiter is +made for each detector, and a fit is done to a 2D elliptical +Gaussian to derive the source amplitude. This makes up the +data that are used for each observation in the averaging. +During the averaging, each individual map is read in. After +removing a baseline from each pass over the source in azimuth, +the RMS noise is calculated from an annulus of the map well +away from the source. Since the solid angle of Jupiter varies +over the course of the observations, we scale the data from the +individual observations for each detector to a fiducial reference +solid angle Ωref. For this we use an equatorial angular diameter +of 48′′. The data are further scaled by eτi, where τi is +the optical depth in the atmosphere, at each observation i, +calculated from the zenith opacity as a function of precipitable +water vapor (PWV) using the atmospheric model of [18], +and then multiplied by the cosecant of the elevation of each +detector to adjust the opacity at zenith to the opacity at the +observing elevation. +Once all of the data for each detector are read in and +processed, observations with an RMS noise of over three times +the median for all observations are excluded. Out of a total of +85 observations, we retain an average of 75 observations per +detector. The surviving data are discretized onto a 6◦ radius + +Tapered Nb +Short Si Beam +Microstrip +(Ballistic G) +PdAu +Bias Lead Filter +Pd +Metal Contact +MoAu +Bilayer +1 +Bias Leads on Long Legs +SEMHV:30.0kV +WD:15.54mm +VEGA3TESCAN +View field: 563 μm +Det: BSE +100 μm +GsFcDetectorDevelopmentLab4EOR1A-01 +3 +(a) +(b) +(c) +Fig. 2: The upgraded CLASS 90 GHz detector pixel (a), and zoom-ins showing the Magic Tee (b) and terminated vialess +crossover (c) with the revised PdAu circuit termination. The new meandered PdAu termination was realized as a lossy stepped +impedance transition between Nb to PdAu. This design strategy leads to a reduction of the total meander length, device +footprint, and sensitivity to detailed implementation. +5 +0 +5 +X (deg) +5 +0 +5 +Y (deg) +10 +3 +5 × 10 +4 +0 +5 × 10 +4 +10 +3 +10 +2 +10 +1 +1 +Normalized Power +(a) +0.2 +0.4 +0.6 +0.8 +1.0 +Optical Efficiency +0 +20 +40 +60 +Count +Upgraded detectors +Original detectors +(b) +Fig. 3: (a) A co-added map of Jupiter obtained from 85 observations of Jupiter with the upgraded 90 GHz focal plane. The map +is produced by co-adding signal from all 345 optical detectors, and is normalized to a peak amplitude of unity. The colorbar +scale is linear from −10−3 to 10−3 and logarithmic above 10−3. (b) A comparison of optical efficiencies between detectors +on the currently fielded 90 GHz focal plane. The focal plane contains four wafers with the upgraded design, and three wafers +with the original design. +regular grid at 0.05◦ resolution and a weighted average is +calculated to yield the final averaged map for each detector: +Mapavg = +�nobs +i=1 wi(Ωref/Ωi)eτiMapi +�nobs +i=1 wi +, +(3) +where Ωi is the solid angle obtained from the equatorial +angular diameter of Jupiter at the time of each observation and +wi is the signal-to-noise ratio calculated from the initial fitted +amplitude and the RMS noise. The averaged map is then fit to +a 2D elliptical Gaussian, yielding a peak amplitude Aobs, and +the beam solid angle ΩB is measured by integrating the nor- +malized map out to a radius of three times the full width at half +maximum (FWHM) of the beam from the peak. The efficiency +can then be calculated, using Ωref corrected for the oblateness +of Jupiter, according to the method described in [19], as ΩJ in +(2). Using the 2.974◦ average observed latitude for the center +of the disk of Jupiter yields ΩJ = 3.978x10−8 sr. +Fig. 3a shows the observed map of Jupiter, composed of +co-added maps from all 345 optical detectors. In Fig. 3b, we +show the distribution of optical efficiencies across the currently +fielded focal plane. The detectors with the updated design +(four of the seven modules) are shown in blue, while the + +Magic Tee +Terminated +Antenna +Vialess +Probe +TES -45° +Crossover +Vialess +Bandpass +Crossover +Filter +TES +45° +SEM HV: 30.0 kV +WD: 15.74 mm +VEGA3 TESCAN +View field: 8.53 mm +Det: BSE +2 mm +GsFcDetectorDevelopmentLabPdAu +Termination +SEMHV:30.0kV +WD:15.54mm +VEGA3 TESCAN +Viewfield:1.42mm +Det:BSE +200 μm +GsFc Detector Development LabPdAu +Termination +PdAu +人 +Termination +SEM HV: 30.0 kV +WD: 15.54 mm +VEGA3 TESCAN +View field: 1.32 mm +Det: BSE +200 μm +GsFc Detector Development Lab4EOR1A-01 +4 +previous generation of detectors (three of the seven modules) +are shown in red. We observe a marked improvement in the +distribution of optical efficiencies for the detectors with the +updated design. Furthermore, we note that the three wafers +from the original design were the best-performing wafers +of the original 90 GHz focal plane, i.e., the four upgraded +detector wafers replaced the four least-performing wafers of +the original focal plane. Therefore, the distribution of the full +original W1 focal plane favors even lower optical efficiencies +than are shown for the original W1 detectors in Fig. 3b. The +optical efficiency distribution for the original 90 GHz focal +plane is shown in [6, Fig. 3]. +IV. CONCLUSION +In conjunction with [7], we have provided in-lab charac- +terization (electrothermal parameters, bandpasses, dark noise +measurements) and on-sky performance (optical efficiencies) +results for the new detectors of the CLASS 90 GHz focal +plane. These detectors obtained first light in the austral winter +of 2022, and replaced four of the seven wafers of the original +90 GHz focal plane. The detectors were redesigned with +three primary changes to the TES aimed at improving optical +efficiency and detector stability. In addition, changes to the +terminations of the Magic Tee and terminated vialess crossover +were implemented to reduce their reflectance, sensitivity to +fabrication details, and the fidelity of the impedance match +seen by the Magic Tee and crossover circuits. We demonstrate +improvements in optical efficiency between the former wafer +design and current wafer design, by comparing expected vs. +observed amplitude measurements of Jupiter. A subsequent +publication will provide further characterization and on-sky +performance analysis of the upgraded 90 GHz focal plane. +ACKNOWLEDGMENT +We acknowledge the National Science Foundation Division +of Astronomical Sciences for their support of CLASS un- +der Grant Numbers 0959349, 1429236, 1636634, 1654494, +2034400, and 2109311. We thank Johns Hopkins University +President R. Daniels and the Krieger School of Arts and +Sciences Deans for their steadfast support of CLASS. We +further acknowledge the very generous support of Jim and +Heather Murren (JHU A&S ’88), Matthew Polk (JHU A&S +Physics BS ’71), David Nicholson, and Michael Bloomberg +(JHU Engineering ’64). The CLASS project employs detec- +tor technology developed in collaboration between JHU and +Goddard Space Flight Center under several previous and on- +going NASA grants. Detector development work at JHU was +funded by NASA cooperative agreement 80NSSC19M0005. +Kyle Helson is supported by NASA under award number +80GSFC17M0002. Zhilei Xu is supported by the Gordon +and Betty Moore Foundation through grant GBMF5215 to +the Massachusetts Institute of Technology. We acknowledge +scientific and engineering contributions from Max Abitbol, +Fletcher Boone, David Carcamo, Lance Corbett, Ted Grun- +berg, Saianeesh Haridas, Jake Hassan, Connor Henley, Ben +Keller, Lindsay Lowry, Nick Mehrle, Sasha Novak, Diva +Parekh, Isu Ravi, Gary Rhodes, Daniel Swartz, Bingjie Wang, +Qinan Wang, Tiffany Wei, Zi´ang Yan, and Zhuo Zhang. We +thank Miguel Angel D´ıaz, Jill Hanson, William Deysher, +and Chantal Boisvert for logistical support. We acknowledge +productive collaboration with Dean Carpenter and the JHU +Physical Sciences Machine Shop team. CLASS is located in +the Parque Astron´omico Atacama in northern Chile under +the auspices of the Agencia Nacional de Investigaci´on y +Desarrollo (ANID). +REFERENCES +[1] T. Essinger-Hileman, A. Ali, M. Amiri, J. W. Appel, D. Araujo, C. L. +Bennett, F. Boone, M. Chan, H.-M. Cho, D. T. Chuss, F. Colazo, +E. Crowe, K. Denis, R. D¨unner, J. Eimer, D. Gothe, M. Halpern, +K. Harrington, G. C. Hilton, G. F. Hinshaw, C. 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Wright, “Seven-year Wilkinson Microwave Anisotropy Probe +(WMAP) Observations: Planets and Celestial Calibration Sources,” The +Astrophysical Journal, Supplement, vol. 192, no. 2, p. 19, Feb. 2011. + diff --git a/u9AzT4oBgHgl3EQfdPx0/content/tmp_files/load_file.txt b/u9AzT4oBgHgl3EQfdPx0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c8b71f68a6519e738baf5c5a2f448bcb1c6ee0c --- /dev/null +++ b/u9AzT4oBgHgl3EQfdPx0/content/tmp_files/load_file.txt @@ -0,0 +1,885 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf,len=884 +page_content='4EOR1A-01 1 On-sky performance of new 90 GHz detectors for the Cosmology Large Angular Scale Surveyor (CLASS) Carolina N´u˜nez, John W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Appel, Michael K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Brewer, Sarah Marie Bruno, Rahul Datta, Charles L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Bennett, Ricardo Bustos, David T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Chuss, Sumit Dahal, Kevin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Denis, Joseph Eimer, Thomas Essinger-Hileman, Kyle Helson, Tobias Marriage, Carolina Morales P´erez, Ivan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Padilla, Matthew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Petroff, Karwan Rostem, Duncan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Watts, Edward J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Wollack, and Zhilei Xu (徐智磊) Abstract—The Cosmology Large Angular Scale Surveyor (CLASS) is a polarization-sensitive telescope array located at an altitude of 5,200 m in the Chilean Atacama Desert and designed to measure the polarized Cosmic Microwave Background (CMB) over large angular scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The CLASS array is currently observ- ing with three telescopes covering four frequency bands: one at 40 GHz (Q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' one at 90 GHz (W1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' and one dichroic system at 150/220 GHz (HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' During the austral winter of 2022, we upgraded the first 90 GHz telescope (W1) by replacing four of the seven focal plane modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' These new modules contain detector wafers with an updated design, aimed at improving the optical efficiency and detector stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We present a description of the design changes and measurements of on-sky optical efficiencies derived from observations of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Index Terms—Transition-edge sensors (TES) devices, Mi- crowave detectors, Millimeter wave detectors, Microwave antenna arrays, Superconducting bolometers I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' INTRODUCTION T HE Cosmology Large Angular Scale Surveyor (CLASS) is a four-telescope polarization-sensitive array located on Cerro Toco at 5,200 m in the Chilean Atacama Desert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The CLASS telescopes are designed to measure “E-mode” (even parity) and “B-mode” (odd parity) polarization patterns in the Cosmic Microwave Background (CMB) over large angular scales (> 1°), with the goal of improving our understanding of inflation, reionization, and dark matter [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The CLASS array design consists of four telescopes: one at 40 GHz (Q), two at 90 GHz (W1 & W2), and one dichroic Carolina N´u˜nez, John W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Appel, Charles L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Bennett, Michael K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Brewer, Sarah Marie Bruno, Rahul Datta, Joseph Eimer, Tobias Marriage, Car- olina Morales P´erez, and Ivan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Padilla are with the Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Ricardo Bustos is with the Departamento de Ingenier´ıa El´ectrica, Univer- sidad Cat´olica de la Sant´ısima Concepci´on, Concepci´on, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Sumit Dahal, Kevin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Denis, Thomas Essinger-Hileman, Karwan Rosten, and Edward J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Wollack are with the NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Kyle Helson is with the Center for Space Sciences and Technology, University of Maryland, Baltimore County, Baltimore, MD 21250, USA, and also with the NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Matthew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Petroff is with the Center for Astrophysics, Harvard & Smithsonian, Cambridge, MA 02138, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Duncan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Watts is with the Institute of Theoretical Astrophysics, University of Oslo, Oslo, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Zhilei Xu (徐智磊) is with the MIT Kavli Institute, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' system at 150/220 GHz (HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The CLASS array is currently observing with three telescopes across all four targeted fre- quency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The Q-band instrument was deployed in 2016, W1 was deployed in 2018, and HF was deployed in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' CLASS focal planes consist of arrays of highly sensitive feedhorn-coupled transition-edge sensor (TES) bolometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The CLASS TES bolometers provide the background-limited sensitivity required to achieve the experiment’s science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Characterization and on-sky performance of these focal planes are presented in [3]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' During the austral winter of 2022, we upgraded the first 90 GHz telescope (W1) by replacing four (out of seven) of the focal plane modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' These new modules contain detector wafers with an upgraded design, aimed at improving the optical efficiency and detector stability performance issues described in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In-lab testing and characterization of these detectors, including electrothermal parameters, bandpass mea- surements, and dark noise performance, are described in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' As a supplement to this previous work, this paper presents preliminary on-sky performance of these new detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In § II, we describe the upgraded wafer design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In § III, we describe optical efficiency measurements derived from dedicated obser- vations of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' DESIGN The CLASS 90 GHz detector wafers, which integrate 37 detector pixels, were fabricated at NASA Goddard Space Flight Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Each of the detector pixels consists of a sym- metric planar ortho-mode transducer (OMT), which reads out two orthogonal linear polarizations to two TES bolometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Signals from opposing antenna probes are routed through a vialess crossover and a terminated vialess crossover, which symmetrizes the response between both OMT signal paths, and then coherently combined onto a single microstrip trans- mission line using the difference output of a Magic Tee, which transmits a single mode [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' On-chip filtering and micro- machined silicon packaging define the signal bandpass [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Finally, the signal from each of the two orthogonal polariza- tions is passed to a MoAu bilayer TES bolometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' For a full description of the original CLASS 90 GHz wafer, see [10]– [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' As summarized in [7], we present the design changes of the new 90 GHz detector wafers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The updated wafer includes the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='01417v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='IM] 4 Jan 2023 4EOR1A-01 2 following main design changes to the original CLASS 90 GHz detector design: 1) a simplified absorber that terminates power from the sky (brought in via a Nb microstrip) onto the TES island, with a resistive PdAu meander that consists of a stepped impedance transition from Nb to PdAu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 2) the addition of a direct normal-metal connection be- tween the TES and the heat capacity element formed by the Pd, to effectively lump the electronic heat capacity into a single element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3) the revision of the choke filter circuit design to extend onto the membrane’s diffusive bolometer legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 4) the addition of a revised absorber at the Magic Tee and at the terminated vialess crossover, realized as a lossy stepped impedance transition between Nb to PdAu that decreased the total meander length, device footprint, and sensitivity to detailed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' These design changes were introduced in order to improve optical efficiency (1, 3) and stability of the TES transition (2), which were described in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The redesigned absorber at the Magic Tee and at the terminated crossovers (4) may have also improved the optical efficiency by achieving lower reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The use of discrete interfaces along the length of absorber (rather than along the length of a taper) minimizes the uncertainty in the microwave propagation parameters arising from proximization [13] at the superconducting and normal metal interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The updated TES is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The full detector pixel and zoom-ins of the Magic Tee and terminated vialess crossover are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' ON-SKY PERFORMANCE: OPTICAL EFFICIENCIES We measure the detector optical efficiencies via observations of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Specifically, we scan back and forth in azimuth at a constant elevation, while Jupiter rises or sets through the telescope beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We can then measure the optical efficiency by comparing the observed and expected amplitude of Jupiter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=', the peak power received by the telescope when pointed directly at Jupiter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In the Rayleigh-Jeans limit, the expected amplitude of Jupiter in this microwave frequency range is given by: AJ = kBTJ∆νBdil , (1) where kB is the Boltzmann constant, TJ is the Jupiter bright- ness temperature at 90 GHz (172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='8 K) as reported by the WMAP team [14], ∆ν is the observing bandwidth of 31 GHz, and Bdil is the beam dilution factor given by the ratio of the oblateness-corrected solid angle of Jupiter (ΩJ) and the convolution of the detector beam with Jupiter (ΩB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We can then obtain the absolute efficiency (η) by taking the ratio of the measured amplitude Aobs with the expected amplitude AJ: η = Aobs AJ = AobsΩB kB∆νTJΩJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' (2) To determine Aobs, we do the following: initially, raw data from the Jupiter observations are converted from DAC units to units of power using the I-V bin calibration method described in [15] and low-pass filtered to remove the emission Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 1: Upgraded CLASS TES showing three primary design changes described in § II: 1) a simplified absorber with a resistive PdAu meander;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 2) a direct normal-metal contact between the MoAu TES and the Pd heat capacity element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3) the revised choke filter circuit implementation extending onto the membrane’s diffusive bolometer legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Power from the sky is brought in via the tapered Nb microstrip and terminates through the absorber onto the TES island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The short Si beam sets the thermal conductance (G) and regulates the flow of power between the TES island and the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The MoAu bilayer determines the superconducting critical temperature (Tc) of the TES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In-lab characterization of electrothermal parameters for the TES can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' signal from the variable-delay polarization modulator (VPM) [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The data are de-projected from the sky into the instrument frame, a 10° radius map centered on Jupiter is made for each detector, and a fit is done to a 2D elliptical Gaussian to derive the source amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' This makes up the data that are used for each observation in the averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' During the averaging, each individual map is read in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' After removing a baseline from each pass over the source in azimuth, the RMS noise is calculated from an annulus of the map well away from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Since the solid angle of Jupiter varies over the course of the observations, we scale the data from the individual observations for each detector to a fiducial reference solid angle Ωref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' For this we use an equatorial angular diameter of 48′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The data are further scaled by eτi, where τi is the optical depth in the atmosphere, at each observation i, calculated from the zenith opacity as a function of precipitable water vapor (PWV) using the atmospheric model of [18], and then multiplied by the cosecant of the elevation of each detector to adjust the opacity at zenith to the opacity at the observing elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Once all of the data for each detector are read in and processed, observations with an RMS noise of over three times the median for all observations are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Out of a total of 85 observations, we retain an average of 75 observations per detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The surviving data are discretized onto a 6◦ radius Tapered Nb Short Si Beam Microstrip (Ballistic G) PdAu Bias Lead Filter Pd Metal Contact MoAu Bilayer 1 Bias Leads on Long Legs SEMHV:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='0kV WD:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='54mm VEGA3TESCAN View field: 563 μm Det: BSE 100 μm GsFcDetectorDevelopmentLab4EOR1A-01 3 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 2: The upgraded CLASS 90 GHz detector pixel (a), and zoom-ins showing the Magic Tee (b) and terminated vialess crossover (c) with the revised PdAu circuit termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The new meandered PdAu termination was realized as a lossy stepped impedance transition between Nb to PdAu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' This design strategy leads to a reduction of the total meander length, device footprint, and sensitivity to detailed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 5 0 5 X (deg) 5 0 5 Y (deg) 10 3 5 × 10 4 0 5 × 10 4 10 3 10 2 10 1 1 Normalized Power (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='0 Optical Efficiency 0 20 40 60 Count Upgraded detectors Original detectors (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3: (a) A co-added map of Jupiter obtained from 85 observations of Jupiter with the upgraded 90 GHz focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The map is produced by co-adding signal from all 345 optical detectors, and is normalized to a peak amplitude of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The colorbar scale is linear from −10−3 to 10−3 and logarithmic above 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' (b) A comparison of optical efficiencies between detectors on the currently fielded 90 GHz focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The focal plane contains four wafers with the upgraded design, and three wafers with the original design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' regular grid at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='05◦ resolution and a weighted average is calculated to yield the final averaged map for each detector: Mapavg = �nobs i=1 wi(Ωref/Ωi)eτiMapi �nobs i=1 wi , (3) where Ωi is the solid angle obtained from the equatorial angular diameter of Jupiter at the time of each observation and wi is the signal-to-noise ratio calculated from the initial fitted amplitude and the RMS noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The averaged map is then fit to a 2D elliptical Gaussian, yielding a peak amplitude Aobs, and the beam solid angle ΩB is measured by integrating the nor- malized map out to a radius of three times the full width at half maximum (FWHM) of the beam from the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The efficiency can then be calculated, using Ωref corrected for the oblateness of Jupiter, according to the method described in [19], as ΩJ in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Using the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='974◦ average observed latitude for the center of the disk of Jupiter yields ΩJ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='978x10−8 sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3a shows the observed map of Jupiter, composed of co-added maps from all 345 optical detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3b, we show the distribution of optical efficiencies across the currently fielded focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The detectors with the updated design (four of the seven modules) are shown in blue, while the Magic Tee Terminated Antenna Vialess Probe TES -45° Crossover Vialess Bandpass Crossover Filter TES +45° SEM HV: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='0 kV WD: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='74 mm VEGA3 TESCAN View field: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='53 mm Det: BSE 2 mm GsFcDetectorDevelopmentLabPdAu Termination SEMHV:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='0kV WD:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='54mm VEGA3 TESCAN Viewfield:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='42mm Det:BSE 200 μm GsFc Detector Development LabPdAu Termination PdAu 人 Termination SEM HV: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='0 kV WD: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='54 mm VEGA3 TESCAN View field: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='32 mm Det: BSE 200 μm GsFc Detector Development Lab4EOR1A-01 4 previous generation of detectors (three of the seven modules) are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We observe a marked improvement in the distribution of optical efficiencies for the detectors with the updated design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Furthermore, we note that the three wafers from the original design were the best-performing wafers of the original 90 GHz focal plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=', the four upgraded detector wafers replaced the four least-performing wafers of the original focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Therefore, the distribution of the full original W1 focal plane favors even lower optical efficiencies than are shown for the original W1 detectors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The optical efficiency distribution for the original 90 GHz focal plane is shown in [6, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' CONCLUSION In conjunction with [7], we have provided in-lab charac- terization (electrothermal parameters, bandpasses, dark noise measurements) and on-sky performance (optical efficiencies) results for the new detectors of the CLASS 90 GHz focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' These detectors obtained first light in the austral winter of 2022, and replaced four of the seven wafers of the original 90 GHz focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The detectors were redesigned with three primary changes to the TES aimed at improving optical efficiency and detector stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' In addition, changes to the terminations of the Magic Tee and terminated vialess crossover were implemented to reduce their reflectance, sensitivity to fabrication details, and the fidelity of the impedance match seen by the Magic Tee and crossover circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We demonstrate improvements in optical efficiency between the former wafer design and current wafer design, by comparing expected vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' observed amplitude measurements of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' A subsequent publication will provide further characterization and on-sky performance analysis of the upgraded 90 GHz focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' ACKNOWLEDGMENT We acknowledge the National Science Foundation Division of Astronomical Sciences for their support of CLASS un- der Grant Numbers 0959349, 1429236, 1636634, 1654494, 2034400, and 2109311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We thank Johns Hopkins University President R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Daniels and the Krieger School of Arts and Sciences Deans for their steadfast support of CLASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We further acknowledge the very generous support of Jim and Heather Murren (JHU A&S ’88), Matthew Polk (JHU A&S Physics BS ’71), David Nicholson, and Michael Bloomberg (JHU Engineering ’64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' The CLASS project employs detec- tor technology developed in collaboration between JHU and Goddard Space Flight Center under several previous and on- going NASA grants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Detector development work at JHU was funded by NASA cooperative agreement 80NSSC19M0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Kyle Helson is supported by NASA under award number 80GSFC17M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Zhilei Xu is supported by the Gordon and Betty Moore Foundation through grant GBMF5215 to the Massachusetts Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We acknowledge scientific and engineering contributions from Max Abitbol, Fletcher Boone, David Carcamo, Lance Corbett, Ted Grun- berg, Saianeesh Haridas, Jake Hassan, Connor Henley, Ben Keller, Lindsay Lowry, Nick Mehrle, Sasha Novak, Diva Parekh, Isu Ravi, Gary Rhodes, Daniel Swartz, Bingjie Wang, Qinan Wang, Tiffany Wei, Zi´ang Yan, and Zhuo Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We thank Miguel Angel D´ıaz, Jill Hanson, William Deysher, and Chantal Boisvert for logistical support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' We acknowledge productive collaboration with Dean Carpenter and the JHU Physical Sciences Machine Shop team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' CLASS is located in the Parque Astron´omico Atacama in northern Chile under the auspices of the Agencia Nacional de Investigaci´on y Desarrollo (ANID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' REFERENCES [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Essinger-Hileman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Ali, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Amiri, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Appel, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Zeng, “CLASS: the cosmology large angular scale surveyor,” in Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Holland and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AzT4oBgHgl3EQfdPx0/content/2301.01417v1.pdf'} +page_content=' Zmuidzinas, Eds.' metadata={'source': 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sha256:f8e1508b90f3daf01057da2540393d312b4d606b87d347da2dcc79aa0c7ab1d4 +size 3997741 diff --git a/xdE4T4oBgHgl3EQfYAwp/content/tmp_files/2301.05045v1.pdf.txt b/xdE4T4oBgHgl3EQfYAwp/content/tmp_files/2301.05045v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..998a3a0c1da23dc910154e36aaa127a75075aafe --- /dev/null +++ b/xdE4T4oBgHgl3EQfYAwp/content/tmp_files/2301.05045v1.pdf.txt @@ -0,0 +1,903 @@ +arXiv:2301.05045v1 [math.FA] 12 Jan 2023 +Characterization of (weak) phase retrieval +dual frames +Fahimeh Arabyani-Neyshaburi1, Ali Akbar Arefijamaal2 and Rajab +Ali Kamyabi-Gol3,∗ +1 Department of Mathematical sciences, Ferdowsi University of Mashhad, +Mashhad, Iran. +2 Department of Mathematics and Computer Sciences, Hakim Sabzevari +University, Sabzevar, Iran. +3 Department of Mathematical sciences, Faculty of Math, Ferdowsi University of +Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS), +Mashhad, Iran. Email: kamyabi@um.ac.ir +Abstract. Recovering a signal up to a unimodular constant from the mag- +nitudes of linear measurements has been popular and well studied in recent +years. However, numerous unsolved problems regarding phase retrieval still +exist. Given a phase retrieval frame, may the family of phase retrieval dual +frames be classified? And is such a family dense in the set of dual frames? Can +we present the equivalent conditions for a family of vectors to do weak phase +retrieval in complex Hilbert space case? What is the connection between phase, +weak phase and norm retrieval? In this context, we aim to deal with these open +problems concerning phase retrieval dual frames, weak phase retrieval frames, +and specially investigate equivalent conditions for identifying these features. +We provide some characterizations of alternate dual frames of a phase retrieval +frame which yield phase retrieval in finite dimensional Hilbert spaces. More- +over, for some classes of frames, we show that the family of phase retrieval dual +frames is open and dense in the set of dual frames. Then, we study weak phase +retrieval problem. Among other things, we obtain some equivalent conditions +on a family of vectors to do phase retrieval in terms of weak phase retrieval. +Mathematics Subject Classification 2020. Primary 42C15; Secondary +41A58. +Keywords. Phase retrieval, weak phase retrieval, dual frames. +∗ Corresponding author. + +2 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +1. Introduction and Preliminaries +Signal reconstruction without using phase is a longstanding conjecture, specially +with regard to speech recognition systems which was first introduced from mathe- +matical point of view by R. Balan, P. Casazza and D. Edidin in [8], and has been +a very popular topic in recent years due to so many applications of X-ray, electron +microscopy, optics, image processing and much more [5, 14, 20, 21]. To present the +problem in a more precise approach, we first briefly state some basic definitions and +preliminaries in relevant areas. Then we propound a number of problems to address +phase retrieval frames in some new aspects. +Throughout this paper, we suppose that H denotes a separable Hilbert space, +Hn denote an n-dimensional real or complex Hilbert space and we use Rn and Cn +whenever it is necessary to differentiate between the two. Also, we consider the +notations; Im = {1, 2, ..., m} and {δi}i∈In as the standard orthonormal basis of Hn. +A family of vectors Φ := {φi}i∈I in H is called a frame if there exist the +constants 0 < AΦ ≤ BΦ < ∞ such that +AΦ∥f∥2 ≤ +� +i∈I +|f, φi⟩|2 ≤ BΦ∥f∥2, +(f ∈ H). +(1.1) +The constants AΦ and BΦ are called frame bounds. The sequence {φi}i∈I is said to +be a Bessel sequence whenever the right hand side of (1.1) holds. A frame {φi}i∈I +is called A-tight frame if A = AΦ = BΦ, and in the case of AΦ = BΦ = 1 it is called +a Parseval frame. Given a frame Φ = {φi}i∈I, its Grammian matrix formed by +the inner product of the frame vectors is as GΦ = [⟨φi, φj⟩]i,j. The frame operator +is defined by SΦf = � +i∈I⟨f, φi⟩φi. It is a bounded, invertible, and self-adjoint +operator [13]. Also, the synthesis operator TΦ : l2 → H is defined by TΦ{ci} = +� +i∈I ciφi. The frame operator can be written as SΦ = TφT ∗ +φ where T ∗ +Φ : H → l2, +the adjoint of T , given by T ∗ +Φf = {⟨f, φi⟩}i∈I is called the analysis operator. The +family {S−1 +Φ fi}i∈I is also a frame for H, called the canonical dual frame. In general, +a frame {ψi}i∈I ⊆ H is called an alternate dual or simply a dual for {φi}i∈I if +f = � +i∈I⟨f, ψi⟩φi, for f ∈ H. All frames have at least a dual, the canonical dual, +and redundant frames have an infinite number of alternate dual frames. We denote +the excess of a frame Φ by E(Φ). It is known that every dual frame is of the form +{S−1 +Φ φi +ui}i∈I, where {ui}i∈I is a Bessel sequence that satisfies � +i∈I⟨f, ui⟩φi = 0, +for all f ∈ H. Also recall that, two frames Φ and Ψ are equivalent if there exists +an invertible operator U on H so that Ψ = UΦ. See [13, 16] for more detailed +information on frame theory and [3, 4, 7, 17, 18, 19] for the importance of duality +principle. +Consider a frame {φi}i∈Im in a Hilbert space Hn. A finite set of indices σ ⊂ Im +satisfies the minimal redundancy condition (MRC) whenever {φi}i∈σc remains to +be a frame for Hn. Furthermore, we say Φ satisfies MRC for r-erasures if every +subset σ ⊂ Im, |σ| = r satisfies MRC for Φ. The spark of a matrix is the size +of the smallest linearly dependent subset of the columns and the spark of a family +{φi}i∈Im is defined as the spark of its synthesis matrix Φ. Also, the family {φi}i∈Im, +m ≥ n is said full spark if it has the spark n + 1. It is shown that if m ≥ n, then +the set of full spark frames is open and dense in the set of all frames [2]. + +Characterization of (weak) phase retrieval dual frames +3 +Suppose now the nonlinear mapping +MΦ : H → l2(I), +MΦ(f) = {|⟨f, φi⟩|2}i∈I +(1.2) +obtained by taking the absolute value element wise of the analysis operator. Let us +denote by H = H/ ∼ considered by identifying two vectors which are different in a +phase factor, i.e., f ∼ g whenever there exists a scalar θ with |θ| = 1 so that g = θf. +Obviously in a real Hilbert space we have H = H/{1, −1} and in the complex case +H = H/T, where T is the complex unit circle. So, the mapping MΦ can be extended +to H as MΦ( ˆf) = {|⟨f, φi⟩|2}i∈I, f ∈ ˆf = {g ∈ H : g ∼ f}. The injectivity of +the nonlinear mapping MΦ leads to the reconstruction of every signal in H up +to a constant phase factor from the modula of its frame coefficients. In [8], the +authors investigated the injectivity of MΦ in finite dimensional real Hilbert spaces +and moreover, it was proven that 4n − 2 measurements suffice for the injectivity in +n-dimensional complex Hilbert spaces. Indeed, the injectivity of the mapping MΦ +is equivalent to the following definition: +Definition 1.1. A family of vectors Φ = {φi}i∈I in H does phase retrieval if +whenever f, g ∈ H satisfy +|⟨f, φi⟩| = |⟨g, φi⟩|, +(i ∈ I) +(1.3) +then there exists a scalar θ with |θ| = 1 so that f = θg. +If for every f, g ∈ H, which satisfy (1.3) we get ∥f∥ = ∥g∥, then it said Φ to +do norm retrieval. Clearly, if Φ does phase (norm) retrieval, then so does αifi for +every 0 < αi, i ∈ I. Also, tight frames do norm retrieval. Moreover, phase retrieval +implies norm retrieval, but the converse fails. For example every orthonormal basis +does norm retrieval, but fails at phase retrieval. +The following result states that for two equivalent frames Φ and ψ, the injec- +tivity of MΦ and MΨ are the same. +Theorem 1.2. [8] A family Φ = {φi}i∈I in H does phase retrieval if and only if +{Uφi}i∈I does phase retrieval for every invertible operator U on H. +Applying the above theorem, shows that a frame does phase retrieval if and +only if its canonical dual does phase retrieval. +Definition 1.3. [8] A family of vectors Φ = {φi}i∈I in H has the complement +property if for every σ ⊂ I either span{φi}i∈σ = H or span{φi}i∈σc = H. +As we stated, a fundamental classification of frames which do phase retrieval +was presented for finite dimensional real case in [8] and then for infinite dimensional +case in [11] as follows: +Theorem 1.4. A family Φ = {φi}i∈I in a real Hilbert space H does phase retrieval +if and only if it has the complement property. +As an immediate result of the above theorem, every phase retrieval frame in +real Hilbert space H is satisfied in MRC for (n − 1)-erasures. +Proposition 1.5. [8] If Φ = {φi}i∈Im does phase retrieval in Rn, then m ≥ 2n−1. +If m ≥ 2n−1 and Φ is full spark then φ does phase retrieval. Moreover, {φi}i∈I2n−1 +does phase retrieval if and only if Φ is full spark. + +4 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +Two vectors x = {xi}i∈In and y = {yi}i∈In in Hn weakly have the same phase +if there is a |α| = 1 so that phase(xi) = αphase(yi), for all i ∈ In which xi ̸= 0 ̸= yi. +Definition 1.6. A family Φ = {φi}i∈Im in Hn does weak phase retrieval if for any +x, y ∈ Hn with |⟨x, φi⟩| = |⟨y, φi⟩| for all i ∈ Im, then x and y weakly have the same +phase. +It is shown that if Φ = {φi}i∈Im does weak phase retrieval in Rn, then m ≥ +2n − 2 [10]. Also, clearly phase retrieval implies weak phase retrieval property, +although the converse does not hold in general. As a simple example {(1, 1), (1, −1)} +does weak phase retrieval for R2, see[1], but clearly does not phase retrieval. +Now, we state the concept of a generic set; A subset Ω ⊆ Rn is called generic +whenever there exists a nonzero polynomial p(x1, ..., xn) so that +Ωc ⊆ {(x1, ..., xn) ∈ Rn : p(x1, ..., xn) = 0}. +It is known that generic sets are open, dense and full measure. Furthermore, a +generic set in Cn is defined as a generic set in R2n. +Applying the linear operator introduced in [6] gives an equivalent condition +for injectivity of MΦ, defined by (1.2). More precisely, let {φi}i∈Im be a family of +vectors in Cn and Hn×n denotes the space of all n×n Hermitian matrices. Consider +the operator ΛΦ : Hn×n → Rm by ΛΦ(A) = {⟨A, φi ⊗ φi⟩}i∈Im, in which φi ⊗ φi is +the rank one projection onto span{φi}. It is easily proven that ΛΦ(f ⊗ f) = MΦ(f) +and this equality has a key role for characterizing the injectivity of MΦ, [6, 12], +See also [15]. So, we get the following equivalent conditions for a frame to do phase +retrieval. +Corollary 1.7. Let Φ = {φi}i∈Im be a frame in Hn, then the followings are equiv- +alent; +(i) Φ does phase retrieval. +(ii) MΦ is injective. +(iii) ΛΦ|B1 is injective, where B1 denotes rank one matrices. +(iv) There exists no rank 2 matrix in the null space of ΛΦ. +Proof. (i) ⇔ (ii) is obvious by the definition. For (ii) ⇔ (iv) we just note that, due +to the completeness of Φ in Hn, the rank one matrices B1 = {f ⊗f : +f ∈ Hn} can +not be in the null space of ΛΦ. Indeed, ΛΦ(f ⊗f) = MΦ(f) = 0 implies that f ⊥ φi, +for all i ∈ Im and so f = 0. So, as a result of Lemma 5.5 of [6], the injectivity of MΦ +is equivalent to the statement that; there is no rank 2 matrix in the null space of +ΛΦ. On the other hand, the injectivity of MΦ along with the fact that the mapping +γ : Hn → B1, γ(f) = f ⊗ f is invertible, deduce that the linear operator ΛΦ|B1 +is also injective. Moreover, for a left inverse LΦ of MΦ, the mapping γLΦ is a left +inverse for ΛΦ|B1. Conversely, for a left inverse ΓΦ of ΛΦ|B1, the mapping γ−1ΓΦ is +a left inverse for MΦ, this completes the proof of (ii) ⇔ (iii). +□ +This paper is organized as follows: Section 2 is devoted to characterizing phase +retrieval dual frames and full spark dual frames. In this section, for some classes of +frames, we show that the family of all ( full spark) phase retrieval dual frames is +open and dense in the set of all dual frames. Moreover, we present several examples + +Characterization of (weak) phase retrieval dual frames +5 +in this regard. In Section 3, we survey weak phase retrieval problem and investigate +some equivalent conditions for identifying phase and weak phase retrieval frames. +We also, obtain a sufficient conditions on a family of weak phase retrieval frames to +constitute a frame and its canonical dual yields weak phase retrieval, as well. +2. Phase Retrieval Dual Frames +In this section, we address the problem that, given a phase retrieval frame Φ, charac- +terize phase retrieval dual frames of Φ in finite dimensional Hilbert spaces. For some +classes of frames we show that phase retrieval dual frames of a given frame are dense +in the set of all dual frames. For a frame Φ we denote the set of all its dual frames +of Φ by DΦ and the subset of phase retrieval dual frames is denoted by PDΦ. We +first investigate the relationship between phase retrieval duals of equivalent frames, +which is a very useful tool for the main results of this section. +Lemma 2.1. Suppose that Φ = {φi}i∈Im is a frame for Hn and T is an invertible +operator on Hn. Then, +(i) DT Φ = (T ∗)−1DΦ. +(ii) PDT Φ = (T ∗)−1PDΦ. +Proof. It is known that T Φ is a frame with ST Φ = T SΦT ∗ [13], and so a simple +computation assures that DT Φ = {(T ∗)−1G : G ∈ DΦ} = (T ∗)−1DΦ. Moreover, +(ii) is given as an immediate result of (i) along with Theorem 1.2. +□ +Remark 2.2. If Φ = {φi}i∈Im is a frame for Hn so that E(Φ) = k, which E(Φ) +denotes the excess of Φ, then there exist i1, ..., ik so that Φ \ {φij}k +j=1 constitutes +a Riesz basis for Hn. Therefore, applying the above notations and without loss of +generality, we may consider Φ = {φi}i∈In∪{φi}m +i=n+1, where {φi}i∈In is a Riesz basis +for Hn. In this point of view, Φ is indeed equivalent to a form as {δi}i∈In ∪{˜φi}m +i=n+1 +with redundant elements {˜φi}m +i=n+1. +In the next theorem we identify the set of dual frames of a frame {φi}i∈I2n−1 +in n-dimensional real space and show that PDφ is an open and dense subset in Dφ. +Theorem 2.3. Let φ = {φi}i∈I2n−1 be a phase retrieval frame in Rn. Then PDφ +is an open and dense subset in Dφ. +Proof. First let {φi}i∈In be the standard orthonormal basis of Rn and φj = � +i∈In aj +iφi, +j = n + 1, ..., 2n − 1, for some non-zero coefficients {aj +i}i∈In. Due to the fact that +DΦ = + + +{S−1 +Φ φi + ui}i∈I2n−1 : +� +i∈I2n−1 +⟨f, φi⟩ui = 0, for all f ∈ Rn + + + +by putting f = φj, j = 1, ..., 2n−1 we observe that (2n−1)×n matrix [u1|u2|...|u2n−1]T +is in the null space of GT +Φ, the transposed Gram matrix of Φ. The fact that, + +6 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +dim(nullGΦ) = n − 1; assures that just n − 1 vectors of {ui}i∈I2n−1 can be in- +dependent. More precisely, +u1 + +2n−1 +� +i=n+1 +⟨φ1, φi⟩ui = 0, +... +un + +2n−1 +� +i=n+1 +⟨φn, φi⟩ui = 0. +So, by choosing {uj}2n−1 +j=n+1 we get +ui = − +2n−1 +� +j=n+1 +aj +iuj, +(i ∈ In), +(2.1) +in which aj +i = ⟨φi, φj⟩, for n + 1 ≤ j ≤ 2n − 1. Hence, every dual frame is uniquely +constructed by the following vector +U = [un+1, ..., u2n−1] ∈ Rn(n−1). +(2.2) +Considering DΦ as a metric space by d(G, H) = � +i∈I2n−1 ∥gi − hi∥, we define the +mapping ξ : DΦ → Rn(n−1) by ξ(G) = Ug, where Ug ∈ Rn(n−1) is the unique +sequence associated to G ∈ DΦ as in (2.2). Then, clearly ξ is well-defined and +injective. Also, take A ∈ Rn(n−1), then put un+1 = {A1, ..., An},..., u2n−1 = +{An2−2n+1, ..., An(n−1)} and construct ui, i ∈ In by (2.1). Thus we get {S−1 +Φ φi + +ui}i∈I2n−1 ∈ DΦ, i.e., ξ is a bijective map. Moreover, ξ is a Lipschitz function. Let +G = {S−1 +Φ φi + ui}i∈I2n−1 and H = {S−1 +Φ φi + vi}i∈I2n−1 be dual frames of Φ with +the associated Ug and Uh obtained as in (2.2), respectively. Then, +∥ξ(G) − ξ(H)∥ = ∥Ug − Uh∥ = +2n−1 +� +i=n+1 +∥ui − vi∥ ≤ +� +i∈I2n−1 +∥ui − vi∥ = d(G, H). +What is more, ξ is a bi-Lipschitz function, but we will not need this fact. Now, +we note that the only cases in which a dual frame {gi}i∈I2n−1 fails to do phase +retrieval is associated to det[gi1| +... +|gin] = 0 for some index set {ij}n +j=1 ⊂ I2n−1, +by Theorem 1.4. Multiplying these equations yields an n(n−1)-variable polynomial +in terms of un+1, ..., u2n−1, denoted by Pφ. Therefore, +PDΦ = ξ−1{U ∈ Rn(n−1) : Pφ(U) ̸= 0}, +(2.3) +that means PDΦ is open in DΦ. To show PDΦ is, moreover, dense in Dφ, let ǫ > 0 +and G = {gi}i∈I2n−1 = {S−1 +Φ Φi + ui}i∈I2n−1 be a dual frame in DΦ \ PDΦ. Then +G is dependent just in vectors {ui}2n−1 +i=n+1 such that Pφ({ui}2n−1 +i=n+1) = 0. Take a +sequence {{vk +i }2n−1 +i=n+1}k∈N out of the roots of the polynomial Pφ in Rn(n−1) so that +limk→∞{vk +i }2n−1 +i=n+1 = {ui}2n−1 +i=n+1. Also, put vk +i = − �2n−1 +j=n+1 aj +ivk +j , for all 1 ≤ i ≤ n +and k ∈ N. Then, {vk +i }i∈I2n−1 satisfies (2.1) and so the sequence {hk +i }k∈N,i∈I2n−1 +associated to {{vk +i }2n−1 +i=n+1}k∈N defined by hk +i = S−1 +Φ Φi + vk +i , i ∈ I2n−1 constitutes a +dual frame of Φ, for all k ∈ N. Furthermore, there exists k0 ∈ N so that for k ≥ k0 +d +� +{vk +i }2n−1 +i=n+1}, {ui}2n−1 +i=n+1} +� += +2n−1 +� +i=n+1 +∥vk +i − ui∥ < ǫ. + +Characterization of (weak) phase retrieval dual frames +7 +Hence, we can write +d({hk +i }k∈N,i∈I2n−1, {gi}i∈I2n−1) += +� +i∈I2n−1 +∥vk +i − ui∥ += +� +i∈In +∥vk +i − ui∥ + +2n−1 +� +i=n+1 +∥vk +i − ui∥ += +� +i∈In +∥ − +2n−1 +� +j=n+1 +aj +i(vk +j − uj)∥ + +2n−1 +� +i=n+1 +∥vk +i − ui∥ +≤ +� +i∈In +2n−1 +� +j=n+1 +|aj +i|∥vk +j − uj∥ + +2n−1 +� +i=n+1 +∥vk +i − ui∥ +≤ +(α + 1) +2n−1 +� +j=n+1 +∥vk +j − uj∥ < ǫ(α + 1), +for k ≥ k0, i.e., PDΦ is dense in DΦ. +Now, applying Lemma 2.1 and Remark 2.2 gives the result in general case. +In fact, every frame Ψ = {ψi}i∈I2n−1 is equivalent to a frame in the form of Φ = +{δi}i∈In ∪ {φi}2n−1 +i=n+1, as we discussed in Remark 2.2. So, there exists an invertible +operator T on Rn so that Ψ = T Φ, consequently by Lemma 2.1 we get +PDΨ = PDT Φ = (T ∗)−1PDΦ ⊇ (T ∗)−1PDΦ = (T ∗)−1DΦ = DΨ, +i.e., PDΨ = DΨ. And using (2.3) +PDΨ = (T ∗)−1PDΦ = (T ∗)−1ξ−1{U ∈ Rn(n−1) : Pφ(U) ̸= 0}, +i.e., PDΨ is open in DΨ. This completes the proof. +□ +An analogous approach to the proof of Theorem 2.3 deduces that for any full +spark frame Φ in Rn with E(Φ) = 1, the set of all full spark dual frames is embedded +into a generic set in Rn. +Corollary 2.4. Let Φ = {φi}n+1 +i=1 be a full spark frame for Rn. Then the set of all +full spark dual frames of Φ is an open and dense subset of DΦ and is embedded into +a generic set in Rn. +Proof. Applying Remark 2.2, a full spark frame Φ = {φi}n+1 +i=1 of Rn is equivalent +to ˜Φ := {δi}n +i=1 ∪ {�n +i=1 αiδi} for non-zero scalars αi, i ∈ In, and in this case the +frame operator is as follows: +S˜Φ = + + +1 + α2 +1 +α1α2 +... +α1αn +α1α2 +1 + α2 +2 +... +α2αn +. +. +α1αn +α2αn +... +1 + α2 +n + + + +8 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +Also, every dual frame of ˜Φ is in the form of {S−1 +˜Φ ˜φi + ui}n+1 +i=1 so that TuT ∗ +˜Φ = 0. +By taking un+1 = [x1, ..., xn]T , we get ui = −αiun+1 for all i ∈ In. Hence, D˜Φ is +the set of all {gi}n+1 +i=1 so that +gk +i = +� +−ααiαk − αixk +k ̸= i, +α(1 + � +j̸=i α2 +j) − αixk +k = i, +where α = +1 +detS˜Φ += +1 +1 + �n +i=1 α2 +i +, gk +i denotes kth coordinate of gi, i ∈ In, and +gn+1 = {ααi + xi}n +i=1. +This shows that, every dual frame is associated to n-variable {xi}i∈In and a dual +frame is full spark frame except the cases that the sequence un+1 = {xi}i∈In belongs +to the roots of the polynomial constructed by det[gi1| +... +|gin] = 0, for some +index set {ij}n +j=1 ⊂ In+1. Multiplying these n + 1 polynomials yield an n-variable +polynomial P˜Φ(x1, ..., xn). So, every full spark dual frame of ˜Φ is obtained by a +generic choice of un+1 out of the roots of the polynomial P˜Φ. Hence, by a similar +approach to the proof of Theorem 2.3, and considering the bijection map ξ : D˜Φ → +Rn defined as ξ(G) = un+1, where G = {S−1 +˜Φ ˜φi + ui}n+1 +i=1 , the complement of all full +spark dual frames of Φ in DΦ is equivalent to the set of roots of P˜Φ, and so, the set +of full spark dual frames of Φ is embedded into a generic set in Rn by ξ. In general +case, if Φ = T ˜Φ, for an invertible operator T on Rn by using Lemma 2.1, the set of +full spark dual frames of Φ, FDΦ, is the same (T ∗)−1FD˜Φ, and this completes the +proof. +□ +2.1. Examples +Example 2.5. Suppose that φ = {φi}3 +i=1 is a full spark frame for R2. Since E(φ) = +1 and equivalent frames do the same phase retrieval we assume that φ is equivalent +to {δ1, δ2, α1δ1 + α2δ2}, in which both α1 and α2 are non-zero. In this case, +Dφ = + + + + + +α(1 + α2 +2) − α1x +−αα1α2 − α1y + + , + + +−αα1α2 − α2x +α(1 + α2 +1) − α2y + + , + + +αα1 + x +αα2 + y + + ; x, y ∈ R + + + +where α = +1 +1 + α2 +1 + α2 +2 +. It is worthy of note that, all dual frames in this case are +full spark and so phase retrieval except dual frames obtained by (x, y) ∈ R2 on +three distinct lines with slopes of 0, ∞ and −α1/α2 such as x = −αα1, y = −αα2 +and α1x + α2y = α. Considering +(x, y) ∈ R2 \ {(x, y) : (x + αα1)(y + αα2)(α1x + α2y − α) = 0} +we get a generic choice of {ui}i∈I4 in R2. The fact that all dual frames of φ are +translations of this family by {S−1 +φ φi}i∈I4 shows that full spark dual frames are +dense in DΦ and full measure. As a special case, considering α1 = α2 = 1 we get +the phase retrieval frame φ = {δ1, δ2, δ1 + δ2} and we can see that dual frames of +Φ do phase retrieval for all (x, y) ∈ R2 except on the lines x = −1 +3 , y = −1 +3 +and + +Characterization of (weak) phase retrieval dual frames +9 +y = 1 +3 − x. Put x = 0, y = 2/3 we get a phase retrieval dual Ψ and x = 1, y = −2 +3 , +satisfying in y = 1 +3 − x, gives a non-phase retrieval dual frame G. See Figure 1. +Example 2.6. Let φ = {φi}4 +i=1 be a full spark frame for R3. In this case φ is +equivalent to the frame +φ := {δ1, δ2, δ3, α1δ1 + α2δ2 + α3δ3 +α1, α2, α3 ̸= 0}. +The set of all dual frames of φ is in the form of Dφ = {g1, g2, g3, g4} where +g1 = + + +α(1 + α2 +2 + α2 +3) − α1x +−αα1α2 − α1y +−αα1α3 − α1z + + +, g2 = + + +−αα1α2 − α2x +α(1 + α2 +1 + α2 +3) − α2y +−αα2α3 − α2z + + +, +g3 = + + +−αα1α3 − α3x +−αα2α3 − α3y +α(1 + α2 +1 + α2 +3) − α3z + + +, g4 = + + +αα1 + x +αα2 + y +αα3 + z + + +where α = +1 +1 + α2 +1 + α2 +2 + α2 +3 +and x, y, z ∈ R are arbitrary. All dual frames of Φ are +full spark except the cases det[gi|gj|gk] = 0, for i, j, k ∈ I4, in which (x, y, z) is belong +to the four distinct planes x = −αα1, y = −αα2, z = −αα3 and α1x+α2y+α3z = α. +In a similar way, by choosing +(x, y, z) ∈ R3 \ {(x, y, z) : (x + αα1)(y + αα2)(z + αα3)(α1x + α2y + α3z − α) = 0} +we can say that full spark dual frames are translations of the canonical dual by a +generic choice of the family {ui}i∈I4. + +Figure1 +2 +1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +FrameΦ +Phaseretrievaldual +Non-phase retrieval dual G +-2 +-2 +-1.5 +-1 +0.5 +0 +0.5 +1 +1.5 +210 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +Example 2.7. Consider φ = {δ1, δ2, δ3, �3 +i=1 δi, δ1−δ2+δ3} as a full spark frame for +R3. In this case, all dual frames are constructed by a generic choice of +� u4 +u5 +� +∈ R6. +In fact, by putting u4 = [x1 +y1 +z1]T and u5 = [x2 +y2 +z2]T , the set of all dual +frames of φ, are given by +g1 = + + +3 +5 − x1 − x2 +−y1 − y2 +−2 +5 − z1 − z2 + + +, g2 = + + +x2 − x1 +1 +3 + y2 − y1 +−z2 − z1 + + +, +g3 = + + +−2 +5 − x1 − x2 +−y1 − y2 +3 +5 − z1 − z2 + + +, g4 = + + +1 +5 + x1 +1 +3 + y1 +1 +5 + z1 + + +, g5 = + + +1 +5 + x1 +−1 +3 + y2 +1 +5 + z2 + + +where x1, x2, y1, y2, z1, z2 are obtained arbitrarily from R. And all cases in which +a dual frame fails to do phase retrieval is associated to the roots of a 6-variable +polynomial in R6 given by multiplying of the polynomials as det[gi1|gi2|gi3] = 0, for +all index set {ij}3 +j=1 ⊂ I5. As one case, det[g2|g4|g5] = 0 deduces that +z1 + x2 − 3x1 +5 +− 3z2 +5 ++ 3x2z1 − 3x1z2 = 0, +and the roots of this equation, which are associated to a family of non-phase retrieval +dual frames, constitute a surface in R3. +3. Weak Phase Retrieval +The main result of this section is to obtain some equivalent conditions on a family +of vectors to do phase retrieval in terms of weak phase retrieval. This also derives a +relationship between, phase, weak phase and norm retrieval. First we present some +properties of a family of vectors to do weak phase retrieval. +Proposition 3.1. Assume that Φ = {φi}i∈Im is a frame in Hn. Then Φ does weak +phase retrieval if and only if PΦ does weak phase retrieval, for every 2-dimensional +orthogonal projection P on Hn. +Proof. In case Φ does weak phase retrieval, it is simple to see that PΦ does weak +phase retrieval, for every orthogonal projection P on Hn, as well, see also [1]. Con- +versely, let x, y ∈ Hn and |⟨x, φi⟩| = |⟨y, φi⟩|. Then by the assumption that PΦ +does weak retrieval for the projection P of Hn onto the closed subspace span{x, y}, +implies that x and y weakly have the same phase. +□ + +Characterization of (weak) phase retrieval dual frames +11 +It is shown that [1] a weak phase retrieval family in Rn spans the space that +means such a family constitutes a frame for Rn. In the complex case Cn there is +no result available. In the following, we present sufficient condition in this regard +which is also useful for the main result of this section. +Proposition 3.2. Let {φi}i∈Im be a family of vectors in Cn so that UΦ does weak +phase retrieval for every unitary operator U on Cn. Then {φi}i∈Im is a frame for +Cn. +Proof. Assume that there exists an element x ∈ {φi}⊥ +i∈Im, and get an ONB for +Cn as {ei}i∈In with e1 = +x +∥x∥. Also, take 0 ̸= y ∈ span{φi}i∈Im so there exists +{βi}n +i=2 ⊂ C, with some non-zero elements such that y = �n +i=2 βiei. Define +µ : Cn → Cn; +µ(ei) = δi +where {δi}i∈In is the standard ONB of Cn. Clearly µ is a unitary operator and +µ(x + y) = µ(x) + µ(y) = ∥x∥δ1 + +n +� +i=2 +βiδi, +µ(−x + y) = −∥x∥δ1 + +n +� +i=2 +βiδi. +On the other hand, +|⟨µ(x + y), µφi⟩| += +|⟨x + y, φi⟩| += +|⟨−x + y, φi⟩| += +|⟨µ(−x + y), µφi⟩|, +for all i ∈ Im. The assumption that µΦ does weak phase retrieval deduces that +µ(x + y) and µ(−x + y) weakly have the same phase that is impossible, unless +x = 0. Therefore, Φ is a frame for Cn. +□ +Theorem 3.3. Let Φ = {φi}i∈Im be a family of vectors in Hn. Then the followings +are equivalent; +(i) Φ does phase retrieval. +(ii) UΦ does phase retrieval for every invertible operator U on Hn. +(iii) UΦ does norm retrieval for every invertible operator U on Hn. +(iv) UΦ does weak phase retrieval for every invertible operator U on Hn. +Proof. The equivalency of (i), (ii), and (iii) was proven in [9]. Also, we clearly have +(i) ⇒ (ii) ⇒ (iv). So, it is sufficient to show that (iv) ⇒ (i). Assuming that UΦ +does weak phase retrieval for all invertible operators on Hn, specially we imply that +φ does weak phase retrieval. Moreover, Φ is a frame for Hn, i.e., Φ spans Hn by +Proposition 3.2. In the sequel, we are going to show that Φ yields phase retrieval. +On the contrary, let there exist non-zero elements x and y in Hn so that +|⟨x, φi⟩| = |⟨y, φi⟩|, +(i ∈ Im) +(3.1) +but y ̸= cx for any |c| = 1. Since Φ does weak phase retrieval, there exists some |θ| = +1 so that θphase(xj) = phase(yj) for all j ∈ In. The assumption that y ̸= θx implies +|xj| ̸= |yj| for some j ∈ In. Since Φ is a frame, the equality in (3.1) is non-zero for +some i then y = cx immediately implies that x and y are linearly independent and + +12 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +we can consider a basis for Hn containing x and y as {x, y, e3, ..., en}. We face the +following cases +Case 1. If x and y are disjointly supported, then there exist no non-zero com- +mon coordinates. Without loss of generality we let x1, y2 ̸= 0, and so x2 = y1 = 0. +Define U : Hn → Hn so that Ux = x − y, Uy = x + y and Uek = ek, 3 ≤ k ≤ n. +Then, U is a bounded invertible operator on Hn and +|⟨Ux, (U −1)∗φi⟩| = |⟨Uy, (U −1)∗φi⟩|, +(i ∈ Im), +by (3.1) and the invertibility of U. However, Ux and Uy have not weakly the same +phase due to phase(Ux)1 = phase(Uy)1 and phase(Ux)2 = eiπphase(Uy)2. This +contradicts the assumption that (U −1)∗Φ does weak phase retrieval. +Case 2. Let x and y have one non-zero coordinate in common in ith index. If +yl = 0 = xl for all l ̸= i then by (3.1) and using the assumption |xi| = |yi|, i.e., +y = θx that is a contradiction. Thus, there exists l ̸= i so that yl ̸= 0 or xl ̸= 0. +Without loss of generality we let yl ̸= 0 and |xl| +|yl| < |xi| +|yi| . In fact, if in all common +non-zero elements |xi| +|yi| = c then y = θ +cx, which contradicts by the assumption. +So, we get ǫ > 0 so that |xl| +|yl| < ǫ < |xi| +|yi| . Define U : Hn → Hn so that +Ux = θx − ǫy, Uy = θx + ǫy and Uek = ek, 3 ≤ k ≤ n. Moreover, +(Ux)j = (|xj| − ǫ|yj|)αjθ, +(Uy)j = (|xj| + ǫ|yj|)αjθ. +(j ∈ In) +where αj = phase(xj). Hence, we obtain phase(Ux)i = phase(Uy)i, however +phase(Ux)l = eiπphase(Uy)l. That leads to a contradiction similar to the previ- +ous case, as required. +□ +The following result gives a sufficient condition on a family of frames so that +their canonical dual also yields weak phase retrieval. +Proposition 3.4. Let Φ = {φi}i∈Im be a frame in Hn with diagonal frame operator. +Then Φ does weak phase retrieval if and only if its canonical dual does so. +Proof. Suppose that α1, ..., αn be the diagonal elements of SΦ, respectively. Ob- +viously, αi > 0, for all i ∈ In. Let Φ does weak phase retrieval, take x, y ∈ Hn +such that |⟨x, S−1 +Φ φi⟩| = |⟨y, S−1 +Φ φi⟩|. Then, we get S−1 +Φ x = (x1/α1, ..., xn/αn) and +S−1 +Φ y = (y1/α1, ..., yn/αn) weakly have the same phase. Consequently, x and y +weakly have the same phase, as well. The converse is implied by a similar explana- +tion. +□ +Funding. The authors have not disclosed any funding. +Data Availability Statement. Data sharing not applicable to this article +as no datasets were generated during the preparation of this paper. +Conflict-of-interest. This work does not have any conflicts of interest. + +Characterization of (weak) phase retrieval dual frames +13 +References +1. F. Akrami, P.G. Casazza, A. Rahimi, M.A. Hasankhanifard and B. Daraby, A note +on (weak) phase and norm retrievable real Hilbert space frames and projections. +arXiv:2110.06868v1 math.FA. +2. B. Alexeev, J. Cahill, D. Mixon, Full spark frames, J. Fourier Anal. Appl. 18(6) (2012), +1167–1194. +3. F. Arabyani Neyshaburi, A. Arefijamaal and Gh. Sadeghi, Extreme points and identi- +fication of optimal alternate dual frames. Linear Algebra Appl. 549 (2018), 123–135. +4. F. Arabyani Neyshaburi, A. Arefijamaal and Gh. 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Han, Optimal dual frames for erasures. Linear Algebra Appl. 432(1) +(2010), 471–482. +18. J. Kovacevic, M. Puschel, Real, tight frames with maximal robustness to erasures. Book +Chapter, in: J.A. Storer, M. Cohn (Eds.), Proceedings of DCC 2005: Data Compression +Conference: The Institute of Electrical and Electronics Engineers, Inc., Los Alamitos, +CA, (2005), 63–72. +19. S. Pehlivan, D. Han and R. Mohapatra, Linearly connected sequences and spectrally +optimal dual frames for erasures. J. Functional Anal. 265(11) (2013), 2855-2876. +20. H.L. van Trees, Optimum Array Processing. Wiley, New York, 2002. + +14 +F. Arabyani-Neyshaburi, A. Arefijamaal and R.Kamyabi-Gol +21. Y. Wang, Z. Xu, Phase retrieval for sparse signals. Appl. Comput. Harmon. Anal. 37 +(2014), 531–544. +Fahimeh Arabyani-Neyshaburi1 +e-mail: f.arabiani@um.ac.ir, fahimeh.arabyani@gmail.com +Ali Akbar Arefijamaal2 +e-mail: arefijamaal@hsu.ac.ir;arefijamaal@gmail.com +Rajab Ali Kamyabi-Gol3,∗ +e-mail: kamyabi@um.ac.ir + diff --git a/xdE4T4oBgHgl3EQfYAwp/content/tmp_files/load_file.txt b/xdE4T4oBgHgl3EQfYAwp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3d09523d395a644d981ea50bf5fde16bb710833 --- /dev/null +++ b/xdE4T4oBgHgl3EQfYAwp/content/tmp_files/load_file.txt @@ -0,0 +1,633 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf,len=632 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='05045v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='FA] 12 Jan 2023 Characterization of (weak) phase retrieval dual frames Fahimeh Arabyani-Neyshaburi1, Ali Akbar Arefijamaal2 and Rajab Ali Kamyabi-Gol3,∗ 1 Department of Mathematical sciences, Ferdowsi University of Mashhad, Mashhad, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 2 Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 3 Department of Mathematical sciences, Faculty of Math, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS), Mashhad, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Email: kamyabi@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='ir Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Recovering a signal up to a unimodular constant from the mag- nitudes of linear measurements has been popular and well studied in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' However, numerous unsolved problems regarding phase retrieval still exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Given a phase retrieval frame, may the family of phase retrieval dual frames be classified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' And is such a family dense in the set of dual frames?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Can we present the equivalent conditions for a family of vectors to do weak phase retrieval in complex Hilbert space case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' What is the connection between phase, weak phase and norm retrieval?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In this context, we aim to deal with these open problems concerning phase retrieval dual frames, weak phase retrieval frames, and specially investigate equivalent conditions for identifying these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' We provide some characterizations of alternate dual frames of a phase retrieval frame which yield phase retrieval in finite dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' More- over, for some classes of frames, we show that the family of phase retrieval dual frames is open and dense in the set of dual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, we study weak phase retrieval problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Among other things, we obtain some equivalent conditions on a family of vectors to do phase retrieval in terms of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Mathematics Subject Classification 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Primary 42C15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Secondary 41A58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Phase retrieval, weak phase retrieval, dual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Introduction and Preliminaries Signal reconstruction without using phase is a longstanding conjecture, specially with regard to speech recognition systems which was first introduced from mathe- matical point of view by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Balan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Casazza and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Edidin in [8], and has been a very popular topic in recent years due to so many applications of X-ray, electron microscopy, optics, image processing and much more [5, 14, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' To present the problem in a more precise approach, we first briefly state some basic definitions and preliminaries in relevant areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then we propound a number of problems to address phase retrieval frames in some new aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Throughout this paper, we suppose that H denotes a separable Hilbert space, Hn denote an n-dimensional real or complex Hilbert space and we use Rn and Cn whenever it is necessary to differentiate between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, we consider the notations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Im = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', m} and {δi}i∈In as the standard orthonormal basis of Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A family of vectors Φ := {φi}i∈I in H is called a frame if there exist the constants 0 < AΦ ≤ BΦ < ∞ such that AΦ∥f∥2 ≤ � i∈I |f, φi⟩|2 ≤ BΦ∥f∥2, (f ∈ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) The constants AΦ and BΦ are called frame bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The sequence {φi}i∈I is said to be a Bessel sequence whenever the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A frame {φi}i∈I is called A-tight frame if A = AΦ = BΦ, and in the case of AΦ = BΦ = 1 it is called a Parseval frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Given a frame Φ = {φi}i∈I, its Grammian matrix formed by the inner product of the frame vectors is as GΦ = [⟨φi, φj⟩]i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The frame operator is defined by SΦf = � i∈I⟨f, φi⟩φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is a bounded, invertible, and self-adjoint operator [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, the synthesis operator TΦ : l2 → H is defined by TΦ{ci} = � i∈I ciφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The frame operator can be written as SΦ = TφT ∗ φ where T ∗ Φ : H → l2, the adjoint of T , given by T ∗ Φf = {⟨f, φi⟩}i∈I is called the analysis operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The family {S−1 Φ fi}i∈I is also a frame for H, called the canonical dual frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In general, a frame {ψi}i∈I ⊆ H is called an alternate dual or simply a dual for {φi}i∈I if f = � i∈I⟨f, ψi⟩φi, for f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' All frames have at least a dual, the canonical dual, and redundant frames have an infinite number of alternate dual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' We denote the excess of a frame Φ by E(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is known that every dual frame is of the form {S−1 Φ φi +ui}i∈I, where {ui}i∈I is a Bessel sequence that satisfies � i∈I⟨f, ui⟩φi = 0, for all f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also recall that, two frames Φ and Ψ are equivalent if there exists an invertible operator U on H so that Ψ = UΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' See [13, 16] for more detailed information on frame theory and [3, 4, 7, 17, 18, 19] for the importance of duality principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Consider a frame {φi}i∈Im in a Hilbert space Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A finite set of indices σ ⊂ Im satisfies the minimal redundancy condition (MRC) whenever {φi}i∈σc remains to be a frame for Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Furthermore, we say Φ satisfies MRC for r-erasures if every subset σ ⊂ Im, |σ| = r satisfies MRC for Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The spark of a matrix is the size of the smallest linearly dependent subset of the columns and the spark of a family {φi}i∈Im is defined as the spark of its synthesis matrix Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, the family {φi}i∈Im, m ≥ n is said full spark if it has the spark n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is shown that if m ≥ n, then the set of full spark frames is open and dense in the set of all frames [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Characterization of (weak) phase retrieval dual frames 3 Suppose now the nonlinear mapping MΦ : H → l2(I), MΦ(f) = {|⟨f, φi⟩|2}i∈I (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2) obtained by taking the absolute value element wise of the analysis operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let us denote by H = H/ ∼ considered by identifying two vectors which are different in a phase factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', f ∼ g whenever there exists a scalar θ with |θ| = 1 so that g = θf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Obviously in a real Hilbert space we have H = H/{1, −1} and in the complex case H = H/T, where T is the complex unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, the mapping MΦ can be extended to H as MΦ( ˆf) = {|⟨f, φi⟩|2}i∈I, f ∈ ˆf = {g ∈ H : g ∼ f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The injectivity of the nonlinear mapping MΦ leads to the reconstruction of every signal in H up to a constant phase factor from the modula of its frame coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In [8], the authors investigated the injectivity of MΦ in finite dimensional real Hilbert spaces and moreover, it was proven that 4n − 2 measurements suffice for the injectivity in n-dimensional complex Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Indeed, the injectivity of the mapping MΦ is equivalent to the following definition: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A family of vectors Φ = {φi}i∈I in H does phase retrieval if whenever f, g ∈ H satisfy |⟨f, φi⟩| = |⟨g, φi⟩|, (i ∈ I) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3) then there exists a scalar θ with |θ| = 1 so that f = θg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' If for every f, g ∈ H, which satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3) we get ∥f∥ = ∥g∥, then it said Φ to do norm retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Clearly, if Φ does phase (norm) retrieval, then so does αifi for every 0 < αi, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, tight frames do norm retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, phase retrieval implies norm retrieval, but the converse fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' For example every orthonormal basis does norm retrieval, but fails at phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The following result states that for two equivalent frames Φ and ψ, the injec- tivity of MΦ and MΨ are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' [8] A family Φ = {φi}i∈I in H does phase retrieval if and only if {Uφi}i∈I does phase retrieval for every invertible operator U on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Applying the above theorem, shows that a frame does phase retrieval if and only if its canonical dual does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' [8] A family of vectors Φ = {φi}i∈I in H has the complement property if for every σ ⊂ I either span{φi}i∈σ = H or span{φi}i∈σc = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' As we stated, a fundamental classification of frames which do phase retrieval was presented for finite dimensional real case in [8] and then for infinite dimensional case in [11] as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A family Φ = {φi}i∈I in a real Hilbert space H does phase retrieval if and only if it has the complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' As an immediate result of the above theorem, every phase retrieval frame in real Hilbert space H is satisfied in MRC for (n − 1)-erasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' [8] If Φ = {φi}i∈Im does phase retrieval in Rn, then m ≥ 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' If m ≥ 2n−1 and Φ is full spark then φ does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, {φi}i∈I2n−1 does phase retrieval if and only if Φ is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol Two vectors x = {xi}i∈In and y = {yi}i∈In in Hn weakly have the same phase if there is a |α| = 1 so that phase(xi) = αphase(yi), for all i ∈ In which xi ̸= 0 ̸= yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A family Φ = {φi}i∈Im in Hn does weak phase retrieval if for any x, y ∈ Hn with |⟨x, φi⟩| = |⟨y, φi⟩| for all i ∈ Im, then x and y weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is shown that if Φ = {φi}i∈Im does weak phase retrieval in Rn, then m ≥ 2n − 2 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, clearly phase retrieval implies weak phase retrieval property, although the converse does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' As a simple example {(1, 1), (1, −1)} does weak phase retrieval for R2, see[1], but clearly does not phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Now, we state the concept of a generic set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' A subset Ω ⊆ Rn is called generic whenever there exists a nonzero polynomial p(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', xn) so that Ωc ⊆ {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', xn) ∈ Rn : p(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', xn) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is known that generic sets are open, dense and full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Furthermore, a generic set in Cn is defined as a generic set in R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Applying the linear operator introduced in [6] gives an equivalent condition for injectivity of MΦ, defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' More precisely, let {φi}i∈Im be a family of vectors in Cn and Hn×n denotes the space of all n×n Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Consider the operator ΛΦ : Hn×n → Rm by ΛΦ(A) = {⟨A, φi ⊗ φi⟩}i∈Im, in which φi ⊗ φi is the rank one projection onto span{φi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is easily proven that ΛΦ(f ⊗ f) = MΦ(f) and this equality has a key role for characterizing the injectivity of MΦ, [6, 12], See also [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, we get the following equivalent conditions for a frame to do phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let Φ = {φi}i∈Im be a frame in Hn, then the followings are equiv- alent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (i) Φ does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (ii) MΦ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (iii) ΛΦ|B1 is injective, where B1 denotes rank one matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (iv) There exists no rank 2 matrix in the null space of ΛΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (i) ⇔ (ii) is obvious by the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' For (ii) ⇔ (iv) we just note that, due to the completeness of Φ in Hn, the rank one matrices B1 = {f ⊗f : f ∈ Hn} can not be in the null space of ΛΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Indeed, ΛΦ(f ⊗f) = MΦ(f) = 0 implies that f ⊥ φi, for all i ∈ Im and so f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, as a result of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 of [6], the injectivity of MΦ is equivalent to the statement that;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' there is no rank 2 matrix in the null space of ΛΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' On the other hand, the injectivity of MΦ along with the fact that the mapping γ : Hn → B1, γ(f) = f ⊗ f is invertible, deduce that the linear operator ΛΦ|B1 is also injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, for a left inverse LΦ of MΦ, the mapping γLΦ is a left inverse for ΛΦ|B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Conversely, for a left inverse ΓΦ of ΛΦ|B1, the mapping γ−1ΓΦ is a left inverse for MΦ, this completes the proof of (ii) ⇔ (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ This paper is organized as follows: Section 2 is devoted to characterizing phase retrieval dual frames and full spark dual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In this section, for some classes of frames, we show that the family of all ( full spark) phase retrieval dual frames is open and dense in the set of all dual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, we present several examples Characterization of (weak) phase retrieval dual frames 5 in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In Section 3, we survey weak phase retrieval problem and investigate some equivalent conditions for identifying phase and weak phase retrieval frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' We also, obtain a sufficient conditions on a family of weak phase retrieval frames to constitute a frame and its canonical dual yields weak phase retrieval, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Phase Retrieval Dual Frames In this section, we address the problem that, given a phase retrieval frame Φ, charac- terize phase retrieval dual frames of Φ in finite dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' For some classes of frames we show that phase retrieval dual frames of a given frame are dense in the set of all dual frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' For a frame Φ we denote the set of all its dual frames of Φ by DΦ and the subset of phase retrieval dual frames is denoted by PDΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' We first investigate the relationship between phase retrieval duals of equivalent frames, which is a very useful tool for the main results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Suppose that Φ = {φi}i∈Im is a frame for Hn and T is an invertible operator on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, (i) DT Φ = (T ∗)−1DΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (ii) PDT Φ = (T ∗)−1PDΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is known that T Φ is a frame with ST Φ = T SΦT ∗ [13], and so a simple computation assures that DT Φ = {(T ∗)−1G : G ∈ DΦ} = (T ∗)−1DΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, (ii) is given as an immediate result of (i) along with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' If Φ = {φi}i∈Im is a frame for Hn so that E(Φ) = k, which E(Φ) denotes the excess of Φ, then there exist i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', ik so that Φ \\ {φij}k j=1 constitutes a Riesz basis for Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Therefore, applying the above notations and without loss of generality, we may consider Φ = {φi}i∈In∪{φi}m i=n+1, where {φi}i∈In is a Riesz basis for Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In this point of view, Φ is indeed equivalent to a form as {δi}i∈In ∪{˜φi}m i=n+1 with redundant elements {˜φi}m i=n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In the next theorem we identify the set of dual frames of a frame {φi}i∈I2n−1 in n-dimensional real space and show that PDφ is an open and dense subset in Dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let φ = {φi}i∈I2n−1 be a phase retrieval frame in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then PDφ is an open and dense subset in Dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' First let {φi}i∈In be the standard orthonormal basis of Rn and φj = � i∈In aj iφi, j = n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', 2n − 1, for some non-zero coefficients {aj i}i∈In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Due to the fact that DΦ = \uf8f1 \uf8f2 \uf8f3{S−1 Φ φi + ui}i∈I2n−1 : � i∈I2n−1 ⟨f, φi⟩ui = 0, for all f ∈ Rn \uf8fc \uf8fd \uf8fe by putting f = φj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', 2n−1 we observe that (2n−1)×n matrix [u1|u2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='|u2n−1]T is in the null space of GT Φ, the transposed Gram matrix of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The fact that, 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol dim(nullGΦ) = n − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' assures that just n − 1 vectors of {ui}i∈I2n−1 can be in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' More precisely, u1 + 2n−1 � i=n+1 ⟨φ1, φi⟩ui = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' un + 2n−1 � i=n+1 ⟨φn, φi⟩ui = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, by choosing {uj}2n−1 j=n+1 we get ui = − 2n−1 � j=n+1 aj iuj, (i ∈ In), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) in which aj i = ⟨φi, φj⟩, for n + 1 ≤ j ≤ 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Hence, every dual frame is uniquely constructed by the following vector U = [un+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', u2n−1] ∈ Rn(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2) Considering DΦ as a metric space by d(G, H) = � i∈I2n−1 ∥gi − hi∥, we define the mapping ξ : DΦ → Rn(n−1) by ξ(G) = Ug, where Ug ∈ Rn(n−1) is the unique sequence associated to G ∈ DΦ as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, clearly ξ is well-defined and injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, take A ∈ Rn(n−1), then put un+1 = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', An},.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', u2n−1 = {An2−2n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', An(n−1)} and construct ui, i ∈ In by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Thus we get {S−1 Φ φi + ui}i∈I2n−1 ∈ DΦ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', ξ is a bijective map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, ξ is a Lipschitz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let G = {S−1 Φ φi + ui}i∈I2n−1 and H = {S−1 Φ φi + vi}i∈I2n−1 be dual frames of Φ with the associated Ug and Uh obtained as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, ∥ξ(G) − ξ(H)∥ = ∥Ug − Uh∥ = 2n−1 � i=n+1 ∥ui − vi∥ ≤ � i∈I2n−1 ∥ui − vi∥ = d(G, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' What is more, ξ is a bi-Lipschitz function, but we will not need this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Now, we note that the only cases in which a dual frame {gi}i∈I2n−1 fails to do phase retrieval is associated to det[gi1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' |gin] = 0 for some index set {ij}n j=1 ⊂ I2n−1, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Multiplying these equations yields an n(n−1)-variable polynomial in terms of un+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', u2n−1, denoted by Pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Therefore, PDΦ = ξ−1{U ∈ Rn(n−1) : Pφ(U) ̸= 0}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3) that means PDΦ is open in DΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' To show PDΦ is, moreover, dense in Dφ, let ǫ > 0 and G = {gi}i∈I2n−1 = {S−1 Φ Φi + ui}i∈I2n−1 be a dual frame in DΦ \\ PDΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then G is dependent just in vectors {ui}2n−1 i=n+1 such that Pφ({ui}2n−1 i=n+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Take a sequence {{vk i }2n−1 i=n+1}k∈N out of the roots of the polynomial Pφ in Rn(n−1) so that limk→∞{vk i }2n−1 i=n+1 = {ui}2n−1 i=n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, put vk i = − �2n−1 j=n+1 aj ivk j , for all 1 ≤ i ≤ n and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, {vk i }i∈I2n−1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) and so the sequence {hk i }k∈N,i∈I2n−1 associated to {{vk i }2n−1 i=n+1}k∈N defined by hk i = S−1 Φ Φi + vk i , i ∈ I2n−1 constitutes a dual frame of Φ, for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Furthermore, there exists k0 ∈ N so that for k ≥ k0 d � {vk i }2n−1 i=n+1}, {ui}2n−1 i=n+1} � = 2n−1 � i=n+1 ∥vk i − ui∥ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Characterization of (weak) phase retrieval dual frames 7 Hence, we can write d({hk i }k∈N,i∈I2n−1, {gi}i∈I2n−1) = � i∈I2n−1 ∥vk i − ui∥ = � i∈In ∥vk i − ui∥ + 2n−1 � i=n+1 ∥vk i − ui∥ = � i∈In ∥ − 2n−1 � j=n+1 aj i(vk j − uj)∥ + 2n−1 � i=n+1 ∥vk i − ui∥ ≤ � i∈In 2n−1 � j=n+1 |aj i|∥vk j − uj∥ + 2n−1 � i=n+1 ∥vk i − ui∥ ≤ (α + 1) 2n−1 � j=n+1 ∥vk j − uj∥ < ǫ(α + 1), for k ≥ k0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', PDΦ is dense in DΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Now, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2 gives the result in general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In fact, every frame Ψ = {ψi}i∈I2n−1 is equivalent to a frame in the form of Φ = {δi}i∈In ∪ {φi}2n−1 i=n+1, as we discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, there exists an invertible operator T on Rn so that Ψ = T Φ, consequently by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1 we get PDΨ = PDT Φ = (T ∗)−1PDΦ ⊇ (T ∗)−1PDΦ = (T ∗)−1DΦ = DΨ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', PDΨ = DΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' And using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3) PDΨ = (T ∗)−1PDΦ = (T ∗)−1ξ−1{U ∈ Rn(n−1) : Pφ(U) ̸= 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', PDΨ is open in DΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ An analogous approach to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3 deduces that for any full spark frame Φ in Rn with E(Φ) = 1, the set of all full spark dual frames is embedded into a generic set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let Φ = {φi}n+1 i=1 be a full spark frame for Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then the set of all full spark dual frames of Φ is an open and dense subset of DΦ and is embedded into a generic set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Applying Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2, a full spark frame Φ = {φi}n+1 i=1 of Rn is equivalent to ˜Φ := {δi}n i=1 ∪ {�n i=1 αiδi} for non-zero scalars αi, i ∈ In, and in this case the frame operator is as follows: S˜Φ = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 + α2 1 α1α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' α1αn α1α2 1 + α2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' α2αn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' α1αn α2αn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 1 + α2 n \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol Also, every dual frame of ˜Φ is in the form of {S−1 ˜Φ ˜φi + ui}n+1 i=1 so that TuT ∗ ˜Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' By taking un+1 = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', xn]T , we get ui = −αiun+1 for all i ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Hence, D˜Φ is the set of all {gi}n+1 i=1 so that gk i = � −ααiαk − αixk k ̸= i, α(1 + � j̸=i α2 j) − αixk k = i, where α = 1 detS˜Φ = 1 1 + �n i=1 α2 i , gk i denotes kth coordinate of gi, i ∈ In, and gn+1 = {ααi + xi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' This shows that, every dual frame is associated to n-variable {xi}i∈In and a dual frame is full spark frame except the cases that the sequence un+1 = {xi}i∈In belongs to the roots of the polynomial constructed by det[gi1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' |gin] = 0, for some index set {ij}n j=1 ⊂ In+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Multiplying these n + 1 polynomials yield an n-variable polynomial P˜Φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, every full spark dual frame of ˜Φ is obtained by a generic choice of un+1 out of the roots of the polynomial P˜Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Hence, by a similar approach to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3, and considering the bijection map ξ : D˜Φ → Rn defined as ξ(G) = un+1, where G = {S−1 ˜Φ ˜φi + ui}n+1 i=1 , the complement of all full spark dual frames of Φ in DΦ is equivalent to the set of roots of P˜Φ, and so, the set of full spark dual frames of Φ is embedded into a generic set in Rn by ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In general case, if Φ = T ˜Φ, for an invertible operator T on Rn by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1, the set of full spark dual frames of Φ, FDΦ, is the same (T ∗)−1FD˜Φ, and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Examples Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Suppose that φ = {φi}3 i=1 is a full spark frame for R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Since E(φ) = 1 and equivalent frames do the same phase retrieval we assume that φ is equivalent to {δ1, δ2, α1δ1 + α2δ2}, in which both α1 and α2 are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In this case, Dφ = \uf8f1 \uf8f2 \uf8f3 \uf8ee \uf8f0 α(1 + α2 2) − α1x −αα1α2 − α1y \uf8f9 \uf8fb , \uf8ee \uf8f0 −αα1α2 − α2x α(1 + α2 1) − α2y \uf8f9 \uf8fb , \uf8ee \uf8f0 αα1 + x αα2 + y \uf8f9 \uf8fb ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' x, y ∈ R \uf8fc \uf8fd \uf8fe where α = 1 1 + α2 1 + α2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' It is worthy of note that, all dual frames in this case are full spark and so phase retrieval except dual frames obtained by (x, y) ∈ R2 on three distinct lines with slopes of 0, ∞ and −α1/α2 such as x = −αα1, y = −αα2 and α1x + α2y = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Considering (x, y) ∈ R2 \\ {(x, y) : (x + αα1)(y + αα2)(α1x + α2y − α) = 0} we get a generic choice of {ui}i∈I4 in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The fact that all dual frames of φ are translations of this family by {S−1 φ φi}i∈I4 shows that full spark dual frames are dense in DΦ and full measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' As a special case, considering α1 = α2 = 1 we get the phase retrieval frame φ = {δ1, δ2, δ1 + δ2} and we can see that dual frames of Φ do phase retrieval for all (x, y) ∈ R2 except on the lines x = −1 3 , y = −1 3 and Characterization of (weak) phase retrieval dual frames 9 y = 1 3 − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Put x = 0, y = 2/3 we get a phase retrieval dual Ψ and x = 1, y = −2 3 , satisfying in y = 1 3 − x, gives a non-phase retrieval dual frame G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let φ = {φi}4 i=1 be a full spark frame for R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In this case φ is equivalent to the frame φ := {δ1, δ2, δ3, α1δ1 + α2δ2 + α3δ3 α1, α2, α3 ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The set of all dual frames of φ is in the form of Dφ = {g1, g2, g3, g4} where g1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 α(1 + α2 2 + α2 3) − α1x −αα1α2 − α1y −αα1α3 − α1z \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , g2 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −αα1α2 − α2x α(1 + α2 1 + α2 3) − α2y −αα2α3 − α2z \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , g3 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −αα1α3 − α3x −αα2α3 − α3y α(1 + α2 1 + α2 3) − α3z \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , g4 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 αα1 + x αα2 + y αα3 + z \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb where α = 1 1 + α2 1 + α2 2 + α2 3 and x, y, z ∈ R are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' All dual frames of Φ are full spark except the cases det[gi|gj|gk] = 0, for i, j, k ∈ I4, in which (x, y, z) is belong to the four distinct planes x = −αα1, y = −αα2, z = −αα3 and α1x+α2y+α3z = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In a similar way, by choosing (x, y, z) ∈ R3 \\ {(x, y, z) : (x + αα1)(y + αα2)(z + αα3)(α1x + α2y + α3z − α) = 0} we can say that full spark dual frames are translations of the canonical dual by a generic choice of the family {ui}i∈I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Figure1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 FrameΦ Phaseretrievaldual Non-phase retrieval dual G 2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='5 210 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Consider φ = {δ1, δ2, δ3, �3 i=1 δi, δ1−δ2+δ3} as a full spark frame for R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In this case, all dual frames are constructed by a generic choice of � u4 u5 � ∈ R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' by putting u4 = [x1 y1 z1]T and u5 = [x2 y2 z2]T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' the set of all dual frames of φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' are given by g1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 3 5 − x1 − x2 −y1 − y2 −2 5 − z1 − z2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' g2 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 x2 − x1 1 3 + y2 − y1 −z2 − z1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' g3 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −2 5 − x1 − x2 −y1 − y2 3 5 − z1 − z2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' g4 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 5 + x1 1 3 + y1 1 5 + z1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' g5 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 5 + x1 −1 3 + y2 1 5 + z2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb where x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' y2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' z2 are obtained arbitrarily from R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' And all cases in which a dual frame fails to do phase retrieval is associated to the roots of a 6-variable polynomial in R6 given by multiplying of the polynomials as det[gi1|gi2|gi3] = 0, for all index set {ij}3 j=1 ⊂ I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' As one case, det[g2|g4|g5] = 0 deduces that z1 + x2 − 3x1 5 − 3z2 5 + 3x2z1 − 3x1z2 = 0, and the roots of this equation, which are associated to a family of non-phase retrieval dual frames, constitute a surface in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Weak Phase Retrieval The main result of this section is to obtain some equivalent conditions on a family of vectors to do phase retrieval in terms of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' This also derives a relationship between, phase, weak phase and norm retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' First we present some properties of a family of vectors to do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Assume that Φ = {φi}i∈Im is a frame in Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then Φ does weak phase retrieval if and only if PΦ does weak phase retrieval, for every 2-dimensional orthogonal projection P on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In case Φ does weak phase retrieval, it is simple to see that PΦ does weak phase retrieval, for every orthogonal projection P on Hn, as well, see also [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Con- versely, let x, y ∈ Hn and |⟨x, φi⟩| = |⟨y, φi⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then by the assumption that PΦ does weak retrieval for the projection P of Hn onto the closed subspace span{x, y}, implies that x and y weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ Characterization of (weak) phase retrieval dual frames 11 It is shown that [1] a weak phase retrieval family in Rn spans the space that means such a family constitutes a frame for Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In the complex case Cn there is no result available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In the following, we present sufficient condition in this regard which is also useful for the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let {φi}i∈Im be a family of vectors in Cn so that UΦ does weak phase retrieval for every unitary operator U on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then {φi}i∈Im is a frame for Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Assume that there exists an element x ∈ {φi}⊥ i∈Im, and get an ONB for Cn as {ei}i∈In with e1 = x ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, take 0 ̸= y ∈ span{φi}i∈Im so there exists {βi}n i=2 ⊂ C, with some non-zero elements such that y = �n i=2 βiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Define µ : Cn → Cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' µ(ei) = δi where {δi}i∈In is the standard ONB of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Clearly µ is a unitary operator and µ(x + y) = µ(x) + µ(y) = ∥x∥δ1 + n � i=2 βiδi, µ(−x + y) = −∥x∥δ1 + n � i=2 βiδi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' On the other hand, |⟨µ(x + y), µφi⟩| = |⟨x + y, φi⟩| = |⟨−x + y, φi⟩| = |⟨µ(−x + y), µφi⟩|, for all i ∈ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The assumption that µΦ does weak phase retrieval deduces that µ(x + y) and µ(−x + y) weakly have the same phase that is impossible, unless x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Therefore, Φ is a frame for Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let Φ = {φi}i∈Im be a family of vectors in Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then the followings are equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (i) Φ does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (ii) UΦ does phase retrieval for every invertible operator U on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (iii) UΦ does norm retrieval for every invertible operator U on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (iv) UΦ does weak phase retrieval for every invertible operator U on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The equivalency of (i), (ii), and (iii) was proven in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Also, we clearly have (i) ⇒ (ii) ⇒ (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, it is sufficient to show that (iv) ⇒ (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Assuming that UΦ does weak phase retrieval for all invertible operators on Hn, specially we imply that φ does weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, Φ is a frame for Hn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', Φ spans Hn by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In the sequel, we are going to show that Φ yields phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' On the contrary, let there exist non-zero elements x and y in Hn so that |⟨x, φi⟩| = |⟨y, φi⟩|, (i ∈ Im) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) but y ̸= cx for any |c| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Since Φ does weak phase retrieval, there exists some |θ| = 1 so that θphase(xj) = phase(yj) for all j ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The assumption that y ̸= θx implies |xj| ̸= |yj| for some j ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Since Φ is a frame, the equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) is non-zero for some i then y = cx immediately implies that x and y are linearly independent and 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol we can consider a basis for Hn containing x and y as {x, y, e3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', en}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' We face the following cases Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' If x and y are disjointly supported, then there exist no non-zero com- mon coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Without loss of generality we let x1, y2 ̸= 0, and so x2 = y1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Define U : Hn → Hn so that Ux = x − y, Uy = x + y and Uek = ek, 3 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, U is a bounded invertible operator on Hn and |⟨Ux, (U −1)∗φi⟩| = |⟨Uy, (U −1)∗φi⟩|, (i ∈ Im), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) and the invertibility of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' However, Ux and Uy have not weakly the same phase due to phase(Ux)1 = phase(Uy)1 and phase(Ux)2 = eiπphase(Uy)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' This contradicts the assumption that (U −1)∗Φ does weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let x and y have one non-zero coordinate in common in ith index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' If yl = 0 = xl for all l ̸= i then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='1) and using the assumption |xi| = |yi|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', y = θx that is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Thus, there exists l ̸= i so that yl ̸= 0 or xl ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Without loss of generality we let yl ̸= 0 and |xl| |yl| < |xi| |yi| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' In fact, if in all common non-zero elements |xi| |yi| = c then y = θ cx, which contradicts by the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' So, we get ǫ > 0 so that |xl| |yl| < ǫ < |xi| |yi| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Define U : Hn → Hn so that Ux = θx − ǫy, Uy = θx + ǫy and Uek = ek, 3 ≤ k ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Moreover, (Ux)j = (|xj| − ǫ|yj|)αjθ, (Uy)j = (|xj| + ǫ|yj|)αjθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' (j ∈ In) where αj = phase(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Hence, we obtain phase(Ux)i = phase(Uy)i, however phase(Ux)l = eiπphase(Uy)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' That leads to a contradiction similar to the previ- ous case, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ The following result gives a sufficient condition on a family of frames so that their canonical dual also yields weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let Φ = {φi}i∈Im be a frame in Hn with diagonal frame operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then Φ does weak phase retrieval if and only if its canonical dual does so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Suppose that α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', αn be the diagonal elements of SΦ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Ob- viously, αi > 0, for all i ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Let Φ does weak phase retrieval, take x, y ∈ Hn such that |⟨x, S−1 Φ φi⟩| = |⟨y, S−1 Φ φi⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Then, we get S−1 Φ x = (x1/α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', xn/αn) and S−1 Φ y = (y1/α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=', yn/αn) weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Consequently, x and y weakly have the same phase, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The converse is implied by a similar explana- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' □ Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' The authors have not disclosed any funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Data Availability Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Data sharing not applicable to this article as no datasets were generated during the preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Conflict-of-interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' This work does not have any conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Characterization of (weak) phase retrieval dual frames 13 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Akrami, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Sadeghi, Numerically and spectrally optimal dual frames in Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='604:52-71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 265(11) (2013), 2855-2876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' van Trees, Optimum Array Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Wiley, New York, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arabyani-Neyshaburi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Arefijamaal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='Kamyabi-Gol 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Xu, Phase retrieval for sparse signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Harmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' 37 (2014), 531–544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content=' Fahimeh Arabyani-Neyshaburi1 e-mail: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='arabiani@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='ir, fahimeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='arabyani@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='com Ali Akbar Arefijamaal2 e-mail: arefijamaal@hsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE4T4oBgHgl3EQfYAwp/content/2301.05045v1.pdf'} +page_content='ir;' metadata={'source': 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